3.888 \(\int \frac{(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=444 \[ -\frac{a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{7/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{7/4} \sqrt{c-d x^2} (b c-a d)}-\frac{c e^3 (e x)^{3/2}}{d \sqrt{c-d x^2} (b c-a d)} \]

[Out]

-((c*e^3*(e*x)^(3/2))/(d*(b*c - a*d)*Sqrt[c - d*x^2])) + (c^(3/4)*(3*b*c - 2*a*d
)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt
[e])], -1])/(b*d^(7/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (c^(3/4)*(3*b*c - 2*a*d)*e
^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]
)], -1])/(b*d^(7/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (a^(3/2)*c^(1/4)*e^(9/2)*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])/(b^(3/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x
^2]) + (a^(3/2)*c^(1/4)*e^(9/2)*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])/(b^(3/2)
*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.20844, antiderivative size = 444, 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 \[ -\frac{a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{7/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{7/4} \sqrt{c-d x^2} (b c-a d)}-\frac{c e^3 (e x)^{3/2}}{d \sqrt{c-d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(9/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

-((c*e^3*(e*x)^(3/2))/(d*(b*c - a*d)*Sqrt[c - d*x^2])) + (c^(3/4)*(3*b*c - 2*a*d
)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt
[e])], -1])/(b*d^(7/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (c^(3/4)*(3*b*c - 2*a*d)*e
^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]
)], -1])/(b*d^(7/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (a^(3/2)*c^(1/4)*e^(9/2)*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])/(b^(3/2)*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x
^2]) + (a^(3/2)*c^(1/4)*e^(9/2)*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])/(b^(3/2)
*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)**(9/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.721983, size = 424, normalized size = 0.95 \[ \frac{c (e x)^{9/2} \left (\frac{49 a^2 c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{11 a \left (7 a c-2 a d x^2-4 b c x^2\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-14 x^2 \left (b x^2-a\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}\right )}{7 d x^3 \sqrt{c-d x^2} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(9/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

(c*(e*x)^(9/2)*((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, 1
1/4, (d*x^2)/c, (b*x^2)/a]))) + (11*a*(7*a*c - 4*b*c*x^2 - 2*a*d*x^2)*AppellF1[7
/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a] - 14*x^2*(-a + b*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]))/((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*d*(-(b*c) + a*d)*x^
3*Sqrt[c - d*x^2])

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Maple [B]  time = 0.066, size = 1041, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(9/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)

[Out]

-1/2*(4*((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^2*b*c*d^2-10*((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^2*c^2*d+6*((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^3*c^3-2*
((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^2*b*c*d^2+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)*EllipticF(((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*c^2*d-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)*
EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^3*c^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)^(1/2)*(c*d)^(1/2)*a^2*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)*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)^(1/2)*(c*d)^(1/2)*a^2
*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)*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*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)*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^2*b*c*d^2-2*x^2*a*b^2*c*d^2+2
*x^2*b^3*c^2*d)*(-d*x^2+c)^(1/2)*e^4*(e*x)^(1/2)/x/d/((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)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\left (e x\right )^{\frac{9}{2}}}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(9/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="maxima")

[Out]

-integrate((e*x)^(9/2)/((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)^(9/2)/((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)**(9/2)/(-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. \[ \int -\frac{\left (e x\right )^{\frac{9}{2}}}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(9/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="giac")

[Out]

integrate(-(e*x)^(9/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), x)