Integrand size = 32, antiderivative size = 388 \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {A c^2-a B d+c (B c-A d) x^2}{2 a \left (c^3-a d^2\right ) x \sqrt {a-c x^4}}-\frac {\left (3 A c^3-a B c d-2 a A d^2\right ) \sqrt {a-c x^4}}{2 a^2 c \left (c^3-a d^2\right ) x}-\frac {\left (3 A c^3-a B c d-2 a A d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{5/4} c^{3/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\left (3 A c^{3/2}+\sqrt {a} (B c+2 A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{5/4} c^{3/4} \left (c^{3/2}+\sqrt {a} d\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} d^2 (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*(A*c^2-B*a*d+c*(-A*d+B*c)*x^2)/a/(-a*d^2+c^3)/x/(-c*x^4+a)^(1/2)-1/2*( -2*A*a*d^2+3*A*c^3-B*a*c*d)*(-c*x^4+a)^(1/2)/a^2/c/(-a*d^2+c^3)/x-1/2*(-2* A*a*d^2+3*A*c^3-B*a*c*d)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/ a^(5/4)/c^(3/4)/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)+1/2*(3*A*c^(3/2)+a^(1/2)*(2* A*d+B*c))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(5/4)/c^(3/4) /(c^(3/2)+a^(1/2)*d)/(-c*x^4+a)^(1/2)-a^(1/4)*d^2*(-A*d+B*c)*(1-c*x^4/a)^( 1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I)/c^(9/4)/(-a*d^2+c^ 3)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.28 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {2 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4-2 a^2 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2-a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 x^2+a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^2-3 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^5 x^4+a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^4+2 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x^4+i \sqrt {a} c^{3/2} \left (-3 A c^3+a B c d+2 a A d^2\right ) x \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {a} c^{3/2} \left (-c^{3/2}+\sqrt {a} d\right ) \left (3 A c^{3/2}+\sqrt {a} (B c+2 A d)\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-2 i a^2 B c d^2 x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+2 i a^2 A d^3 x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 a^2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 \left (-c^3+a d^2\right ) x \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2)/(x^2*(c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(2*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^4 - 2*a^2*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d ^2 - a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^4*x^2 + a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c ^3*d*x^2 - 3*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^5*x^4 + a*B*Sqrt[-(Sqrt[c]/Sqrt[ a])]*c^3*d*x^4 + 2*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^2*x^4 + I*Sqrt[a]*c^ (3/2)*(-3*A*c^3 + a*B*c*d + 2*a*A*d^2)*x*Sqrt[1 - (c*x^4)/a]*EllipticE[I*A rcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*Sqrt[a]*c^(3/2)*(-c^(3/2) + Sq rt[a]*d)*(3*A*c^(3/2) + Sqrt[a]*(B*c + 2*A*d))*x*Sqrt[1 - (c*x^4)/a]*Ellip ticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (2*I)*a^2*B*c*d^2*x*Sqrt [1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c ]/Sqrt[a])]*x], -1] + (2*I)*a^2*A*d^3*x*Sqrt[1 - (c*x^4)/a]*EllipticPi[-(( Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*a^2*Sq rt[-(Sqrt[c]/Sqrt[a])]*c^2*(-c^3 + a*d^2)*x*Sqrt[a - c*x^4])
Time = 0.81 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2249, 2009}
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.
