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

Optimal. Leaf size=514 \[ \frac{d \sqrt{e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{6 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a c^{7/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-13 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-13 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b \sqrt{e x}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d \sqrt{e x} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d)*Sqrt[e*x])/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*Sqr
t[e*x])/(2*a*(b*c - a*d)*e*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 17*a
*b*c*d - 5*a^2*d^2)*Sqrt[e*x])/(6*a*c^2*(b*c - a*d)^3*e*Sqrt[c - d*x^2]) + (d^(3
/4)*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d
^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*a*c^(7/4)*(b*c - a*d)^3*Sqrt[e]*Sq
rt[c - d*x^2]) + (b^2*c^(1/4)*(3*b*c - 13*a*d)*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*a^2*d^(1/4)*(b*c - a*d)^3*Sqrt[e]*Sqrt[c - d*x^2]) + (b^2*c^(1/4)*(
3*b*c - 13*a*d)*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*a^2*d^(1/4)*(b*c - a*
d)^3*Sqrt[e]*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.48185, antiderivative size = 514, 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 \[ \frac{d \sqrt{e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{6 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a c^{7/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-13 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-13 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b \sqrt{e x}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d \sqrt{e x} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(3*b*c + 2*a*d)*Sqrt[e*x])/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*Sqr
t[e*x])/(2*a*(b*c - a*d)*e*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 17*a
*b*c*d - 5*a^2*d^2)*Sqrt[e*x])/(6*a*c^2*(b*c - a*d)^3*e*Sqrt[c - d*x^2]) + (d^(3
/4)*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d
^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*a*c^(7/4)*(b*c - a*d)^3*Sqrt[e]*Sq
rt[c - d*x^2]) + (b^2*c^(1/4)*(3*b*c - 13*a*d)*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*a^2*d^(1/4)*(b*c - a*d)^3*Sqrt[e]*Sqrt[c - d*x^2]) + (b^2*c^(1/4)*(
3*b*c - 13*a*d)*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*a^2*d^(1/4)*(b*c - a*
d)^3*Sqrt[e]*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)**(1/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.79476, size = 629, normalized size = 1.22 \[ \frac{x \left (\frac{10 x^2 \left (a^3 d^3 \left (5 d x^2-7 c\right )+a^2 b d^2 \left (19 c^2-10 c d x^2-5 d^2 x^4\right )+a b^2 c d^2 x^2 \left (17 d x^2-19 c\right )+3 b^3 c^2 \left (c-d x^2\right )^2\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-9 a c \left (5 a^3 d^3 \left (7 c-5 d x^2\right )+5 a^2 b d^2 \left (-19 c^2+9 c d x^2+6 d^2 x^4\right )+2 a b^2 c d^2 x^2 \left (56 c-51 d x^2\right )-3 b^3 c^2 \left (5 c^2-11 c d x^2+6 d^2 x^4\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{a \left (d x^2-c\right ) (a d-b c)^3 \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{25 c \left (-5 a^3 d^3+17 a^2 b c d^2-36 a b^2 c^2 d+9 b^3 c^3\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{(b c-a d)^3 \left (2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}\right )}{30 c^2 \sqrt{e x} \left (a-b x^2\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((25*c*(9*b^3*c^3 - 36*a*b^2*c^2*d + 17*a^2*b*c*d^2 - 5*a^3*d^3)*AppellF1[1/4
, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/((b*c - a*d)^3*(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
*(2*a*b^2*c*d^2*x^2*(56*c - 51*d*x^2) + 5*a^3*d^3*(7*c - 5*d*x^2) - 3*b^3*c^2*(5
*c^2 - 11*c*d*x^2 + 6*d^2*x^4) + 5*a^2*b*d^2*(-19*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] + 10*x^2*(3*b^3*c^2*(c - d*x^2)
^2 + a^3*d^3*(-7*c + 5*d*x^2) + a*b^2*c*d^2*x^2*(-19*c + 17*d*x^2) + a^2*b*d^2*(
19*c^2 - 10*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]))/(a*(-(b*c)
+ a*d)^3*(-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*c^2*Sqrt[e*x]*(a - b*x^2)*Sqrt[c
- d*x^2])

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Maple [B]  time = 0.067, size = 4776, normalized size = 9.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(-d*x^2+c)^(1/2)*b*d*(-39*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^3*d*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+44*EllipticF(((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3*b*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^
(1/2)*(a*b)^(1/2)+9*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)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+20*x^3*a^4*d^5*(a*b)^(1/2)-12*x*b^4*c^5*(a*b)^
(1/2)-16*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^2*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2
)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+34*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^3*c^3*d*((-d*x+(
c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-39*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*a
*b^3*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d
*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+39*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^3*c^3*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d
/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(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))*2^(1/2)*x^2*a*b^4*c^4*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d
)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-9*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*b^4*c^4*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(
1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(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^4*d^4*((-d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+6*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*b^4*c^4*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(
-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^
(1/2)+9*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*b^3*c^4*((-d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^
(1/2)*(a*b)^(1/2)+9*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*b^3*c^4*((-d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-9*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^5*c^4*d*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)+9*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^5*c^4*d*((-d*x+(
c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2)+39*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^3*c^4*d*((-d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2)-39*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^3*c^4*d*((-d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-
20*x^5*a^3*b*d^5*(a*b)^(1/2)-9*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*b^4*c^4*((-
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)*(c*d)^(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^4*c*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(
a*b)^(1/2)-6*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2
)*a*b^3*c^4*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*
x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+39*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*a*b^4*c^3*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c
*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-39*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*a*b^4*c^3*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)
^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-39*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^3*c^3*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^
(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-30*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^4*c^4*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2)
)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-9*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*b^5*c^5*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-9*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*b^4*c^
5*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)+9*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*b^4*c^5*((-d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2)-39*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^2*c^3*d*((-d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(c*d)^(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^2*b^2*c^3*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/
(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(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^3
*b*d^4*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-6*EllipticF(((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2),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)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(c*d)^(1/2)*(a*b)^(1/2)+9*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)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*
d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-56*x^5*a*b^3*c^2*d^3*(a*b)^(1/2)-60*x^3*
a^3*b*c*d^4*(a*b)^(1/2)-36*x^3*a^2*b^2*c^2*d^3*(a*b)^(1/2)+52*x^3*a*b^3*c^3*d^2*
(a*b)^(1/2)+104*x*a^3*b*c^2*d^3*(a*b)^(1/2)-76*x*a^2*b^2*c^3*d^2*(a*b)^(1/2)+12*
x*a*b^3*c^4*d*(a*b)^(1/2)+9*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*b^5*c^5*((-d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*
d)^(1/2))^(1/2)+88*x^5*a^2*b^2*c*d^4*(a*b)^(1/2)-39*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*a*b^3*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2)
)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+44*Ellipti
cF(((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^2*c*d^3*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*(c*d)^(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*a*b^3*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c
*d)^(1/2)*(a*b)^(1/2)+39*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^2*d^2*(
(-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+30*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^3*d*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2
)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+39*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^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(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))*2^(1/2)*x^2*a*b^3*c^3*d*((-d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
)*(c*d)^(1/2)*(a*b)^(1/2)-34*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*b*c*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(
c*d)^(1/2))^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-
12*x^5*b^4*c^3*d^2*(a*b)^(1/2)+24*x^3*b^4*c^4*d*(a*b)^(1/2)-28*x*a^4*c*d^4*(a*b)
^(1/2))/(e*x)^(1/2)/(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/a/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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