3.923 \(\int \frac{1}{\sqrt{e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=426 \[ \frac{3 b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{3 b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{3/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b \sqrt{e x}}{2 a e \left (a-b x^2\right ) \sqrt{c-d x^2} (b c-a d)}+\frac{d \sqrt{e x} (2 a d+b c)}{2 a c e \sqrt{c-d x^2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 1.91928, antiderivative size = 426, 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 \[ \frac{3 b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{3 b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{3/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b \sqrt{e x}}{2 a e \left (a-b x^2\right ) \sqrt{c-d x^2} (b c-a d)}+\frac{d \sqrt{e x} (2 a d+b c)}{2 a c e \sqrt{c-d x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 1.32074, size = 472, normalized size = 1.11 \[ \frac{x \left (\frac{25 \left (2 a^2 d^2-8 a b c d+3 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{9 a c \left (10 a^2 d^2-12 a b d^2 x^2+b^2 c \left (5 c-6 d x^2\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-10 x^2 \left (-2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (d x^2-c\right )\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{a c \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}\right )}{10 \sqrt{e x} \left (a-b x^2\right ) \sqrt{c-d x^2} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]

[Out]

(x*((25*(3*b^2*c^2 - 8*a*b*c*d + 2*a^2*d^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*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*(10*a^2*d^2 - 12*a*b*d^2*x^2 + b^2*c*(5*
c - 6*d*x^2))*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] - 10*x^2*(-2*a^2*
d^2 + 2*a*b*d^2*x^2 + b^2*c*(-c + d*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]))/(a*
c*(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*AppellF
1[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])))))/(10*(b*c - a*d)^2*Sqrt[e*x]*(a - b*x^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.056, size = 2554, normalized size = 6. \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

1/8*b*d*(-2*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^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)+2*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^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)+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*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)-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*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)-8*x*a^3*d^3*(a*b)^(1/2)+8*x*a^2*b*c*d^2*(a*b)^(1/2)-4*x*a*b^2*c^2*
d*(a*b)^(1/2)-2*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*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)-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*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)
-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*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)+3*(c*d)^(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)*Ellipti
cPi(((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*(c*d)^(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)*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^2-3*(c*d)^(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)*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*(c*d)^(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)*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^2-3*((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^4*c^3+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^4*c^3+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^3*c^3-3*((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^3*c^3+8*x^3*a^2*b*d^3*(a*b)^(1/2)-4*x^3*b^3*c^2*d*(a*b)^(1/2)
+4*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*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)+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^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)+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^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)+2*El
lipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),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)+4*x*b^3*c^3*(a*b)^(1/2)-4*x^3*a*b^2*c
*d^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^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)-4*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),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)-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^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))*(-d*x^2+c)^(1/2)/c/a/((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)^
2/(d*x^2-c)/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*sqrt(e*x)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*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)^(3/2)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*sqrt(e*x)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*sqrt(e*x)), x)