3.917 \(\int \frac{1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=429 \[ \frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (7 b c-4 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 c^{3/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (7 b c-4 a d)}{6 a^2 c e (e x)^{3/2} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right ) (b c-a d)} \]

[Out]

-((7*b*c - 4*a*d)*Sqrt[c - d*x^2])/(6*a^2*c*(b*c - a*d)*e*(e*x)^(3/2)) + (b*Sqrt
[c - d*x^2])/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)) + (d^(3/4)*(7*b*c - 4*a
*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
-1])/(6*a^2*c^(3/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c -
9*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), Ar
cSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^3*d^(1/4)*(b*c - a*d)*e^(
5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c - 9*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticP
i[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[
e])], -1])/(4*a^3*d^(1/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.12532, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (7 b c-4 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 c^{3/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (7 b c-4 a d)}{6 a^2 c e (e x)^{3/2} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((7*b*c - 4*a*d)*Sqrt[c - d*x^2])/(6*a^2*c*(b*c - a*d)*e*(e*x)^(3/2)) + (b*Sqrt
[c - d*x^2])/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)) + (d^(3/4)*(7*b*c - 4*a
*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
-1])/(6*a^2*c^(3/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c -
9*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), Ar
cSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^3*d^(1/4)*(b*c - a*d)*e^(
5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c - 9*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticP
i[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[
e])], -1])/(4*a^3*d^(1/4)*(b*c - a*d)*e^(5/2)*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(1/(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 = 1.47613, size = 390, normalized size = 0.91 \[ \frac{x \left (\frac{25 a x^2 \left (4 a^2 d^2+20 a b c d-21 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{5 \left (c-d x^2\right ) \left (4 a^2 d-4 a b \left (c+d x^2\right )+7 b^2 c x^2\right )}{c}+\frac{9 a b d x^4 (7 b c-4 a d) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{30 a^2 (e x)^{5/2} \left (a-b x^2\right ) \sqrt{c-d x^2} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-5*(c - d*x^2)*(4*a^2*d + 7*b^2*c*x^2 - 4*a*b*(c + d*x^2)))/c + (25*a*(-21*
b^2*c^2 + 20*a*b*c*d + 4*a^2*d^2)*x^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*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*b*d*(7*b*c - 4*a*d)*x^4*AppellF1[5/4, 1/2, 1, 9/
4, (d*x^2)/c, (b*x^2)/a])/(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*AppellF
1[9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a]))))/(30*a^2*(-(b*c) + a*d)*(e*x)^(5/2
)*(a - b*x^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.048, size = 2622, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b*d*(21*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*a*b^2*c^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)-14*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*
2^(1/2))*2^(1/2)*x*a*b^2*c^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)+28*x^
2*a*b^2*c^2*d*(a*b)^(1/2)-44*x^4*a*b^2*c*d^2*(a*b)^(1/2)-32*a^2*b*c^2*d*(a*b)^(1
/2)+28*x^4*b^3*c^2*d*(a*b)^(1/2)-28*x^2*b^3*c^3*(a*b)^(1/2)+16*x^2*a^2*b*c*d^2*(
a*b)^(1/2)+16*x^4*a^2*b*d^3*(a*b)^(1/2)+21*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^3
*b^4*c^3*((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)-21*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^3*b^4*c^3*
((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)+8*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1
/2))*2^(1/2)*x^3*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)+21*Elli
pticPi(((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*a*b^2*c^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)-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))*2^(1/2)*x*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)+22*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^
(1/2))*2^(1/2)*x*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)+27*Elli
pticPi(((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^3*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)+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))*2^(1/2)*x^3*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)-22*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/
2*2^(1/2))*2^(1/2)*x^3*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)-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*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)+16*a*b^2*c^3*(a*b)^(1/2)-21*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*a*
b^3*c^3*((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)+21*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*a*b^3*c^3*(
(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)+16*a^3*c*d^2*(a*b)^(1/2)-16*x^2*a^3*d^3*(a*b)^(1/2)-21*Elli
pticPi(((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^3*b^3*c^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)+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))*2^(1/2)*x^3*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)-21*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^3*b^3*c^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)+14*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2)
)*2^(1/2)*x^3*b^3*c^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)-27*EllipticP
i(((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^3*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)+27*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*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)-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))*2^(1/2)*x*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)-8*EllipticF(((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x*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)/x/c/a^2/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/((a*
b)^(1/2)*d+(c*d)^(1/2)*b)/(a*b)^(1/2)/(b*x^2-a)/(a*d-b*c)/(d*x^2-c)/e^2/(e*x)^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - a\right )}^{2} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(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(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(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(1/(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. \[ \int \frac{1}{{\left (b x^{2} - a\right )}^{2} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(5/2)), x)