Integrand size = 22, antiderivative size = 713 \[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\frac {c x \left (d-e x^2\right )}{2 a \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3 \sqrt {a-c x^4}}+\frac {e^2 \left (6 c d^2+a e^2\right ) x \sqrt {a-c x^4}}{6 a d \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^3}+\frac {e^2 \left (36 c^2 d^4+39 a c d^2 e^2-5 a^2 e^4\right ) x \sqrt {a-c x^4}}{24 a d^2 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )^2}+\frac {e^2 \left (32 c^3 d^6+121 a c^2 d^4 e^2-18 a^2 c d^2 e^4+5 a^3 e^6\right ) x \sqrt {a-c x^4}}{16 a d^3 \left (c d^2-a e^2\right )^4 \left (d+e x^2\right )}+\frac {\sqrt [4]{c} e \left (32 c^3 d^6+121 a c^2 d^4 e^2-18 a^2 c d^2 e^4+5 a^3 e^6\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 \sqrt [4]{a} d^3 \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} \left (24 c^3 d^6-72 \sqrt {a} c^{5/2} d^5 e+285 a c^2 d^4 e^2-78 a^{3/2} c^{3/2} d^3 e^3-44 a^2 c d^2 e^4+10 a^{5/2} \sqrt {c} d e^5+15 a^3 e^6\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{48 a^{3/4} d^3 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} e^2 \left (231 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}} \] Output:
1/2*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(e*x^2+d)^3/(-c*x^4+a)^(1/2)+1/6*e^2*( a*e^2+6*c*d^2)*x*(-c*x^4+a)^(1/2)/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)^3+1/24*e^ 2*(-5*a^2*e^4+39*a*c*d^2*e^2+36*c^2*d^4)*x*(-c*x^4+a)^(1/2)/a/d^2/(-a*e^2+ c*d^2)^3/(e*x^2+d)^2+1/16*e^2*(5*a^3*e^6-18*a^2*c*d^2*e^4+121*a*c^2*d^4*e^ 2+32*c^3*d^6)*x*(-c*x^4+a)^(1/2)/a/d^3/(-a*e^2+c*d^2)^4/(e*x^2+d)+1/16*c^( 1/4)*e*(5*a^3*e^6-18*a^2*c*d^2*e^4+121*a*c^2*d^4*e^2+32*c^3*d^6)*(1-c*x^4/ a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/d^3/(-a*e^2+c*d^2)^4/(-c*x ^4+a)^(1/2)+1/48*c^(1/4)*(24*c^3*d^6-72*a^(1/2)*c^(5/2)*d^5*e+285*a*c^2*d^ 4*e^2-78*a^(3/2)*c^(3/2)*d^3*e^3-44*a^2*c*d^2*e^4+10*a^(5/2)*c^(1/2)*d*e^5 +15*a^3*e^6)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/d^3/ (c^(1/2)*d-a^(1/2)*e)^3/(c^(1/2)*d+a^(1/2)*e)^4/(-c*x^4+a)^(1/2)-1/16*a^(1 /4)*e^2*(-5*a^3*e^6+21*a^2*c*d^2*e^4+33*a*c^2*d^4*e^2+231*c^3*d^6)*(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^ 4/(-a*e^2+c*d^2)^4/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.55 (sec) , antiderivative size = 602, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d x \left (8 a e^4 \left (c d^3-a d e^2\right )^2 \left (a-c x^4\right )+2 a d e^4 \left (-c d^2+a e^2\right ) \left (-27 c d^2+5 a e^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )+3 a e^4 \left (89 c^2 d^4-18 a c d^2 e^2+5 a^2 e^4\right ) \left (d+e x^2\right )^2 \left (a-c x^4\right )+24 c^2 d^3 \left (d+e x^2\right )^3 \left (a^2 e^4+c^2 d^3 \left (d-4 e x^2\right )+2 a c d e^2 \left (3 d-2 e x^2\right )\right )\right )-i \left (d+e x^2\right )^3 \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (32 c^3 d^6+121 a c^2 d^4 e^2-18 a^2 c d^2 e^4+5 a^3 e^6\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (24 c^{7/2} d^7-96 \sqrt {a} c^3 d^6 e+357 a c^{5/2} d^5 e^2-363 a^{3/2} c^2 d^4 e^3+34 a^2 c^{3/2} d^3 e^4+54 a^{5/2} c d^2 e^5+5 a^3 \sqrt {c} d e^6-15 a^{7/2} e^7\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 a e^2 \left (-231 c^3 d^6-33 a c^2 d^4 e^2-21 a^2 c d^2 e^4+5 a^3 e^6\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{48 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (c d^3-a d e^2\right )^4 \left (d+e x^2\right )^3 \sqrt {a-c x^4}} \] Input:
Integrate[1/((d + e*x^2)^4*(a - c*x^4)^(3/2)),x]
Output:
(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*x*(8*a*e^4*(c*d^3 - a*d*e^2)^2*(a - c*x^4) + 2 *a*d*e^4*(-(c*d^2) + a*e^2)*(-27*c*d^2 + 5*a*e^2)*(d + e*x^2)*(a - c*x^4) + 3*a*e^4*(89*c^2*d^4 - 18*a*c*d^2*e^2 + 5*a^2*e^4)*(d + e*x^2)^2*(a - c*x ^4) + 24*c^2*d^3*(d + e*x^2)^3*(a^2*e^4 + c^2*d^3*(d - 4*e*x^2) + 2*a*c*d* e^2*(3*d - 2*e*x^2))) - I*(d + e*x^2)^3*Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*Sqr t[c]*d*e*(32*c^3*d^6 + 121*a*c^2*d^4*e^2 - 18*a^2*c*d^2*e^4 + 5*a^3*e^6)*E llipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*d*(24*c^(7/2 )*d^7 - 96*Sqrt[a]*c^3*d^6*e + 357*a*c^(5/2)*d^5*e^2 - 363*a^(3/2)*c^2*d^4 *e^3 + 34*a^2*c^(3/2)*d^3*e^4 + 54*a^(5/2)*c*d^2*e^5 + 5*a^3*Sqrt[c]*d*e^6 - 15*a^(7/2)*e^7)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 3*a*e^2*(-231*c^3*d^6 - 33*a*c^2*d^4*e^2 - 21*a^2*c*d^2*e^4 + 5*a^3*e^6)*E llipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x ], -1]))/(48*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*(c*d^3 - a*d*e^2)^4*(d + e*x^2)^3* Sqrt[a - c*x^4])
Leaf count is larger than twice the leaf count of optimal. \(1700\) vs. \(2(713)=1426\).
Time = 5.70 (sec) , antiderivative size = 1700, normalized size of antiderivative = 2.38, 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.
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1557 |
| \(\displaystyle \int \left (\frac {c^2 \left (a^2 e^4+4 c d e x^2 \left (-a e^2-c d^2\right )+6 a c d^2 e^2+c^2 d^4\right )}{\left (a-c x^4\right )^{3/2} \left (a e^2-c d^2\right )^4}+\frac {4 c^2 d e^2 \left (-a e^2-c d^2\right )}{\sqrt {a-c x^4} \left (d+e x^2\right ) \left (a e^2-c d^2\right )^4}-\frac {c e^2 \left (-a e^2-3 c d^2\right )}{\sqrt {a-c x^4} \left (d+e x^2\right )^2 \left (a e^2-c d^2\right )^3}-\frac {2 c d e^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^3 \left (a e^2-c d^2\right )^2}+\frac {e^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^4 \left (a e^2-c d^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 c \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4} e^4}{4 d \left (c d^2-a e^2\right )^4 \left (e x^2+d\right )}+\frac {c \left (3 c d^2+a e^2\right ) x \sqrt {a-c x^4} e^4}{2 d \left (c d^2-a e^2\right )^4 \left (e x^2+d\right )}+\frac {\left (29 c^2 d^4-14 a c e^2 d^2+5 a^2 e^4\right ) x \sqrt {a-c x^4} e^4}{16 d^3 \left (c d^2-a e^2\right )^4 \left (e x^2+d\right )}+\frac {5 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4} e^4}{24 d^2 \left (c d^2-a e^2\right )^3 \left (e x^2+d\right )^2}+\frac {c x \sqrt {a-c x^4} e^4}{2 \left (c d^2-a e^2\right )^3 \left (e x^2+d\right )^2}+\frac {x \sqrt {a-c x^4} e^4}{6 d \left (c d^2-a e^2\right )^2 \left (e x^2+d\right )^3}+\frac {3 a^{3/4} c^{5/4} \left (3 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 ) e^3}{4 d \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}+\frac {a^{3/4} c^{5/4} \left (3 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 ) e^3}{2 d \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt [4]{c} \left (29 c^2 d^4-14 a c e^2 d^2+5 a^2 e^4\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 ) e^3}{16 d^3 \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{5/4} \left (7 c d^2-2 \sqrt {a} \sqrt {c} e d-3 