3.910 \(\int \frac{(e x)^{9/2}}{\left (a-b x^2\right )^2 \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=484 \[ \frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b^2 d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b^2 d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{a e^3 (e x)^{3/2} \sqrt{c-d x^2}}{2 b \left (a-b x^2\right ) (b c-a d)} \]

[Out]

(a*e^3*(e*x)^(3/2)*Sqrt[c - d*x^2])/(2*b*(b*c - a*d)*(a - b*x^2)) + (c^(3/4)*(4*
b*c - 5*a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c
^(1/4)*Sqrt[e])], -1])/(2*b^2*d^(3/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (c^(3/4)*(4
*b*c - 5*a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(
c^(1/4)*Sqrt[e])], -1])/(2*b^2*d^(3/4)*(b*c - a*d)*Sqrt[c - d*x^2]) + (Sqrt[a]*c
^(1/4)*(7*b*c - 5*a*d)*e^(9/2)*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*b^(
5/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (Sqrt[a]*c^(1/4)*(7*b*c - 5*a*d)*e^(
9/2)*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*b^(5/2)*d^(1/4)*(b*c - a*d)*Sqrt
[c - d*x^2])

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Rubi [A]  time = 2.29223, antiderivative size = 484, 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 \[ \frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 b^{5/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b^2 d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 b^2 d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{a e^3 (e x)^{3/2} \sqrt{c-d x^2}}{2 b \left (a-b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(9/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]

[Out]

(a*e^3*(e*x)^(3/2)*Sqrt[c - d*x^2])/(2*b*(b*c - a*d)*(a - b*x^2)) + (c^(3/4)*(4*
b*c - 5*a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c
^(1/4)*Sqrt[e])], -1])/(2*b^2*d^(3/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (c^(3/4)*(4
*b*c - 5*a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(
c^(1/4)*Sqrt[e])], -1])/(2*b^2*d^(3/4)*(b*c - a*d)*Sqrt[c - d*x^2]) + (Sqrt[a]*c
^(1/4)*(7*b*c - 5*a*d)*e^(9/2)*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*b^(
5/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (Sqrt[a]*c^(1/4)*(7*b*c - 5*a*d)*e^(
9/2)*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*b^(5/2)*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)**(9/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.587322, size = 414, normalized size = 0.86 \[ \frac{a (e x)^{9/2} \left (\frac{49 a c^2 F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{14 x^2 \left (d x^2-c\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 c \left (-7 a c+2 a d x^2+4 b c x^2\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{14 b x^3 \left (b x^2-a\right ) \sqrt{c-d x^2} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(9/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]

[Out]

(a*(e*x)^(9/2)*((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*c*(-7*a*c + 4*b*c*x^2 + 2*a*d*x^2)*AppellF1[7/4, 1/2, 1, 11
/4, (d*x^2)/c, (b*x^2)/a] + 14*x^2*(-c + d*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]))/(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*Ap
pellF1[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*(b*c - a*d)*x^3*(-a + b*x^2)*Sqrt[c - d*x^2]
)

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Maple [B]  time = 0.051, size = 2956, normalized size = 6.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)

[Out]

1/8*(-16*((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^4*c^3-4*x^4*a^2*b^2*d^3+4*x^4*a*b^3*c*d^2+5*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^2*a^2*b^2*c*d^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)-7*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^2*a*b^3*c^2*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)+5*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^2*a^2*b^2*c*d^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)-7*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^2*a*b^3*c^2*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)+20*((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^3*b*c*d^2-36*((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^2*c^2*d-10*((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^3*b*c*d^2+18*((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^2*c^2*d+4*x^
2*a^2*b^2*c*d^2-4*x^2*a*b^3*c^2*d+7*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^2*a*b^2*
c*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)*(c*d)^(1/2)*(a*b)^(1/2)-7*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^2*a*b^2*c*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)*(c*d)^(1/2)*(a*b)^(1/2)-20*((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^2*b^2*c*d^2+36*((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^3*c^2*d+10*((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^2*b^2*c*d^2-
18*((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^3*c^2*d+8*((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^4*c^3+16*((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^3*c^3-8
*((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^3*c^3-5*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^2*a^2*b*d^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)*(c*d)^(1/2)*(a*b)^(1/2)+5*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^2*a^
2*b*d^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)*(c*d)^(1/2)*(a*b)^(1/2)-7*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)*a^2*b*c*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)*(c*d)^(1/2)*(a*b)^(1/2)+7*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)*a^2*b*c*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)*(c*d)^(1/2)*(a*b)^(1/2)+7*
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)*a^2*b^2*c^2*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)-5*Elliptic
Pi(((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)*a^3*b*c*d^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)+5*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)*a^3*d^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)*(c*d)^(1/2)*(a*b)^(1/2)+7*Elliptic
Pi(((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)*a^2*b^2*c^2*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)-5*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)*a^3*b*c*d^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)-5*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)*a^3*d^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)*(c*d)^(1/2)*(a*b)^(1/2))*(-d*x^2+c)^(1/2)*
e^4*(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^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{9}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(9/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)),x, algorithm="maxima")

[Out]

integrate((e*x)^(9/2)/((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)^(9/2)/((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)**(9/2)/(-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. \[ \int \frac{\left (e x\right )^{\frac{9}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(9/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)),x, algorithm="giac")

[Out]

integrate((e*x)^(9/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)), x)