Integrand size = 35, antiderivative size = 615 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 d^2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a-b x^2}}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}-\frac {(4 B c-7 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{4 a c^3 x}+\frac {\sqrt {b} \left (b c^2 (4 B c-7 A d)+a d \left (8 c^2 C-12 B c d+15 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {a} c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {b} (4 B c-5 A d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (4 A b c^2+8 a c^2 C-12 a B c d+15 a A d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2*d^2*(A*d^2-B*c*d+C*c^2)*(-b*x^2+a)^(1/2)/c^3/(-a*d^2+b*c^2)/(d*x+c)^(1/ 2)-1/2*A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/c^2/x^2-1/4*(-7*A*d+4*B*c)*(d*x+ c)^(1/2)*(-b*x^2+a)^(1/2)/a/c^3/x+1/4*b^(1/2)*(b*c^2*(-7*A*d+4*B*c)+a*d*(1 5*A*d^2-12*B*c*d+8*C*c^2))*(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)/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1 /2)/(-b*x^2+a)^(1/2)-1/4*b^(1/2)*(-5*A*d+4*B*c)*((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/(d*x +c)^(1/2)/(-b*x^2+a)^(1/2)-1/4*(15*A*a*d^2+4*A*b*c^2-12*B*a*c*d+8*C*a*c^2) *((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(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))/a/c^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 32.46 (sec) , antiderivative size = 2647, normalized size of antiderivative = 4.30 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + B*x + C*x^2)/(x^3*(c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*(-1/2*A/(a*c^2*x^2) + (-4*B*c + 7*A*d)/(4*a* c^3*x) - (2*d^2*(c^2*C - B*c*d + A*d^2))/(c^3*(b*c^2 - a*d^2)*(c + d*x))) - (d*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(-4*b^2*B*c^4*Sqrt [-c + (Sqrt[a]*d)/Sqrt[b]] + 7*A*b^2*c^3*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 8*a*b*c^3*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 12*a*b*B*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 15*a*A*b*c*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - ( 4*b^2*B*c^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (7*A*b^2*c^5*d*S qrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (8*a*b*c^5*C*d*Sqrt[-c + (Sqr t[a]*d)/Sqrt[b]])/(c + d*x)^2 + (16*a*b*B*c^4*d^2*Sqrt[-c + (Sqrt[a]*d)/Sq rt[b]])/(c + d*x)^2 - (22*a*A*b*c^3*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (8*a^2*c^3*C*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (12*a^2*B*c^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (15*a^2*A *c*d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (8*b^2*B*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (14*A*b^2*c^4*d*Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]])/(c + d*x) + (16*a*b*c^4*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (24*a*b*B*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) + (30* a*A*b*c^2*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) + (I*Sqrt[b]*c*(Sq rt[b]*c - Sqrt[a]*d)*(b*c^2*(4*B*c - 7*A*d) + a*d*(8*c^2*C - 12*B*c*d + 15 *A*d^2))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt...
Leaf count is larger than twice the leaf count of optimal. \(1401\) vs. \(2(615)=1230\).
Time = 8.58 (sec) , antiderivative size = 1401, normalized size of antiderivative = 2.28, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {2355, 637, 2009, 2352, 2352, 25, 2351, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^3 \sqrt {a-b x^2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2355 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}}dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \sqrt {a-b x^2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {1}{c x^3 \sqrt {c+d x} \sqrt {a-b x^2}}\right )dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^3}{c^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\int \frac {-b \left (B-\frac {c C}{d}\right ) x^2+\frac {2 b c (c C-B d) x}{d^2}+a \left (3 B-\frac {7 c C}{d}\right )}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {-\frac {\sqrt {c+d x} \sqrt {a-b x^2} \left (3 B-\frac {7 c C}{d}\right )}{c x}-\frac {\int -\frac {a b (7 c C-3 B d) x^2-2 a b c \left (B-\frac {c C}{d}\right ) x+a \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right )}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}}{4 a c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {\int \frac {a b (7 c C-3 B d) x^2-2 a b c \left (B-\frac {c C}{d}\right ) x+a \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right )}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {a \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {\frac {2 a b C c^2}{d}-2 a b B c+a b (7 c C-3 B d) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {a b c (5 c C-B d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+a \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a b (7 c C-3 B d) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {a b c (5 c C-B d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+a \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a b (7 c C-3 B d) \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}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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}} a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {b c (5 c C-B d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx a}{d}+\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx a}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {b c (5 c C-B d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx a}{d}+\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx a}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx a-\frac {b c (5 c C-B d) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx a}{d \sqrt {a-b x^2}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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}} a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}+\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx a}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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 ) a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}+\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx a}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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 ) a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx a}{\sqrt {a-b x^2}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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 ) a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \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 a}{\sqrt {a-b x^2}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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 ) a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \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}}} a}{\sqrt {a-b x^2}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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 ) a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \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}}} 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}}}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{2 a c d^2 x^2}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a-b x^2} d^4}{c^3 \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^3}{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 {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^2}{c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {7 \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}{4 \sqrt {a} c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {5 \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}{4 \sqrt {a} c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {7 \sqrt {c+d x} \sqrt {a-b x^2} d}{4 a c^3 x}-\frac {\left (4 b c^2+3 a d^2\right ) \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 )}{4 a c^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 a c^2 x^2}\right )-\frac {\frac {-\frac {2 \sqrt {b} (7 c C-3 B d) \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 ) a^{3/2}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} c (5 c C-B d) \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 ) a^{3/2}}{d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \left (\frac {4 b (c C-B d) c^2}{d^2}+a (7 c C-3 B d)\right ) \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 ) 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}}}}}{2 a c}-\frac {\left (3 B-\frac {7 c C}{d}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{c x}}{4 a c}\) |
Input:
Int[(A + B*x + C*x^2)/(x^3*(c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
((c*C - B*d)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(2*a*c*d^2*x^2) + (A + (c*(c*C - B*d))/d^2)*((-2*d^4*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])/(2*a*c^2*x^2) + (7*d*Sqrt[c + d*x]*Sqr t[a - b*x^2])/(4*a*c^3*x) - (7*Sqrt[b]*d*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)])/(4*Sqrt[a]*c^3*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqr t[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*Sqrt[b]*d^3*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]) + (5*Sqrt[b]*d*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)])/(4*S qrt[a]*c^2*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - (3*d^2*Sqrt[(Sqrt[b]*(c + d*x) )/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/ (c^3*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - ((4*b*c^2 + 3*a*d^2)*Sqrt[(Sqrt[b]*( c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticPi[2, ArcSi n[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[ a]*d)])/(4*a*c^3*Sqrt[c + d*x]*Sqrt[a - b*x^2])) - (-(((3*B - (7*c*C)/d)*S qrt[c + d*x]*Sqrt[a - b*x^2])/(c*x)) + ((-2*a^(3/2)*Sqrt[b]*(7*c*C - 3*...
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[(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_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x _)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* (m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && LtQ[m, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs. \(2(522)=1044\).
Time = 7.41 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.71
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1051\) |
| risch | \(\text {Expression too large to display}\) | \(1420\) |
| default | \(\text {Expression too large to display}\) | \(5536\) |
Input:
int((C*x^2+B*x+A)/x^3/(d*x+c)^(3/2)/(-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*(-b*d*x^2+a*d )/(a*d^2-b*c^2)*d*(A*d^2-B*c*d+C*c^2)/c^3/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-1 /2*A/c^2/a/x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+1/4*(7*A*d-4*B*c)/a/c^3* (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x+2*(b*d*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c ^2)/c^2+1/4*A*b*d/a/c^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*(d^2*b*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)/c^ 3+1/8*(7*A*d-4*B*c)*d/a*b/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)))-1/4*(15*A*a*d^2+4*A*b*c^2-12*B*a*c*d+8*C*a* c^2)/a/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*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*d*EllipticPi(( (x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/c*d,((-c/d...
Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{3} \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((C*x**2+B*x+A)/x**3/(d*x+c)**(3/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral((A + B*x + C*x**2)/(x**3*sqrt(a - b*x**2)*(c + d*x)**(3/2)), x)
\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(3/2)/(-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)^(3/2)*x^3), x)
\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(3/2)/(-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)^(3/2)*x^3), x)
Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^3\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((A + B*x + C*x^2)/(x^3*(a - b*x^2)^(1/2)*(c + d*x)^(3/2)),x)
Output:
int((A + B*x + C*x^2)/(x^3*(a - b*x^2)^(1/2)*(c + d*x)^(3/2)), x)
\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {C \,x^{2}+B x +A}{x^{3} \left (d x +c \right )^{\frac {3}{2}} \sqrt {-b \,x^{2}+a}}d x \] Input:
int((C*x^2+B*x+A)/x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x)
Output:
int((C*x^2+B*x+A)/x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x)