3.929 \(\int \frac{(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=447 \[ \frac{d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (a d+14 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 c^{3/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (9 a d+b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (9 a d+b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e \sqrt{e x} (a d+14 b c)}{6 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e \sqrt{e x}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{e \sqrt{e x}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

[Out]

(5*d*e*Sqrt[e*x])/(6*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (e*Sqrt[e*x])/(2*(b*c -
a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(14*b*c + a*d)*e*Sqrt[e*x])/(6*c*(b*c -
a*d)^3*Sqrt[c - d*x^2]) + (d^(3/4)*(14*b*c + a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*E
llipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*c^(3/4)*(b*c - a
*d)^3*Sqrt[c - d*x^2]) - (b*c^(1/4)*(b*c + 9*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*El
lipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(
1/4)*Sqrt[e])], -1])/(4*a*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (b*c^(1/4)*(b
*c + 9*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sq
rt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a*d^(1/4)*(b*c -
a*d)^3*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.36827, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (a d+14 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 c^{3/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (9 a d+b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (9 a d+b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e \sqrt{e x} (a d+14 b c)}{6 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e \sqrt{e x}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{e \sqrt{e x}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

(5*d*e*Sqrt[e*x])/(6*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (e*Sqrt[e*x])/(2*(b*c -
a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(14*b*c + a*d)*e*Sqrt[e*x])/(6*c*(b*c -
a*d)^3*Sqrt[c - d*x^2]) + (d^(3/4)*(14*b*c + a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*E
llipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*c^(3/4)*(b*c - a
*d)^3*Sqrt[c - d*x^2]) - (b*c^(1/4)*(b*c + 9*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*El
lipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(
1/4)*Sqrt[e])], -1])/(4*a*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (b*c^(1/4)*(b
*c + 9*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sq
rt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a*d^(1/4)*(b*c -
a*d)^3*Sqrt[c - d*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.646, size = 547, normalized size = 1.22 \[ \frac{(e x)^{3/2} \left (\frac{25 a \left (-a^2 d^2+13 a b c d+3 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{9 a c \left (5 a^2 d^2 \left (c+d x^2\right )+a b d \left (-65 c^2+51 c d x^2-6 d^2 x^4\right )+b^2 c \left (-15 c^2+109 c d x^2-84 d^2 x^4\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-10 x^2 \left (-a^2 d^2 \left (c+d x^2\right )+a b d \left (13 c^2-10 c d x^2+d^2 x^4\right )+b^2 c \left (3 c^2-19 c d x^2+14 d^2 x^4\right )\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{c \left (c-d x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}\right )}{30 \left (b x^3-a x\right ) \sqrt{c-d x^2} (b c-a d)^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

((e*x)^(3/2)*((25*a*(3*b^2*c^2 + 13*a*b*c*d - a^2*d^2)*AppellF1[1/4, 1/2, 1, 5/4
, (d*x^2)/c, (b*x^2)/a])/(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a]
+ 2*x^2*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[
5/4, 3/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])) + (9*a*c*(5*a^2*d^2*(c + d*x^2) + b^2*
c*(-15*c^2 + 109*c*d*x^2 - 84*d^2*x^4) + a*b*d*(-65*c^2 + 51*c*d*x^2 - 6*d^2*x^4
))*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] - 10*x^2*(-(a^2*d^2*(c + d*x
^2)) + a*b*d*(13*c^2 - 10*c*d*x^2 + d^2*x^4) + b^2*c*(3*c^2 - 19*c*d*x^2 + 14*d^
2*x^4))*(2*b*c*AppellF1[9/4, 1/2, 2, 13/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[
9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a]))/(c*(c - d*x^2)*(9*a*c*AppellF1[5/4, 1
/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[9/4, 1/2, 2, 13/4, (d*
x^2)/c, (b*x^2)/a] + a*d*AppellF1[9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a])))))/
(30*(b*c - a*d)^3*Sqrt[c - d*x^2]*(-(a*x) + b*x^3))

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Maple [B]  time = 0.06, size = 4403, normalized size = 9.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)

[Out]

