Integrand size = 32, antiderivative size = 319 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A c-\frac {a B d}{c}+(B c-A d) x^2\right )}{2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {a^{3/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 c^{3/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (2 B c^{3/2}+\sqrt {a} B d-A \sqrt {c} d\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 c^{5/4} d \left (c^{3/2}+\sqrt {a} d\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d \left (c^3-a d^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*x*(A*c-a*B*d/c+(-A*d+B*c)*x^2)/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)-1/2*a^(3/ 4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/(-a *d^2+c^3)/(-c*x^4+a)^(1/2)-1/2*a^(1/4)*(2*B*c^(3/2)+a^(1/2)*B*d-A*c^(1/2)* d)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(5/4)/d/(c^(3/2)+a^( 1/2)*d)/(-c*x^4+a)^(1/2)+a^(1/4)*c^(3/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*Elli pticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I)/d/(-a*d^2+c^3)/(-c*x^4+a)^( 1/2)
Result contains complex when optimal does not.
Time = 11.20 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.31 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {-A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x+a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 x-B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x^3+A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2 x^3-i \sqrt {a} \sqrt {c} d (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (-c^{3/2}+\sqrt {a} d\right ) \left (2 B c^{3/2}+\sqrt {a} B d-A \sqrt {c} d\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+2 i B c^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-2 i A c^2 d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (-c^4 d+a c d^3\right ) \sqrt {a-c x^4}} \] Input:
Integrate[(x^4*(A + B*x^2))/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(-(A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d*x) + a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2* x - B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d*x^3 + A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^ 2*x^3 - I*Sqrt[a]*Sqrt[c]*d*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*Ar cSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*(-c^(3/2) + Sqrt[a]*d)*(2*B*c^( 3/2) + Sqrt[a]*B*d - A*Sqrt[c]*d)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[ Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (2*I)*B*c^3*Sqrt[1 - (c*x^4)/a]*Ellipti cPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - ( 2*I)*A*c^2*d*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcS inh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*Sqrt[-(Sqrt[c]/Sqrt[a])]*(-(c^4*d ) + a*c*d^3)*Sqrt[a - c*x^4])
Time = 0.69 (sec) , antiderivative size = 366, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2\right )}{\left (a-c x^4\right )^{3/2} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (-\frac {c^2 (B c-A d)}{d \left (a d^2-c^3\right ) \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {a \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{c \left (c^3-a d^2\right ) \left (a-c x^4\right )^{3/2}}-\frac {B}{c d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 c^{3/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B+A \sqrt {c}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 c^{5/4} \left (\sqrt {a} d+c^{3/2}\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {x \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{2 c \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} d \sqrt {a-c x^4}}\) |
Input:
Int[(x^4*(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*c*(c^3 - a*d^2)*Sqrt[a - c*x^4] ) - (a^(3/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/ a^(1/4)], -1])/(2*c^(3/4)*(c^3 - a*d^2)*Sqrt[a - c*x^4]) - (a^(1/4)*B*Sqrt [1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(5/4)*d*Sqr t[a - c*x^4]) + (a^(1/4)*(Sqrt[a]*B + A*Sqrt[c])*Sqrt[1 - (c*x^4)/a]*Ellip ticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*c^(5/4)*(c^(3/2) + Sqrt[a]*d)*Sq rt[a - c*x^4]) + (a^(1/4)*c^(3/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*Elliptic Pi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d*(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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 849 vs. \(2 (261 ) = 522\).
Time = 1.73 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.66
| method | result | size |
| default | \(\frac {c^{2} \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^{3}}-\frac {-d \left (A d -B c \right ) \left (\frac {x^{3}}{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}}}\, \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 c d \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 )-B \,c^{2} \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 )-B \,d^{2} \left (\frac {x}{2 c \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 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d^{3}}\) | \(850\) |
| elliptic | \(\frac {2 c \left (\frac {\left (A d -B c \right ) x^{3}}{4 c \left (a \,d^{2}-c^{3}\right )}-\frac {\left (A \,c^{2}-B a d \right ) x}{4 c^{2} \left (a \,d^{2}-c^{3}\right )}\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 ) B}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, c d}-\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 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,d^{2}-c^{3}\right )}+\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 B d}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, c \left (a \,d^{2}-c^{3}\right )}+\frac {\sqrt {a}\, \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 {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {a}\, \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 {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{2 \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {c \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 ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {c^{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 ) B}{\left (a \,d^{2}-c^{3}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(867\) |
Input:
int(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
c^2*(A*d-B*c)/d^3*(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)*Ellipti cF(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)*EllipticE(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)))-1/d^3*(-d*(A*d-B*c)*(1/2/a*x^3/(-(x^4-a/c)*c)^(1/2)+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)/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)))+A*c*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)*EllipticF(x*(c^(1/2)/a^(1/2))^( 1/2),I))-B*c^2*(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)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I))-B*d^2*(1/2/c*x/(-(x^4-a/c)*...
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**4*(B*x**2+A)/(d*x**2+c)/(-c*x**4+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*x^4/((-c*x^4 + a)^(3/2)*(d*x^2 + c)), x)
\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*x^4/((-c*x^4 + a)^(3/2)*(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {x^4\,\left (B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((x^4*(A + B*x^2))/((a - c*x^4)^(3/2)*(c + d*x^2)),x)
Output:
int((x^4*(A + B*x^2))/((a - c*x^4)^(3/2)*(c + d*x^2)), x)
\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-c \,x^{4}+a}\, a x -\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a^{3} c +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a^{2} c^{2} x^{4}-\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{6}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a^{2} c d +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{6}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a b \,c^{2}+\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{6}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a \,c^{2} d \,x^{4}-\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{6}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) b \,c^{3} x^{4}-\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a^{3} d +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} d \,x^{10}+c^{3} x^{8}-2 a c d \,x^{6}-2 a \,c^{2} x^{4}+a^{2} d \,x^{2}+a^{2} c}d x \right ) a^{2} c d \,x^{4}}{c^{2} \left (-c \,x^{4}+a \right )} \] Input:
int(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x)
Output:
(sqrt(a - c*x**4)*a*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**3*c + 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*c**2*x**4 - int((sqrt(a - c*x**4)*x**6)/(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*c*d + int((sqrt(a - c*x**4)*x**6)/(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**2 + int((sqrt(a - c*x**4)*x**6)/(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*d*x**4 - int((sqrt(a - c*x**4)*x**6)/(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**1 0),x)*b*c**3*x**4 - 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**3*d + int((sq rt(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**2*c*d*x**4)/(c**2*(a - c*x**4))