3.927 \(\int \frac{(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=454 \[ \frac{5 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{5 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{5 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 e^3 \sqrt{e x} (2 a d+b c)}{6 \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 \sqrt{e x} (3 a d+2 b c)}{6 b \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{a e^3 \sqrt{e x}}{2 b \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

[Out]

((2*b*c + 3*a*d)*e^3*Sqrt[e*x])/(6*b*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (a*e^3*S
qrt[e*x])/(2*b*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (5*(b*c + 2*a*d)*e^3
*Sqrt[e*x])/(6*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (5*c^(1/4)*(b*c + 2*a*d)*e^(7/2)
*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1
])/(6*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (5*c^(1/4)*(b*c + a*d)*e^(7/2)*Sq
rt[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*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2
]) - (5*c^(1/4)*(b*c + a*d)*e^(7/2)*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*d
^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.2747, antiderivative size = 454, 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 \[ \frac{5 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{5 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{5 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 e^3 \sqrt{e x} (2 a d+b c)}{6 \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 \sqrt{e x} (3 a d+2 b c)}{6 b \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{a e^3 \sqrt{e x}}{2 b \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

((2*b*c + 3*a*d)*e^3*Sqrt[e*x])/(6*b*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (a*e^3*S
qrt[e*x])/(2*b*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (5*(b*c + 2*a*d)*e^3
*Sqrt[e*x])/(6*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (5*c^(1/4)*(b*c + 2*a*d)*e^(7/2)
*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1
])/(6*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (5*c^(1/4)*(b*c + a*d)*e^(7/2)*Sq
rt[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*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2
]) - (5*c^(1/4)*(b*c + a*d)*e^(7/2)*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*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)**(7/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.43422, size = 520, normalized size = 1.15 \[ \frac{(e x)^{7/2} \left (\frac{2 x^2 \left (a^2 d \left (5 c-7 d x^2\right )+2 a b \left (5 c^2-8 c d x^2+5 d^2 x^4\right )+b^2 c x^2 \left (5 d x^2-7 c\right )\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c \left (a^2 d \left (5 c-7 d x^2\right )+2 a b \left (5 c^2-9 c d x^2+6 d^2 x^4\right )+2 b^2 c x^2 \left (3 d x^2-4 c\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (d x^2-c\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{25 a^2 c (a d+2 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{6 x^3 \left (a-b x^2\right ) \sqrt{c-d x^2} (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

((e*x)^(7/2)*((25*a^2*c*(2*b*c + a*d)*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*A
ppellF1[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*(a^2*d*(5*c - 7*d*x^2) + 2*b^2*c*x^2*(-4*c + 3
*d*x^2) + 2*a*b*(5*c^2 - 9*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] + 2*x^2*(a^2*d*(5*c - 7*d*x^2) + b^2*c*x^2*(-7*c + 5*d*x^2) +
2*a*b*(5*c^2 - 8*c*d*x^2 + 5*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 +
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*A
ppellF1[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])))))/(6*(-(b*c) + a*d)^3*x^3*(a - b*x^2)*Sqrt[c - d*x^2
])

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Maple [B]  time = 0.082, size = 4403, normalized size = 9.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)

[Out]

1/24*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)*b*(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)-20*x*a^2*b*c^2*d^2*(a*b)^(1/2)+40*x*a*b^2*c^3*d*(a*b)^(
1/2)-20*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)+10*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)+10*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)-1
5*((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^3*b*c^2*d^2+15*((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^3*b*c^2*d^2+15*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+
15*((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*b*c^2*d+20*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)+10*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)+10*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)-20*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)-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)*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^2-30*((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)*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^2*b*c*d^2-15*((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+20*x^5*b^3*c^2*d^2*(a*b)^(1/2)-28*x^3*b^3*c^3*d*(a*b)^(1/2)-20
*x*a^3*c*d^3*(a*b)^(1/2)+15*((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+28*x^3*a^3*d^4*(a*b)^(1/2)-15*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))*((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-15*((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^2*c^2*d+20*x^5*a*b^2*c*d^3*(a*b)^(1/2)+15*((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-15*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+15*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-10*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)-20*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)+10*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)+15*((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^2-15*((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^3*b*c*d^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/((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+15*((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^3*b*c*d^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^
2*b^2*c^2*d^2+15*((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^3*c*d^2+15*((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)*Elliptic
Pi(((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^3*c*d^2-15*((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^3*d^3-15*((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^3*d^3-15*((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^2*b^2*c*d^3-15*((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+15*((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^2*b^2*c*d^3+15*((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^4*a^2*b*d^3+15
*((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^2*b*d^3-40*x^5*a^2*b*d^4*(a*b)^(1/2)+15*((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^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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(7/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="maxima")

[Out]

integrate((e*x)^(7/2)/((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)^(7/2)/((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)**(7/2)/(-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. \[ \int \frac{\left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(7/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="giac")

[Out]

integrate((e*x)^(7/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)