3.2386 \(\int \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \, dx\)

Optimal. Leaf size=95 \[ b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a+\frac{b}{x^{3/2}}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \]

[Out]

(a + b/x^(3/2))^(2/3)*x - (2*b^(2/3)*ArcTan[(1 + (2*b^(1/3))/((a + b/x^(3/2))^(1
/3)*Sqrt[x]))/Sqrt[3]])/Sqrt[3] + b^(2/3)*Log[(a + b/x^(3/2))^(1/3) - b^(1/3)/Sq
rt[x]]

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Rubi [A]  time = 0.190498, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a+\frac{b}{x^{3/2}}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(3/2))^(2/3),x]

[Out]

(a + b/x^(3/2))^(2/3)*x - (2*b^(2/3)*ArcTan[(1 + (2*b^(1/3))/((a + b/x^(3/2))^(1
/3)*Sqrt[x]))/Sqrt[3]])/Sqrt[3] + b^(2/3)*Log[(a + b/x^(3/2))^(1/3) - b^(1/3)/Sq
rt[x]]

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Rubi in Sympy [A]  time = 21.2598, size = 146, normalized size = 1.54 \[ \frac{2 b^{\frac{2}{3}} \log{\left (- \frac{\sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a + \frac{b}{x^{\frac{3}{2}}}}} + 1 \right )}}{3} - \frac{b^{\frac{2}{3}} \log{\left (\frac{b^{\frac{2}{3}}}{x \left (a + \frac{b}{x^{\frac{3}{2}}}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a + \frac{b}{x^{\frac{3}{2}}}}} + 1 \right )}}{3} - \frac{2 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b}}{3 \sqrt{x} \sqrt [3]{a + \frac{b}{x^{\frac{3}{2}}}}} + \frac{1}{3}\right ) \right )}}{3} + x \left (a + \frac{b}{x^{\frac{3}{2}}}\right )^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**(2/3)*log(-b**(1/3)/(sqrt(x)*(a + b/x**(3/2))**(1/3)) + 1)/3 - b**(2/3)*log
(b**(2/3)/(x*(a + b/x**(3/2))**(2/3)) + b**(1/3)/(sqrt(x)*(a + b/x**(3/2))**(1/3
)) + 1)/3 - 2*sqrt(3)*b**(2/3)*atan(sqrt(3)*(2*b**(1/3)/(3*sqrt(x)*(a + b/x**(3/
2))**(1/3)) + 1/3))/3 + x*(a + b/x**(3/2))**(2/3)

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Mathematica [C]  time = 0.0881688, size = 53, normalized size = 0.56 \[ \frac{x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b}{a x^{3/2}}\right )}{\left (\frac{a+\frac{b}{x^{3/2}}}{a}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \[ \int \left ( a+{b{x}^{-{\frac{3}{2}}}} \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(3/2))^(2/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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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((a + b/x^(3/2))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 46.9952, size = 46, normalized size = 0.48 \[ - \frac{2 a^{\frac{2}{3}} x \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{\frac{3}{2}}}} \right )}}{3 \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^(3/2))^(2/3), x)