Integrand size = 35, antiderivative size = 524 \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 d \left (c^2 C-B c d+A d^2\right ) \sqrt {a-b x^2}}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}-\frac {\sqrt {b} \left (2 a c (c C-B d)-A \left (b c^2-3 a d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {A \sqrt {b} \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {(2 B c-3 A d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2*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)^(1/2)- A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/c^2/x-b^(1/2)*(2*a*c*(-B*d+C*c)-A*(-3*a *d^2+b*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^2/(-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)-A*b^(1/2)*((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/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-(-3*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)*Ellipt icPi(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^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.43 (sec) , antiderivative size = 1699, normalized size of antiderivative = 3.24 \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:
Integrate[(A + B*x + C*x^2)/(x^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
-((Sqrt[a - b*x^2]*((c*(2*a*c*d*(-(c*C) + B*d)*x + A*b*c^2*(c + d*x) - a*A *d^2*(c + 3*d*x)))/x + (A*b^2*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 2*a*b*c ^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 2*a*b*B*c^4*d*Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]] - 4*a*A*b*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 2*a^2*c^3*C*d^ 2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 2*a^2*B*c^2*d^3*Sqrt[-c + (Sqrt[a]*d)/S qrt[b]] + 3*a^2*A*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 2*A*b^2*c^4*Sqrt[ -c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 4*a*b*c^4*C*Sqrt[-c + (Sqrt[a]*d)/Sq rt[b]]*(c + d*x) - 4*a*b*B*c^3*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 6*a*A*b*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + A*b^2*c^3*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 2*a*b*c^3*C*Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]]*(c + d*x)^2 + 2*a*b*B*c^2*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 3*a*A*b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqr t[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(2*a*c*(-(c*C) + B*d) + A*(b*c^2 - 3*a*d^2) )*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d) /Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*(Sqrt[b]*c - Sqrt[a]*d)*(A*b*c^2*d + a*d^2*(-2*B*c + 3*A*d) + 2*Sqrt[a]*Sqrt[b]*c*(c^2*C - 2*B*c*d + 3*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x) ^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]...
Leaf count is larger than twice the leaf count of optimal. \(1131\) vs. \(2(524)=1048\).
Time = 5.57 (sec) , antiderivative size = 1131, normalized size of antiderivative = 2.16, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {2355, 637, 2009, 2352, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^2 \sqrt {a-b x^2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2355 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (\frac {d^2}{c^2 (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {d}{c^2 x \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {1}{c x^2 \sqrt {c+d x} \sqrt {a-b x^2}}\right )dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a} \sqrt {b} d^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{a c^2 x}\right )\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle -\frac {\int \frac {b \left (B-\frac {c C}{d}\right ) x^2+a \left (B-\frac {3 c C}{d}\right )}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a} \sqrt {b} d^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{a c^2 x}\right )+\frac {\sqrt {a-b x^2} \sqrt {c+d x} (c C-B d)}{a c d^2 x}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {b \left (B-\frac {c C}{d}\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a} \sqrt {b} d^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{a c^2 x}\right )+\frac {\sqrt {a-b x^2} \sqrt {c+d x} (c C-B d)}{a c d^2 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (B-\frac {c C}{d}\right ) \int \frac {x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a} \sqrt {b} d^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{a c^2 x}\right )+\frac {\sqrt {a-b x^2} \sqrt {c+d x} (c C-B d)}{a c d^2 x}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (B-\frac {c C}{d}\right ) \left (\frac {\int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a c}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {2 \sqrt {a} \sqrt {b} d^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{a c^2 x}\right )+\frac {\sqrt {a-b x^2} \sqrt {c+d x} (c C-B d)}{a c d^2 x}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {2 \sqrt {a-b x^2} d^3}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) d^2}{c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) d}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}\right )-\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (B-\frac {c C}{d}\right ) \left (\frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a c}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (B-\frac {c C}{d}\right ) \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 )}{2 a c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (B-\frac {c C}{d}\right ) \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 )}{2 a c}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {a \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+b \left (B-\frac {c C}{d}\right ) \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 )}{2 a c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{d}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{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}{\sqrt {a-b x^2}}}{2 a c}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{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}{\sqrt {a-b x^2}}}{2 a c}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{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}}}}{\sqrt {a-b x^2}}}{2 a c}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{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}}}}{\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}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt {c+d x} \sqrt {a-b x^2} (c C-B d)}{a c d^2 x}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \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 )-\frac {b \left (B-\frac {c C}{d}\right ) \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 \left (B-\frac {3 c C}{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 )}{\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}\) |
Input:
Int[(A + B*x + C*x^2)/(x^2*(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])/(a*c*d^2*x) + (A + (c*(c*C - B *d))/d^2)*((2*d^3*Sqrt[a - b*x^2])/(c^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - ( Sqrt[c + d*x]*Sqrt[a - b*x^2])/(a*c^2*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^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b ]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*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^2*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (Sqrt[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*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (3*d*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^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])) - (b*(B - (c*C)/d)*((-2*Sqrt[a]*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[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*Sqrt[(Sqrt [b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcS in[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] +...
