3.851 \(\int \frac{x^4}{\left (a-b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{24 a^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a-b x^2}}+\frac{12 x \left (a-b x^2\right )^{3/4}}{5 b^2}+\frac{2 x^3}{b \sqrt [4]{a-b x^2}} \]

[Out]

(2*x^3)/(b*(a - b*x^2)^(1/4)) + (12*x*(a - b*x^2)^(3/4))/(5*b^2) - (24*a^(3/2)*(
1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*b^(5/2)*(a
- b*x^2)^(1/4))

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Rubi [A]  time = 0.104501, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{24 a^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a-b x^2}}+\frac{12 x \left (a-b x^2\right )^{3/4}}{5 b^2}+\frac{2 x^3}{b \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a - b*x^2)^(5/4),x]

[Out]

(2*x^3)/(b*(a - b*x^2)^(1/4)) + (12*x*(a - b*x^2)^(3/4))/(5*b^2) - (24*a^(3/2)*(
1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*b^(5/2)*(a
- b*x^2)^(1/4))

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Rubi in Sympy [A]  time = 15.0392, size = 87, normalized size = 0.86 \[ - \frac{24 a^{\frac{3}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{5 b^{\frac{5}{2}} \sqrt [4]{a - b x^{2}}} + \frac{2 x^{3}}{b \sqrt [4]{a - b x^{2}}} + \frac{12 x \left (a - b x^{2}\right )^{\frac{3}{4}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-b*x**2+a)**(5/4),x)

[Out]

-24*a**(3/2)*(1 - b*x**2/a)**(1/4)*elliptic_e(asin(sqrt(b)*x/sqrt(a))/2, 2)/(5*b
**(5/2)*(a - b*x**2)**(1/4)) + 2*x**3/(b*(a - b*x**2)**(1/4)) + 12*x*(a - b*x**2
)**(3/4)/(5*b**2)

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Mathematica [C]  time = 0.0564367, size = 65, normalized size = 0.64 \[ -\frac{2 x \left (6 a \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-6 a+b x^2\right )}{5 b^2 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a - b*x^2)^(5/4),x]

[Out]

(-2*x*(-6*a + b*x^2 + 6*a*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2,
(b*x^2)/a]))/(5*b^2*(a - b*x^2)^(1/4))

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( -b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-b*x^2+a)^(5/4),x)

[Out]

int(x^4/(-b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(-b*x^2 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate(x^4/(-b*x^2 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x^{4}}{{\left (b x^{2} - a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(-b*x^2 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(-x^4/((b*x^2 - a)*(-b*x^2 + a)^(1/4)), x)

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Sympy [A]  time = 2.99925, size = 29, normalized size = 0.29 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-b*x**2+a)**(5/4),x)

[Out]

x**5*hyper((5/4, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/(5*a**(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(-b*x^2 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^4/(-b*x^2 + a)^(5/4), x)