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 ) e^2}{4 d \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^4 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{5/4} \left (3 c d^2+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 ) e^2}{2 d \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (57 c^2 d^4-30 \sqrt {a} c^{3/2} e d^3-32 a c e^2 d^2+10 a^{3/2} \sqrt {c} e^3 d+15 a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) e^2}{48 d^3 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^4 \sqrt {a-c x^4}}-\frac {4 \sqrt [4]{a} c^{7/4} \left (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 ) e^2}{\left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{3/4} \left (9 c^2 d^4-a^2 e^4\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 ) e^2}{2 d^2 \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}-\frac {3 \sqrt [4]{a} c^{3/4} \left (5 c^2 d^4-2 a c e^2 d^2+a^2 e^4\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 ) e^2}{4 d^2 \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (35 c^3 d^6-7 a c^2 e^2 d^4+17 a^2 c e^4 d^2-5 a^3 e^6\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 ) e^2}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}+\frac {2 c^{9/4} d \left (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 ) e}{\sqrt [4]{a} \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}+\frac {c^{7/4} \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} \left (\sqrt {c} d+\sqrt {a} e\right )^4 \sqrt {a-c x^4}}+\frac {c^2 x \left (c^2 d^4+6 a c e^2 d^2-4 c e \left (c d^2+a e^2\right ) x^2 d+a^2 e^4\right )}{2 a \left (c d^2-a e^2\right )^4 \sqrt {a-c x^4}}\) |
Input:
Int[1/((d + e*x^2)^4*(a - c*x^4)^(3/2)),x]
Output:
(c^2*x*(c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*c*d*e*(c*d^2 + a*e^2)*x^2))/ (2*a*(c*d^2 - a*e^2)^4*Sqrt[a - c*x^4]) + (e^4*x*Sqrt[a - c*x^4])/(6*d*(c* d^2 - a*e^2)^2*(d + e*x^2)^3) + (c*e^4*x*Sqrt[a - c*x^4])/(2*(c*d^2 - a*e^ 2)^3*(d + e*x^2)^2) + (5*e^4*(3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(24*d^2* (c*d^2 - a*e^2)^3*(d + e*x^2)^2) + (3*c*e^4*(3*c*d^2 - a*e^2)*x*Sqrt[a - c *x^4])/(4*d*(c*d^2 - a*e^2)^4*(d + e*x^2)) + (c*e^4*(3*c*d^2 + a*e^2)*x*Sq rt[a - c*x^4])/(2*d*(c*d^2 - a*e^2)^4*(d + e*x^2)) + (e^4*(29*c^2*d^4 - 14 *a*c*d^2*e^2 + 5*a^2*e^4)*x*Sqrt[a - c*x^4])/(16*d^3*(c*d^2 - a*e^2)^4*(d + e*x^2)) + (3*a^(3/4)*c^(5/4)*e^3*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*E llipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(4*d*(c*d^2 - a*e^2)^4*Sqrt[a - c*x^4]) + (2*c^(9/4)*d*e*(c*d^2 + a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[Ar cSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*(c*d^2 - a*e^2)^4*Sqrt[a - c*x^4] ) + (a^(3/4)*c^(5/4)*e^3*(3*c*d^2 + a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[A rcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*(c*d^2 - a*e^2)^4*Sqrt[a - c*x^4]) + (a^(3/4)*c^(1/4)*e^3*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*Sqrt[1 - ( c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(16*d^3*(c*d^2 - a*e ^2)^4*Sqrt[a - c*x^4]) + (c^(7/4)*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)^4*Sqrt[a - c*x^ 4]) + (a^(1/4)*c^(5/4)*e^2*(7*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Sqr t[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(4*d*(Sqrt...
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. 1796 vs. \(2 (631 ) = 1262\).
