3.2271 \(\int \frac{x}{\left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=42 \[ \frac{x^2 \sqrt [3]{a+b x^{3/2}} \, _2F_1\left (1,\frac{5}{3};\frac{7}{3};-\frac{b x^{3/2}}{a}\right )}{2 a} \]

[Out]

(x^2*(a + b*x^(3/2))^(1/3)*Hypergeometric2F1[1, 5/3, 7/3, -((b*x^(3/2))/a)])/(2*
a)

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Rubi [A]  time = 0.0950916, antiderivative size = 57, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x^2 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^{3/2}}{a}\right )}{2 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^(3/2))^(2/3),x]

[Out]

(x^2*(1 + (b*x^(3/2))/a)^(2/3)*Hypergeometric2F1[2/3, 4/3, 7/3, -((b*x^(3/2))/a)
])/(2*(a + b*x^(3/2))^(2/3))

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Rubi in Sympy [A]  time = 9.06406, size = 48, normalized size = 1.14 \[ \frac{x^{2} \sqrt [3]{a + b x^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{- \frac{b x^{\frac{3}{2}}}{a}} \right )}}{2 a \sqrt [3]{1 + \frac{b x^{\frac{3}{2}}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(a + b*x**(3/2))**(1/3)*hyper((2/3, 4/3), (7/3,), -b*x**(3/2)/a)/(2*a*(1 +
b*x**(3/2)/a)**(1/3))

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Mathematica [A]  time = 0.0510622, size = 71, normalized size = 1.69 \[ \frac{\sqrt{x} \left (-a \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^{3/2}}{a}\right )+a+b x^{3/2}\right )}{b \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{x \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(3/2))^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^(3/2) + a)^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/(b*x^(3/2) + a)^(2/3), x)

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Sympy [A]  time = 2.48303, size = 41, normalized size = 0.98 \[ \frac{2 x^{2} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*gamma(4/3)*hyper((2/3, 4/3), (7/3,), b*x**(3/2)*exp_polar(I*pi)/a)/(3*a**
(2/3)*gamma(7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^(3/2) + a)^(2/3), x)