Integrand size = 22, antiderivative size = 420 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\frac {c x \left (d-e x^2\right )}{2 a \left (c d^2-a e^2\right ) \left (d+e x^2\right ) \sqrt {a-c x^4}}+\frac {e^2 \left (2 c d^2+a e^2\right ) x \sqrt {a-c x^4}}{2 a d \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac {\sqrt [4]{c} e \left (2 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} d \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} \left (c d^2-\sqrt {a} \sqrt {c} d e+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} d \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} e^2 \left (7 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \] Output:
1/2*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(e*x^2+d)/(-c*x^4+a)^(1/2)+1/2*e^2*(a* e^2+2*c*d^2)*x*(-c*x^4+a)^(1/2)/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)+1/2*c^(1/4) *e*(a*e^2+2*c*d^2)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4 )/d/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)+1/2*c^(1/4)*(c*d^2-a^(1/2)*c^(1/2)*d *e+a*e^2)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/d/(c^(1 /2)*d+a^(1/2)*e)/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)-1/2*a^(1/4)*e^2*(-a*e^2+7 *c*d^2)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/ d,I)/c^(1/4)/d^2/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.64 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d \left (a e^4 x \left (a-c x^4\right )+c d x \left (d+e x^2\right ) \left (a e^2+c d \left (d-2 e x^2\right )\right )\right )-i \left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} \sqrt {c} d e \left (2 c d^2+a e^2\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (c^{3/2} d^3-2 \sqrt {a} c d^2 e+2 a \sqrt {c} d e^2-a^{3/2} e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+a e^2 \left (-7 c d^2+a e^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{2 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (c d^3-a d e^2\right )^2 \left (d+e x^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[1/((d + e*x^2)^2*(a - c*x^4)^(3/2)),x]
Output:
(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*(a*e^4*x*(a - c*x^4) + c*d*x*(d + e*x^2)*(a*e^ 2 + c*d*(d - 2*e*x^2))) - I*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*(Sqrt[a]*Sqrt[ c]*d*e*(2*c*d^2 + a*e^2)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*d*(c^(3/2)*d^3 - 2*Sqrt[a]*c*d^2*e + 2*a*Sqrt[c]*d*e^2 - a^( 3/2)*e^3)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + a*e^2*(-7 *c*d^2 + a*e^2)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sq rt[c]/Sqrt[a])]*x], -1]))/(2*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*(c*d^3 - a*d*e^2)^ 2*(d + e*x^2)*Sqrt[a - c*x^4])
Time = 1.73 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1557, 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.
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1557 |
| \(\displaystyle \int \left (-\frac {2 c d e^2}{\sqrt {a-c x^4} \left (d+e x^2\right ) \left (a e^2-c d^2\right )^2}+\frac {e^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^2 \left (a e^2-c d^2\right )}-\frac {c \left (-a e^2-c d^2+2 c d e x^2\right )}{\left (a-c x^4\right )^{3/2} \left (a e^2-c d^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^{3/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right )^2}+\frac {a^{3/4} \sqrt [4]{c} e^3 \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {c^{5/4} d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}-\frac {2 \sqrt [4]{a} c^{3/4} e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 d \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (c d^2-a e^2\right )}-\frac {\sqrt [4]{a} e^2 \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {c x \left (a e^2+c d^2-2 c d e x^2\right )}{2 a \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {e^4 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}\) |
Input:
Int[1/((d + e*x^2)^2*(a - c*x^4)^(3/2)),x]
Output:
(c*x*(c*d^2 + a*e^2 - 2*c*d*e*x^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4] ) + (e^4*x*Sqrt[a - c*x^4])/(2*d*(c*d^2 - a*e^2)^2*(d + e*x^2)) + (c^(5/4) *d*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(1 /4)*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) + (a^(3/4)*c^(1/4)*e^3*Sqrt[1 - (c* x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*(c*d^2 - a*e^2)^2 *Sqrt[a - c*x^4]) + (c^(3/4)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4) *x)/a^(1/4)], -1])/(2*a^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[a - c*x^4]) + (a^(1/4)*c^(1/4)*e^2*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^( 1/4)], -1])/(2*d*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) - (2*a^(1/4)*c^(3/4)*e^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqr t[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/((c*d^2 - a*e^2)^2*Sqrt[a - c* x^4]) - (a^(1/4)*e^2*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((S qrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*c^(1/4)*d^2*( c*d^2 - a*e^2)^2*Sqrt[a - c*x^4])
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Mod ule[{aa, cc}, Int[ExpandIntegrand[1/Sqrt[aa + cc*x^4], (d + e*x^2)^q*(aa + cc*x^4)^(p + 1/2), x] /. {aa -> a, cc -> c}, x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (358 ) = 716\).