Time = 1.56 (sec) , antiderivative size = 1797, normalized size of antiderivative = 2.52
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1797\) |
| elliptic | \(\text {Expression too large to display}\) | \(1797\) |
Input:
int(1/(e*x^2+d)^4/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(-c^2*e*d*(a*e^2+c*d^2)/a/(a*e^2-c*d^2)^4*x^3+1/4*c/a*(a^2*e^4+6*a*c*d ^2*e^2+c^2*d^4)/(a*e^2-c*d^2)^4*x)/(-(x^4-1/c*a)*c)^(1/2)+1/6*e^4/d/(a*e^2 -c*d^2)^2*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^3+1/24*e^4*(5*a*e^2-27*c*d^2)/(a*e^ 2-c*d^2)^3/d^2*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^2+1/16*e^4*(5*a^2*e^4-18*a*c*d ^2*e^2+89*c^2*d^4)/(a*e^2-c*d^2)^4/d^3*x*(-c*x^4+a)^(1/2)/(e*x^2+d)-231/16 *e^2/(a*e^2-c*d^2)^4*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)/(1/ a^(1/2)*c^(1/2))^(1/2))*c^3+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)*EllipticF (x*(1/a^(1/2)*c^(1/2))^(1/2),I)*c^4/a/(a*e^2-c*d^2)^4*d^4+5/16*e^8/(a*e^2- c*d^2)^4/d^4/(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)/(1/a^(1/2)*c^(1 /2))^(1/2))*a^3-33/16*e^4/(a*e^2-c*d^2)^4/(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)/(1/a^(1/2)*c^(1/2))^(1/2))*a*c^2+5/48/(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^6*c/d^2/(a*e^2-c*...
\[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{4}} \,d x } \] Input:
integrate(1/(e*x^2+d)^4/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c*x^4 + a)/(c^2*e^4*x^16 + 4*c^2*d*e^3*x^14 + 2*(3*c^2*d^2* e^2 - a*c*e^4)*x^12 + 4*(c^2*d^3*e - 2*a*c*d*e^3)*x^10 + (c^2*d^4 - 12*a*c *d^2*e^2 + a^2*e^4)*x^8 + 4*a^2*d^3*e*x^2 - 4*(2*a*c*d^3*e - a^2*d*e^3)*x^ 6 + a^2*d^4 - 2*(a*c*d^4 - 3*a^2*d^2*e^2)*x^4), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x**2+d)**4/(-c*x**4+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{4}} \,d x } \] Input:
integrate(1/(e*x^2+d)^4/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^4), x)
\[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{4}} \,d x } \] Input:
integrate(1/(e*x^2+d)^4/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^4), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^4} \,d x \] Input:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^4),x)
Output:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^4), x)
\[ \int \frac {1}{\left (d+e x^2\right )^4 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} e^{4} x^{16}+4 c^{2} d \,e^{3} x^{14}-2 a c \,e^{4} x^{12}+6 c^{2} d^{2} e^{2} x^{12}-8 a c d \,e^{3} x^{10}+4 c^{2} d^{3} e \,x^{10}+a^{2} e^{4} x^{8}-12 a c \,d^{2} e^{2} x^{8}+c^{2} d^{4} x^{8}+4 a^{2} d \,e^{3} x^{6}-8 a c \,d^{3} e \,x^{6}+6 a^{2} d^{2} e^{2} x^{4}-2 a c \,d^{4} x^{4}+4 a^{2} d^{3} e \,x^{2}+a^{2} d^{4}}d x \] Input:
int(1/(e*x^2+d)^4/(-c*x^4+a)^(3/2),x)
Output:
int(sqrt(a - c*x**4)/(a**2*d**4 + 4*a**2*d**3*e*x**2 + 6*a**2*d**2*e**2*x* *4 + 4*a**2*d*e**3*x**6 + a**2*e**4*x**8 - 2*a*c*d**4*x**4 - 8*a*c*d**3*e* x**6 - 12*a*c*d**2*e**2*x**8 - 8*a*c*d*e**3*x**10 - 2*a*c*e**4*x**12 + c** 2*d**4*x**8 + 4*c**2*d**3*e*x**10 + 6*c**2*d**2*e**2*x**12 + 4*c**2*d*e**3 *x**14 + c**2*e**4*x**16),x)