3.1026 \(\int \frac{x^4}{\left (a+b x^2\right )^{5/6}} \, dx\)

Optimal. Leaf size=300 \[ \frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{40 b^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{27 a x \sqrt [6]{a+b x^2}}{40 b^2}+\frac{3 x^3 \sqrt [6]{a+b x^2}}{10 b} \]

[Out]

(-27*a*x*(a + b*x^2)^(1/6))/(40*b^2) + (3*x^3*(a + b*x^2)^(1/6))/(10*b) + (27*3^
(3/4)*Sqrt[2 - Sqrt[3]]*a^2*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(
1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2)
)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3]
- (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(40*b^3*x*(a/(a + b*x^2))^(1/3)*Sqrt
[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

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Rubi [A]  time = 0.575122, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{40 b^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{27 a x \sqrt [6]{a+b x^2}}{40 b^2}+\frac{3 x^3 \sqrt [6]{a+b x^2}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(5/6),x]

[Out]

(-27*a*x*(a + b*x^2)^(1/6))/(40*b^2) + (3*x^3*(a + b*x^2)^(1/6))/(10*b) + (27*3^
(3/4)*Sqrt[2 - Sqrt[3]]*a^2*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(
1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2)
)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3]
- (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(40*b^3*x*(a/(a + b*x^2))^(1/3)*Sqrt
[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

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Rubi in Sympy [A]  time = 18.1516, size = 275, normalized size = 0.92 \[ \frac{27 \cdot 3^{\frac{3}{4}} a^{2} \sqrt{\frac{\left (- \frac{b x^{2}}{a + b x^{2}} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [6]{a + b x^{2}} \left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{40 b^{3} x \sqrt [3]{\frac{a}{a + b x^{2}}} \sqrt{\frac{\sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}}} - \frac{27 a x \sqrt [6]{a + b x^{2}}}{40 b^{2}} + \frac{3 x^{3} \sqrt [6]{a + b x^{2}}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**(5/6),x)

[Out]

27*3**(3/4)*a**2*sqrt(((-b*x**2/(a + b*x**2) + 1)**(2/3) + (-b*x**2/(a + b*x**2)
+ 1)**(1/3) + 1)/(-(-b*x**2/(a + b*x**2) + 1)**(1/3) - sqrt(3) + 1)**2)*sqrt(-s
qrt(3) + 2)*(a + b*x**2)**(1/6)*(-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1)*ellipti
c_f(asin((-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1 + sqrt(3))/(-(-b*x**2/(a + b*x*
*2) + 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(40*b**3*x*(a/(a + b*x**2))**(1
/3)*sqrt(((-b*x**2/(a + b*x**2) + 1)**(1/3) - 1)/(-(-b*x**2/(a + b*x**2) + 1)**(
1/3) - sqrt(3) + 1)**2)) - 27*a*x*(a + b*x**2)**(1/6)/(40*b**2) + 3*x**3*(a + b*
x**2)**(1/6)/(10*b)

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Mathematica [C]  time = 0.0552159, size = 79, normalized size = 0.26 \[ \frac{3 \left (9 a^2 x \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )-9 a^2 x-5 a b x^3+4 b^2 x^5\right )}{40 b^2 \left (a+b x^2\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^2)^(5/6),x]

[Out]

(3*(-9*a^2*x - 5*a*b*x^3 + 4*b^2*x^5 + 9*a^2*x*(1 + (b*x^2)/a)^(5/6)*Hypergeomet
ric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)]))/(40*b^2*(a + b*x^2)^(5/6))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(5/6),x)

[Out]

int(x^4/(b*x^2+a)^(5/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(5/6),x, algorithm="maxima")

[Out]

integrate(x^4/(b*x^2 + a)^(5/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{5}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(5/6),x, algorithm="fricas")

[Out]

integral(x^4/(b*x^2 + a)^(5/6), x)

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Sympy [A]  time = 2.7677, size = 27, normalized size = 0.09 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**(5/6),x)

[Out]

x**5*hyper((5/6, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(5/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(5/6),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^2 + a)^(5/6), x)