Integrand size = 35, antiderivative size = 681 \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 d \left (c^2 C-B c d+A d^2\right ) \sqrt {a-b x^2}}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {2 d \left (3 a d^3 (B c-2 A d)+b c^2 \left (4 c^2 C-7 B c d+10 A d^2\right )\right ) \sqrt {a-b x^2}}{3 c^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}+\frac {\sqrt {b} \left (A \left (3 b^2 c^4-26 a b c^2 d^2+15 a^2 d^4\right )-2 a c \left (3 a B d^3+b c^2 (4 c C-7 B d)\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} c^3 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {b} \left (2 a c (c C-B d)-A \left (3 b c^2-5 a d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {(2 B c-5 A d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/3*d*(A*d^2-B*c*d+C*c^2)*(-b*x^2+a)^(1/2)/c^2/(-a*d^2+b*c^2)/(d*x+c)^(3/2 )+2/3*d*(3*a*d^3*(-2*A*d+B*c)+b*c^2*(10*A*d^2-7*B*c*d+4*C*c^2))*(-b*x^2+a) ^(1/2)/c^3/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)-A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2) /a/c^3/x+1/3*b^(1/2)*(A*(15*a^2*d^4-26*a*b*c^2*d^2+3*b^2*c^4)-2*a*c*(3*B*a *d^3+b*c^2*(-7*B*d+4*C*c)))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1 /2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/ 2)*d))^(1/2))/a^(1/2)/c^3/(-a*d^2+b*c^2)^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2))) ^(1/2)/(-b*x^2+a)^(1/2)+1/3*b^(1/2)*(2*a*c*(-B*d+C*c)-A*(-5*a*d^2+3*b*c^2) )*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(1/2 *(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2) *d))^(1/2))/a^(1/2)/c^2/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-(-5* A*d+2*B*c)*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*Elli pticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/ 2)*c+a^(1/2)*d))^(1/2))/c^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 33.27 (sec) , antiderivative size = 2682, normalized size of antiderivative = 3.94 \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + B*x + C*x^2)/(x^2*(c + d*x)^(5/2)*Sqrt[a - b*x^2]),x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*(-(A/(a*c^3*x)) + (2*d*(c^2*C - B*c*d + A*d^ 2))/(3*c^2*(b*c^2 - a*d^2)*(c + d*x)^2) + (2*d*(4*b*c^4*C - 7*b*B*c^3*d + 10*A*b*c^2*d^2 + 3*a*B*c*d^3 - 6*a*A*d^4))/(3*c^3*(b*c^2 - a*d^2)^2*(c + d *x))) - (d*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(-3*A*b^3*c^ 5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 8*a*b^2*c^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]] - 14*a*b^2*B*c^4*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 26*a*A*b^2*c^3*d ^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 6*a^2*b*B*c^2*d^3*Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]] - 15*a^2*A*b*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - (3*A*b^3*c^ 7*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (8*a*b^2*c^7*C*Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (14*a*b^2*B*c^6*d*Sqrt[-c + (Sqrt[a]*d) /Sqrt[b]])/(c + d*x)^2 + (29*a*A*b^2*c^5*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] ])/(c + d*x)^2 - (8*a^2*b*c^5*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d *x)^2 + (20*a^2*b*B*c^4*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (41*a^2*A*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (6*a^3*B *c^2*d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (15*a^3*A*c*d^6*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (6*A*b^3*c^6*Sqrt[-c + (Sqrt[a] *d)/Sqrt[b]])/(c + d*x) - (16*a*b^2*c^6*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/ (c + d*x) + (28*a*b^2*B*c^5*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (52*a*A*b^2*c^4*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (12*a^2*b* B*c^3*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) + (30*a^2*A*b*c^2*d...
Leaf count is larger than twice the leaf count of optimal. \(2117\) vs. \(2(681)=1362\).
