3.2272 \(\int \frac{1}{\left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{\log \left (\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1\right )}{3 b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.200703, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{\log \left (\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1\right )}{3 b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(3/2))^(-2/3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.6116, size = 133, normalized size = 0.95 \[ - \frac{2 \log{\left (- \frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a + b x^{\frac{3}{2}}}} + 1 \right )}}{3 b^{\frac{2}{3}}} + \frac{\log{\left (\frac{b^{\frac{2}{3}} x}{\left (a + b x^{\frac{3}{2}}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a + b x^{\frac{3}{2}}}} + 1 \right )}}{3 b^{\frac{2}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} \sqrt{x}}{3 \sqrt [3]{a + b x^{\frac{3}{2}}}} + \frac{1}{3}\right ) \right )}}{3 b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0259033, size = 53, normalized size = 0.38 \[ \frac{x \left (\frac{a+b x^{3/2}}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^{3/2}}{a}\right )}{\left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.021, 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(1/(a+b*x^(3/2))^(2/3),x)

[Out]

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

_______________________________________________________________________________________

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((b*x^(3/2) + a)^(-2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [A]  time = 3.9904, size = 39, normalized size = 0.28 \[ \frac{2 x \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*gamma(2/3)*hyper((2/3, 2/3), (5/3,), b*x**(3/2)*exp_polar(I*pi)/a)/(3*a**(2/
3)*gamma(5/3))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^(3/2) + a)^(-2/3), x)