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

Optimal. Leaf size=460 \[ \frac{\sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b \sqrt{c-d x^2} (b c-a d)}+\frac{e (e x)^{3/2} \sqrt{c-d x^2}}{2 \left (a-b x^2\right ) (b c-a d)} \]

[Out]

(e*(e*x)^(3/2)*Sqrt[c - d*x^2])/(2*(b*c - a*d)*(a - b*x^2)) - (c^(3/4)*d^(1/4)*e
^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]
)], -1])/(2*b*(b*c - a*d)*Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*e^(5/2)*Sqrt[1 - (
d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*b*(b*
c - a*d)*Sqrt[c - d*x^2]) + (c^(1/4)*(3*b*c - a*d)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*E
llipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^
(1/4)*Sqrt[e])], -1])/(4*Sqrt[a]*b^(3/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) -
(c^(1/4)*(3*b*c - a*d)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/
(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*Sqrt[a
]*b^(3/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.18056, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b \sqrt{c-d x^2} (b c-a d)}+\frac{e (e x)^{3/2} \sqrt{c-d x^2}}{2 \left (a-b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(e*(e*x)^(3/2)*Sqrt[c - d*x^2])/(2*(b*c - a*d)*(a - b*x^2)) - (c^(3/4)*d^(1/4)*e
^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]
)], -1])/(2*b*(b*c - a*d)*Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*e^(5/2)*Sqrt[1 - (
d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*b*(b*
c - a*d)*Sqrt[c - d*x^2]) + (c^(1/4)*(3*b*c - a*d)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*E
llipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^
(1/4)*Sqrt[e])], -1])/(4*Sqrt[a]*b^(3/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) -
(c^(1/4)*(3*b*c - a*d)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/
(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*Sqrt[a
]*b^(3/2)*d^(1/4)*(b*c - a*d)*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)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.410571, size = 325, normalized size = 0.71 \[ \frac{e (e x)^{3/2} \left (-\frac{49 a c^2 F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{11 a c d x^2 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+7 c-7 d x^2\right )}{14 \left (b x^2-a\right ) \sqrt{c-d x^2} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(e*(e*x)^(3/2)*(7*c - 7*d*x^2 - (49*a*c^2*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c,
(b*x^2)/a])/(7*a*c*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b
*c*AppellF1[7/4, 1/2, 2, 11/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[7/4, 3/2, 1,
11/4, (d*x^2)/c, (b*x^2)/a])) + (11*a*c*d*x^2*AppellF1[7/4, 1/2, 1, 11/4, (d*x^
2)/c, (b*x^2)/a])/(11*a*c*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a] + 2*
x^2*(2*b*c*AppellF1[11/4, 1/2, 2, 15/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[11/
4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a]))))/(14*(-(b*c) + a*d)*(-a + b*x^2)*Sqrt[
c - d*x^2])

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Maple [B]  time = 0.039, size = 2548, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)

[Out]

1/8*(4*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2),1/2*2^(1/2))*x^2*b^3*c^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)*Ellipt
icPi(((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^2*c-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/((c*d)^(1/2
)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*(c*d)^(1/2)*x^2*b^2*c+((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^2*c*d+((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)*Ellipt
icPi(((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*b^2*c*d-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)*El
lipticPi(((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*c+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/((c*d)^(1/2
)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*(c*d)^(1/2)*a*b*c-4*x^2*b^3*c^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)*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*b*d+4*x^2*
a*b^2*c*d-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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2),1/2*2^(1/2))*x^2*b^3*c^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))*x^2*b^3*c^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/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2
))*x^2*b^3*c^2-4*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*c^2+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)*EllipticF(
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*c^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^2*c^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/((c*d)^(1/2)*b-(
a*b)^(1/2)*d),1/2*2^(1/2))*a*b^2*c^2-4*((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)*EllipticE((
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2*c*d+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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2
*a*b^2*c*d-4*x^4*a*b^2*d^2+4*x^4*b^3*c*d+((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^2*a*b*d+4*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b*c*d-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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^
(1/2))*a^2*b*c*d+((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^2*d-((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)*a^2*d-((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*c*d-((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))*a^2*b*c*d
)*d*(-d*x^2+c)^(1/2)*e^2*(e*x)^(1/2)/x/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/((a*b)^(1/2
)*d+(c*d)^(1/2)*b)/(b*x^2-a)/(a*d-b*c)/(d*x^2-c)/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [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)^(5/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)),x, algorithm="fricas")

[Out]

Timed out

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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)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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