Integrand size = 29, antiderivative size = 306 \[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A c^2-a B d+c (B c-A d) x^2\right )}{2 a \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{c} (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt [4]{c} \left (c^{3/2}+\sqrt {a} d\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} d (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*x*(A*c^2-B*a*d+c*(-A*d+B*c)*x^2)/a/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)-1/2*c ^(1/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4) /(-a*d^2+c^3)/(-c*x^4+a)^(1/2)+1/2*(a^(1/2)*B+A*c^(1/2))*(1-c*x^4/a)^(1/2) *EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/c^(1/4)/(c^(3/2)+a^(1/2)*d)/(-c*x^ 4+a)^(1/2)+a^(1/4)*d*(-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^(5/4)/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.03 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.33 \[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {-A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 x+a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d x-B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 x^3+A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x^3-i \sqrt {a} c^{3/2} (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \left (\sqrt {a} B+A \sqrt {c}\right ) c \left (-c^{3/2}+\sqrt {a} d\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+2 i a B c d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-2 i a A d^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c \left (-c^3+a d^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2)/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(-(A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*x) + a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*x - B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*x^3 + A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d*x^ 3 - I*Sqrt[a]*c^(3/2)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[ Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*(Sqrt[a]*B + A*Sqrt[c])*c*(-c^(3/2) + Sqrt[a]*d)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a] )]*x], -1] + (2*I)*a*B*c*d*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*A*d^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/S qrt[a])]*x], -1])/(2*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*(-c^3 + a*d^2)*Sqrt[a - c*x^4])
Time = 0.59 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2259, 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.
| \(\displaystyle \int \frac {A+B x^2}{\left (a-c x^4\right )^{3/2} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-a B d+c x^2 (B c-A d)+A c^2}{\left (c^3-a d^2\right ) \left (a-c x^4\right )^{3/2}}-\frac {d (A d-B c)}{\left (c^3-a d^2\right ) \sqrt {a-c x^4} \left (c+d x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B+A \sqrt {c}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt [4]{c} \left (\sqrt {a} d+c^{3/2}\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {x \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{2 a \left (c^3-a d^2\right ) \sqrt {a-c x^4}}\) |
Input:
Int[(A + B*x^2)/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(x*(A*c^2 - a*B*d + c*(B*c - A*d)*x^2))/(2*a*(c^3 - a*d^2)*Sqrt[a - c*x^4] ) - (c^(1/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/ a^(1/4)], -1])/(2*a^(1/4)*(c^3 - a*d^2)*Sqrt[a - c*x^4]) + ((Sqrt[a]*B + A *Sqrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/ (2*a^(3/4)*c^(1/4)*(c^(3/2) + Sqrt[a]*d)*Sqrt[a - c*x^4]) + (a^(1/4)*d*(B* c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^ (1/4)*x)/a^(1/4)], -1])/(c^(5/4)*(c^3 - a*d^2)*Sqrt[a - c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (250 ) = 500\).
Time = 0.66 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(\frac {B \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d}+\frac {\left (A d -B c \right ) \left (\frac {2 c \left (\frac {d \,x^{3}}{4 a \left (a \,d^{2}-c^{3}\right )}-\frac {c x}{4 a \left (a \,d^{2}-c^{3}\right )}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,d^{2}-c^{3}\right ) c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d}\) | \(530\) |
| elliptic | \(\frac {2 c \left (\frac {\left (A d -B c \right ) x^{3}}{4 a \left (a \,d^{2}-c^{3}\right )}-\frac {\left (A \,c^{2}-B a d \right ) x}{4 a \left (a \,d^{2}-c^{3}\right ) c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\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 ) A \,c^{2}}{2 a \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\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 ) B d}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, \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 ) A d}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {c^{\frac {3}{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 ) B}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, \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 ) A d}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {c^{\frac {3}{2}} \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 ) B}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {a}\, \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 ) A}{\left (a \,d^{2}-c^{3}\right ) c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {d \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 ) B}{\left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(800\) |
Input:
int((B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
B/d*(1/2/a*x/(-(x^4-a/c)*c)^(1/2)+1/2/a/(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)*Ellipti cF(x*(c^(1/2)/a^(1/2))^(1/2),I))+(A*d-B*c)/d*(2*c*(1/4/a*d/(a*d^2-c^3)*x^3 -1/4*c/a/(a*d^2-c^3)*x)/(-(x^4-a/c)*c)^(1/2)-1/2*c^2/a/(a*d^2-c^3)/(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*c^(1/2)/a^( 1/2)*d/(a*d^2-c^3)/(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*c^(1/2)/a^(1/2)*d/(a*d^2-c^3)/(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)*El lipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/(a*d^2-c^3)*d^2/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)))
Timed out. \[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{\left (a - c x^{4}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/(d*x**2+c)/(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)/((a - c*x**4)**(3/2)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate((B*x^2+A)/(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)
\[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate((B*x^2+A)/(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)
Timed out. \[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (a-c\,x^4\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2)/((a - c*x^4)^(3/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2)/((a - c*x^4)^(3/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) b \] Input:
int((B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x)
Output:
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 + 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)*b