3.890 \(\int \frac{(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=414 \[ -\frac{c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{d^{3/4} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{e (e x)^{3/2}}{\sqrt{c-d x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 1.99355, antiderivative size = 414, 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 \[ -\frac{c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{d^{3/4} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{e (e x)^{3/2}}{\sqrt{c-d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 0.523401, size = 327, normalized size = 0.79 \[ \frac{e (e x)^{3/2} \left (\frac{49 a^2 c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{11 a b c x^2 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+7\right )}{7 \sqrt{c-d x^2} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

(e*(e*x)^(3/2)*(7 + (49*a^2*c*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a])/
((-a + b*x^2)*(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*a*b*c*x^2*AppellF1[7/4, 1/2, 1, 11/4, (d
*x^2)/c, (b*x^2)/a])/((a - b*x^2)*(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*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])))))/(7*(-(b*c) + a*d)*Sq
rt[c - d*x^2])

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Maple [B]  time = 0.038, size = 839, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)

[Out]

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)*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*c*d-((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*d+((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*c*d+((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)*a*d+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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*
d-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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
),1/2*2^(1/2))*b^2*c^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)*EllipticF(((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*d+((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)*Elliptic
F(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2+2*a*b*d^2*x^2-2*b^2
*c*d*x^2)*(-d*x^2+c)^(1/2)*e^2*(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)/(a*d-b*c)/(d*x^2-c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="maxima")

[Out]

-integrate((e*x)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), 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)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),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)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="giac")

[Out]

integrate(-(e*x)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), x)