-1/24*e*(e*x)^(1/2)*(-d*x^2+c)^(1/2)*b*d*(-36*x^3*a^2*b*c*d^3*(a*b)^(1/2)-36*x^3
*a*b^2*c^2*d^2*(a*b)^(1/2)+56*x*a^2*b*c^2*d^2*(a*b)^(1/2)-40*x*a*b^2*c^3*d*(a*b)
^(1/2)+3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2)
)^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*a*b^3*c^4-3*((d
*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d
)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^2*b^4*c^4-3*EllipticPi(((
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),
1/2*2^(1/2))*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*a*b^3*c^4-12*x*b^3*c^4*(a*b)^(1/2)+2*
EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^4*a^2*b*d
^3*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((
-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-28*EllipticF(((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^4*b^3*c^2*d*(-x*d/(c*d)^(1/2))^(1/2)*
(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2
))^(1/2)*(a*b)^(1/2)+26*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1
/2))*2^(1/2)*a^2*b*c^2*d*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+3*Ellipti
cPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/
2)*d),1/2*2^(1/2))*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*x^2*b^4*c^4-3*EllipticPi(((d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*
2^(1/2))*2^(1/2)*x^4*b^3*c^2*d*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-3*E
llipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*
b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*x^4*b^3*c^2*d*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)
*(a*b)^(1/2)-27*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/(
(c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1
/2)*(c*d)^(1/2)*a^2*b*c^2*d-27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*Ellipti
cPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/
2)*b),1/2*2^(1/2))*(c*d)^(1/2)*a^2*b*c^2*d-2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*a^3*d^3*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/
2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(a*b)^(1/2)+26*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1
/2)*x^4*a*b^2*c*d^2*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-28*EllipticF((
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*a^2*b*c*d^2*(-x*d/
(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d
)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*a*b^2*c^2*d*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1
/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*(a*b)^(1/2)+27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/
2))*(c*d)^(1/2)*x^2*a^2*b*c*d^2+27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)
*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*Ell
ipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)
^(1/2)*d),1/2*2^(1/2))*(c*d)^(1/2)*x^2*a^2*b*c*d^2+27*((d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1
/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*
d+(c*d)^(1/2)*b),1/2*2^(1/2))*a^2*b^2*c^3*d-56*x^5*b^3*c^2*d^2*(a*b)^(1/2)+76*x^
3*b^3*c^3*d*(a*b)^(1/2)-4*x*a^3*c*d^3*(a*b)^(1/2)-27*((d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/
2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((
c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*(c*d)^(1/2)*x^4*a*b^2*c*d^2-4*x^3*a^3*d
^4*(a*b)^(1/2)+30*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b
/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^
(1/2)*(c*d)^(1/2)*x^2*a*b^2*c^2*d+30*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/
2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*E
llipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*
d)^(1/2)*b),1/2*2^(1/2))*(c*d)^(1/2)*x^2*a*b^2*c^2*d+52*x^5*a*b^2*c*d^3*(a*b)^(1
/2)+3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*(c*d)^
(1/2)*x^2*b^3*c^3-30*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^
(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^2
*a*b^3*c^3*d-3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((
c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2
^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/
2)*(c*d)^(1/2)*a*b^2*c^3-3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b
),1/2*2^(1/2))*(c*d)^(1/2)*a*b^2*c^3+3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2)
)^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d
)^(1/2))^(1/2)*(a*b)^(1/2)*(c*d)^(1/2)*x^2*b^3*c^3+30*EllipticPi(((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(-x*d/(c*d)^(1/2))^(1/2)*x^2*a*b^3*c^3*d-27*EllipticPi(((d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(-x*d/(c*d)^(1/2))^(1/2)*a^2*b^2*c^3*d+28*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*b^3*c^3*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*
b)^(1/2)+2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*
a^3*c*d^2*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-28*EllipticF(((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^2*c^3*(-x*d/(c*d)^(1/2))^(1/
2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2)*(a*b)^(1/2)-27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^
(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2
))*x^4*a*b^3*c^2*d^2+27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*
x^2*a^2*b^2*c^2*d^2-27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d
)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x
^2*a^2*b^2*c^2*d^2+27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^
4*a*b^3*c^2*d^2+3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b
/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*2^(1/2)*x^4*b^4*c^3*d*((d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2)
)^(1/2)-3*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^
(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*x^4*b^4*c^3*d*((d*x+(c*d)^(1/2))/(c*
d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)+
4*x^5*a^2*b*d^4*(a*b)^(1/2)-27*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*Ellipti
cPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/
2)*b),1/2*2^(1/2))*(c*d)^(1/2)*x^4*a*b^2*c*d^2)/x/(a*d-b*c)^3/(b*x^2-a)/(a*b)^(1
/2)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/(d*x^2-c)^2/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="maxima")

[Out]

integrate((e*x)^(3/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="giac")

[Out]

integrate((e*x)^(3/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)