Time = 9.90 (sec) , antiderivative size = 2117, normalized size of antiderivative = 3.11, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {2355, 637, 2009, 2355, 27, 636, 25, 27, 637, 2009, 2351, 27, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^2 \sqrt {a-b x^2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2355 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}}dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (\frac {2 d^2}{c^3 (c+d x)^{3/2} \sqrt {a-b x^2}}+\frac {d^2}{c^2 (c+d x)^{5/2} \sqrt {a-b x^2}}-\frac {2 d}{c^3 x \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {1}{c^2 x^2 \sqrt {c+d x} \sqrt {a-b x^2}}\right )dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 2355 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{d^2}+\int \frac {C}{d^2 x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{d^2}+\frac {C \int \frac {1}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{d^2}\) |
| \(\Big \downarrow \) 636 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{d^2}+\frac {C \left (\frac {\int -\frac {d \left (b x^2+a\right )}{c x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{d^2}+\frac {C \left (-\frac {\int \frac {d \left (b x^2+a\right )}{c x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{d^2}+\frac {C \left (-\frac {d \int \frac {b x^2+a}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )+\frac {C \left (-\frac {d \int \frac {b x^2+a}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}-\frac {(2 c C-B d) \int \left (\frac {d^2}{c^2 (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {d}{c^2 x \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {1}{c x^2 \sqrt {c+d x} \sqrt {a-b x^2}}\right )dx}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \int \frac {b x^2+a}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {b x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \int \frac {x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (\frac {\int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (\frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (-\frac {c \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {c \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {c \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )+a \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )+\frac {a \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )+\frac {a \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {4 \sqrt {a-b x^2} d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {8 b \sqrt {a-b x^2} d^3}{3 c \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a-b x^2} d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{3 c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^3 x}\right )-\frac {(2 c C-B d) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )}{d^2}+\frac {C \left (-\frac {d \left (b \left (\frac {2 \sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}\right )}{2 a c}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c x}\right )}{d^2}\) |
Input:
Int[(A + B*x + C*x^2)/(x^2*(c + d*x)^(5/2)*Sqrt[a - b*x^2]),x]
Output:
(A + (c*(c*C - B*d))/d^2)*((2*d^3*Sqrt[a - b*x^2])/(3*c^2*(b*c^2 - a*d^2)* (c + d*x)^(3/2)) + (8*b*d^3*Sqrt[a - b*x^2])/(3*c*(b*c^2 - a*d^2)^2*Sqrt[c + d*x]) + (4*d^3*Sqrt[a - b*x^2])/(c^3*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - ( Sqrt[c + d*x]*Sqrt[a - b*x^2])/(a*c^3*x) + (Sqrt[b]*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d) /((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[a]*c^3*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b ]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (8*Sqrt[a]*b^(3/2)*d^2*Sqrt[c + d*x]* Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2] ], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(3*c*(b*c^2 - a*d^2)^2*Sqrt[(Sqrt[b]* (c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (4*Sqrt[a]*Sqrt[b]* d^2*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x )/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(c^3*(b*c^2 - a*d^2 )*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (Sq rt[b]*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a ]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b] *c)/Sqrt[a] + d)])/(Sqrt[a]*c^2*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (2*Sqrt[a ]*Sqrt[b]*d^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - ( b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/( (Sqrt[b]*c)/Sqrt[a] + d)])/(3*c^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b *x^2]) + (5*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 ...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2] /(a*e*(m + 1))), x] - Simp[1/(2*a*e*(m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2 )*d*(2*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x ^3 - 2*a*(m + 1)*(c + d*x)^(n + 1/2))/x, x], x], x] /; FreeQ[{a, b, c, d, e }, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] I nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} , x] && PolynomialQ[Px, x] && LtQ[n, 0]
Time = 9.64 (sec) , antiderivative size = 1145, normalized size of antiderivative = 1.68
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1145\) |
| risch | \(\text {Expression too large to display}\) | \(1554\) |
| default | \(\text {Expression too large to display}\) | \(11676\) |
Input:
int((C*x^2+B*x+A)/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-2/3/(a*d^2-b*c ^2)/d*(A*d^2-B*c*d+C*c^2)/c^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2 -2/3*(-b*d*x^2+a*d)/(a*d^2-b*c^2)^2*(6*A*a*d^4-10*A*b*c^2*d^2-3*B*a*c*d^3+ 7*B*b*c^3*d-4*C*b*c^4)/c^3/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-A/c^3/a/x*(-b*d* x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(1/3*b*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)/c^ 2-1/3*b/c^2*(6*A*a*d^4-10*A*b*c^2*d^2-3*B*a*c*d^3+7*B*b*c^3*d-4*C*b*c^4)/( a*d^2-b*c^2)^2)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2 )*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/ (-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF (((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*( a*b)^(1/2)))^(1/2))+2*(-1/3*b*d*(6*A*a*d^4-10*A*b*c^2*d^2-3*B*a*c*d^3+7*B* b*c^3*d-4*C*b*c^4)/(a*d^2-b*c^2)^2/c^3-1/2*A*b*d/a/c^3)*(c/d-1/b*(a*b)^(1/ 2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*( a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b* d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/ (c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2) ))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2), ((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))+(5*A*d-2*B*c)/c^4* (c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^ (1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*...
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^2+B*x+A)/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{2} \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((C*x**2+B*x+A)/x**2/(d*x+c)**(5/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral((A + B*x + C*x**2)/(x**2*sqrt(a - b*x**2)*(c + d*x)**(5/2)), x)
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)/(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)*x^2), x)
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)/(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)*x^2), x)
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^2\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int((A + B*x + C*x^2)/(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(5/2)),x)
Output:
int((A + B*x + C*x^2)/(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {C \,x^{2}+B x +A}{x^{2} \left (d x +c \right )^{\frac {5}{2}} \sqrt {-b \,x^{2}+a}}d x \] Input:
int((C*x^2+B*x+A)/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x)
Output:
int((C*x^2+B*x+A)/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x)