| \(\displaystyle \int \frac {A+B x^2}{x^2 \left (a-c x^4\right )^{3/2} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (-\frac {d^2 (B c-A d)}{c \left (c^3-a d^2\right ) \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {c x^2 \left (A c^2-a B d\right )+a c (B c-A d)}{a \left (c^3-a d^2\right ) \left (a-c x^4\right )^{3/2}}+\frac {A}{a c x^2 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B+A \sqrt {c}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{5/4} \left (\sqrt {a} d+c^{3/2}\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \left (A c^2-a B d\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{5/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{a^{5/4} c^{3/4} \sqrt {a-c x^4}}-\frac {A \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{a^{5/4} c^{3/4} \sqrt {a-c x^4}}+\frac {c x \left (x^2 \left (A c^2-a B d\right )+a (B c-A d)\right )}{2 a^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {A \sqrt {a-c x^4}}{a^2 c x}-\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}\) |
Input:
Int[(A + B*x^2)/(x^2*(c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(c*x*(a*(B*c - A*d) + (A*c^2 - a*B*d)*x^2))/(2*a^2*(c^3 - a*d^2)*Sqrt[a - c*x^4]) - (A*Sqrt[a - c*x^4])/(a^2*c*x) - (A*Sqrt[1 - (c*x^4)/a]*EllipticE [ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(5/4)*c^(3/4)*Sqrt[a - c*x^4]) - (c^ (1/4)*(A*c^2 - a*B*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^( 1/4)], -1])/(2*a^(5/4)*(c^3 - a*d^2)*Sqrt[a - c*x^4]) + (A*Sqrt[1 - (c*x^4 )/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(5/4)*c^(3/4)*Sqrt[a - c*x^4]) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticF[ ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(5/4)*(c^(3/2) + Sqrt[a]*d)*Sqrt[a - c*x^4]) - (a^(1/4)*d^2*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqr t[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(9/4)*(c^3 - a*d^2) *Sqrt[a - c*x^4])
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & & PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 4.95 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(-\frac {A \sqrt {-c \,x^{4}+a}}{c \,a^{2} x}-\frac {-\frac {A \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {a^{2} \left (A d -B c \right ) d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,d^{2}-c^{3}\right ) c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {a \,c^{2} \left (\frac {2 c \left (-\frac {\left (A \,c^{2}-B a d \right ) x^{3}}{4 a c}+\frac {\left (A d -B c \right ) x}{4 c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {A d}{2}-\frac {B c}{2}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A \,c^{2}-B a d \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )}{a \,d^{2}-c^{3}}}{a^{2} c}\) | \(486\) |
| default | \(\frac {A \left (\frac {c \,x^{3}}{2 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {-c \,x^{4}+a}}{a^{2} x}+\frac {3 \sqrt {c}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \left (\frac {2 c \left (\frac {d \,x^{3}}{4 a \left (a \,d^{2}-c^{3}\right )}-\frac {c x}{4 a \left (a \,d^{2}-c^{3}\right )}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,d^{2}-c^{3}\right ) c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}\) | \(572\) |
| elliptic | \(\text {Expression too large to display}\) | \(969\) |
Input:
int((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-A*(-c*x^4+a)^(1/2)/c/a^2/x-1/a^2/c*(-A*c^(1/2)*a^(1/2)/(c^(1/2)/a^(1/2))^ (1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+ a)^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1 /2))^(1/2),I))+a^2*(A*d-B*c)*d^2/(a*d^2-c^3)/c/(c^(1/2)/a^(1/2))^(1/2)*(1- c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)* EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*d/c^(3/2),(-c^(1/2)/a^(1/2)) ^(1/2)/(c^(1/2)/a^(1/2))^(1/2))-a*c^2/(a*d^2-c^3)*(2*c*(-1/4*(A*c^2-B*a*d) /a/c*x^3+1/4*(A*d-B*c)/c*x)/(-(x^4-a/c)*c)^(1/2)+(1/2*A*d-1/2*B*c)/(c^(1/2 )/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/ 2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-1/2*(A*c^2-B*a* d)/a^(1/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2 )*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2 ))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))))
Timed out. \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{x^{2} \left (a - c x^{4}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/x**2/(d*x**2+c)/(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)/(x**2*(a - c*x**4)**(3/2)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)/((-c*x^4 + a)^(3/2)*(d*x^2 + c)*x^2), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)/((-c*x^4 + a)^(3/2)*(d*x^2 + c)*x^2), x)
Timed out. \[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{x^2\,{\left (a-c\,x^4\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2)/(x^2*(a - c*x^4)^(3/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2)/(x^2*(a - c*x^4)^(3/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2}{x^2 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {-c \,x^{4}+a}-\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a^{2} d x +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a b c x +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a c d \,x^{5}-\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) b \,c^{2} x^{5}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a c d x -3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) c^{2} d \,x^{5}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a \,c^{2} x -3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) c^{3} x^{5}}{c x \left (-c \,x^{4}+a \right )} \] Input:
int((B*x^2+A)/x^2/(d*x^2+c)/(-c*x^4+a)^(3/2),x)
Output:
( - sqrt(a - c*x**4) - int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c* *2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**2*d*x + int(sqrt( a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x* *8 + c**2*d*x**10),x)*a*b*c*x + int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a*c*d*x**5 - int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x** 6 + c**3*x**8 + c**2*d*x**10),x)*b*c**2*x**5 + 3*int((sqrt(a - c*x**4)*x** 4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2 *d*x**10),x)*a*c*d*x - 3*int((sqrt(a - c*x**4)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*c**2*d*x**5 + 3*int((sqrt(a - c*x**4)*x**2)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2 *a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a*c**2*x - 3*int((sqrt(a - c*x* *4)*x**2)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*c**3*x**5)/(c*x*(a - c*x**4))