Time = 0.69 (sec) , antiderivative size = 865, normalized size of antiderivative = 2.06
| method | result | size |
| default | \(\frac {2 c \left (-\frac {e c d \,x^{3}}{2 a \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) x}{4 a \left (a \,e^{2}-c \,d^{2}\right )^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {e^{4} x \sqrt {-c \,x^{4}+a}}{2 d \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e \,x^{2}+d \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 ) e^{2} c}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\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 ) c^{2} d^{2}}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, a \left (a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, c^{\frac {3}{2}} e d \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, c^{\frac {3}{2}} e d \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, e^{3} \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} d}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, e^{3} \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} d}+\frac {e^{4} \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}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 e^{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}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(865\) |
| elliptic | \(\frac {2 c \left (-\frac {e c d \,x^{3}}{2 a \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) x}{4 a \left (a \,e^{2}-c \,d^{2}\right )^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {e^{4} x \sqrt {-c \,x^{4}+a}}{2 d \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e \,x^{2}+d \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 ) e^{2} c}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\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 ) c^{2} d^{2}}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, a \left (a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, c^{\frac {3}{2}} e d \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, c^{\frac {3}{2}} e d \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, e^{3} \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} d}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, e^{3} \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} d}+\frac {e^{4} \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}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 e^{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}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(865\) |
Input:
int(1/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(-1/2/a*e*c*d/(a*e^2-c*d^2)^2*x^3+1/4*(a*e^2+c*d^2)/a/(a*e^2-c*d^2)^2* x)/(-(x^4-1/c*a)*c)^(1/2)+1/2*e^4/d/(a*e^2-c*d^2)^2*x*(-c*x^4+a)^(1/2)/(e* x^2+d)+1/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^ (1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^ (1/2),I)*e^2*c/(a*e^2-c*d^2)^2+1/2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)* c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*Ellipt icF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)*c^2/a/(a*e^2-c*d^2)^2*d^2-1/a^(1/2)/(1/ a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2 )*x^2)^(1/2)/(-c*x^4+a)^(1/2)*c^(3/2)*e*d/(a*e^2-c*d^2)^2*EllipticF(x*(1/a ^(1/2)*c^(1/2))^(1/2),I)+1/a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)* c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*c^(3/2 )*e*d/(a*e^2-c*d^2)^2*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-1/2*a^(1/2) /(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^ (1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*c^(1/2)*e^3/(a*e^2-c*d^2)^2/d*EllipticF( x*(1/a^(1/2)*c^(1/2))^(1/2),I)+1/2*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/ a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2 )*c^(1/2)*e^3/(a*e^2-c*d^2)^2/d*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+1 /2*e^4/(a*e^2-c*d^2)^2/d^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)* x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*( 1/a^(1/2)*c^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-1/a^(1/2)*c^(1/2))^(1/2...
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a - c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(e*x**2+d)**2/(-c*x**4+a)**(3/2),x)
Output:
Integral(1/((a - c*x**4)**(3/2)*(d + e*x**2)**2), x)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^2),x)
Output:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^2), x)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} e^{2} x^{12}+2 c^{2} d e \,x^{10}-2 a c \,e^{2} x^{8}+c^{2} d^{2} x^{8}-4 a c d e \,x^{6}+a^{2} e^{2} x^{4}-2 a c \,d^{2} x^{4}+2 a^{2} d e \,x^{2}+a^{2} d^{2}}d x \] Input:
int(1/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x)
Output:
int(sqrt(a - c*x**4)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x**4 - 2*a*c *d**2*x**4 - 4*a*c*d*e*x**6 - 2*a*c*e**2*x**8 + c**2*d**2*x**8 + 2*c**2*d* e*x**10 + c**2*e**2*x**12),x)