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[(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. \(959\) vs. \(2(445)=890\).
Time = 6.61 (sec) , antiderivative size = 960, normalized size of antiderivative = 1.83
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (-b d \,x^{2}+d a \right ) \left (A \,d^{2}-B c d +C \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) c^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+d a \right )}}-\frac {A \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{c^{2} a x}-\frac {2 b \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {b d \left (A \,d^{2}-B c d +C \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) c^{2}}-\frac {A b d}{2 a \,c^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {\left (3 A d -2 B c \right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{c^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(960\) |
| risch | \(-\frac {A \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{a \,c^{2} x}-\frac {\left (-\frac {\left (3 A d -2 B c \right ) a \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, 2, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {A d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-2 a \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {2 \left (-b d \,x^{2}+d a \right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+d a \right )}}-\frac {c \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}}{2 c^{2} a \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(978\) |
| default | \(\text {Expression too large to display}\) | \(3739\) |
Input:
int((C*x^2+B*x+A)/x^2/(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)*(A*d^2-B*c*d+C*c^2)/c^2/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-A/ c^2/a/x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2*b*(A*d^2-B*c*d+C*c^2)/(a*d^2- b*c^2)/c*(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*(-b*d*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)/c^2-1/2*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)*((-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)) )+(3*A*d-2*B*c)/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)*d*Ell ipticPi(((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+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^2+B*x+A)/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{2} \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((C*x**2+B*x+A)/x**2/(d*x+c)**(3/2)/(-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)**(3/2)), x)
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/x^2/(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^2), x)
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/x^2/(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^2), x)
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^2\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((A + B*x + C*x^2)/(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(3/2)),x)
Output:
int((A + B*x + C*x^2)/(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(3/2)), x)
\[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}+\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) b c d x +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) b \,d^{2} x^{2}-3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{5}-2 b c d \,x^{4}+a \,d^{2} x^{3}-b \,c^{2} x^{3}+2 a c d \,x^{2}+a \,c^{2} x}d x \right ) a c d x -3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{5}-2 b c d \,x^{4}+a \,d^{2} x^{3}-b \,c^{2} x^{3}+2 a c d \,x^{2}+a \,c^{2} x}d x \right ) a \,d^{2} x^{2}+2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{5}-2 b c d \,x^{4}+a \,d^{2} x^{3}-b \,c^{2} x^{3}+2 a c d \,x^{2}+a \,c^{2} x}d x \right ) b \,c^{2} x +2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{5}-2 b c d \,x^{4}+a \,d^{2} x^{3}-b \,c^{2} x^{3}+2 a c d \,x^{2}+a \,c^{2} x}d x \right ) b c d \,x^{2}+2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) c^{3} x +2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) c^{2} d \,x^{2}}{2 c x \left (d x +c \right )} \] Input:
int((C*x^2+B*x+A)/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a - b*x**2) + int((sqrt(c + d*x)*sqrt(a - b*x**2) *x)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d** 2*x**4),x)*b*c*d*x + int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a*c**2 + 2*a* c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*b*d**2* x**2 - 3*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2*x + 2*a*c*d*x**2 + a *d**2*x**3 - b*c**2*x**3 - 2*b*c*d*x**4 - b*d**2*x**5),x)*a*c*d*x - 3*int( (sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2*x + 2*a*c*d*x**2 + a*d**2*x**3 - b*c**2*x**3 - 2*b*c*d*x**4 - b*d**2*x**5),x)*a*d**2*x**2 + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2*x + 2*a*c*d*x**2 + a*d**2*x**3 - b*c**2*x* *3 - 2*b*c*d*x**4 - b*d**2*x**5),x)*b*c**2*x + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2*x + 2*a*c*d*x**2 + a*d**2*x**3 - b*c**2*x**3 - 2*b*c*d *x**4 - b*d**2*x**5),x)*b*c*d*x**2 + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2) )/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2* x**4),x)*c**3*x + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2 + 2*a*c*d *x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*c**2*d*x** 2)/(2*c*x*(c + d*x))