3.786 \(\int \frac{1}{(c x)^{4/3} \left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0669824, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{3 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(4/3)*(a + b*x^2)^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.80874, size = 51, normalized size = 0.91 \[ - \frac{3 \sqrt [3]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c \sqrt [3]{c x} \sqrt [3]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(4/3)/(b*x**2+a)**(2/3),x)

[Out]

-3*(a + b*x**2)**(1/3)*hyper((2/3, -1/6), (5/6,), -b*x**2/a)/(a*c*(c*x)**(1/3)*(
1 + b*x**2/a)**(1/3))

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Mathematica [A]  time = 0.0626722, size = 74, normalized size = 1.32 \[ \frac{3 x \left (b x^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )-5 \left (a+b x^2\right )\right )}{5 a (c x)^{4/3} \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(4/3)*(a + b*x^2)^(2/3)),x]

[Out]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{4}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(4/3)/(b*x^2+a)^(2/3),x)

[Out]

int(1/(c*x)^(4/3)/(b*x^2+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(4/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(4/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{1}{3}} c x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(4/3)),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(2/3)*(c*x)^(1/3)*c*x), x)

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Sympy [A]  time = 26.7313, size = 48, normalized size = 0.86 \[ \frac{\Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{2}{3} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{2}{3}} c^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{5}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x)**(4/3)/(b*x**2+a)**(2/3),x)

[Out]

gamma(-1/6)*hyper((-1/6, 2/3), (5/6,), b*x**2*exp_polar(I*pi)/a)/(2*a**(2/3)*c**
(4/3)*x**(1/3)*gamma(5/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(4/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(4/3)), x)