3.2694 \(\int x^m \left (a+b x^{-\frac{3}{2} (1+m)}\right )^{2/3} \, dx\)

Optimal. Leaf size=139 \[ \frac{b^{2/3} \log \left (\sqrt [3]{b} x^{\frac{1}{2} (-m-1)}-\sqrt [3]{a+b x^{-\frac{3}{2} (m+1)}}\right )}{m+1}-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{\frac{1}{2} (-m-1)}}{\sqrt [3]{a+b x^{-\frac{3}{2} (m+1)}}}+1}{\sqrt{3}}\right )}{\sqrt{3} (m+1)}+\frac{x^{m+1} \left (a+b x^{-\frac{3}{2} (m+1)}\right )^{2/3}}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.279096, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{b^{2/3} \log \left (\sqrt [3]{b} x^{\frac{1}{2} (-m-1)}-\sqrt [3]{a+b x^{-\frac{3}{2} (m+1)}}\right )}{m+1}-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{\frac{1}{2} (-m-1)}}{\sqrt [3]{a+b x^{-\frac{3}{2} (m+1)}}}+1}{\sqrt{3}}\right )}{\sqrt{3} (m+1)}+\frac{x^{m+1} \left (a+b x^{-\frac{3}{2} (m+1)}\right )^{2/3}}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.7437, size = 107, normalized size = 0.77 \[ \frac{x^{m + 1} \left (a + b x^{- \frac{3 m}{2} - \frac{3}{2}}\right )^{\frac{2}{3}}}{m + 1} - \frac{2 b x^{- \frac{m}{2} - \frac{1}{2}} \left (a + b x^{- \frac{3 m}{2} - \frac{3}{2}}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{- \frac{3 m}{2} - \frac{3}{2}}}{a}} \right )}}{a \left (1 + \frac{b x^{- \frac{3 m}{2} - \frac{3}{2}}}{a}\right )^{\frac{2}{3}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.184405, size = 73, normalized size = 0.53 \[ \frac{x^{m+1} \left (a+b x^{-\frac{3}{2} (m+1)}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b x^{-\frac{3}{2} (m+1)}}{a}\right )}{(m+1) \left (\frac{b x^{-\frac{3}{2} (m+1)}}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+{b \left ({x}^{{\frac{3}{2}}+{\frac{3\,m}{2}}} \right ) ^{-1}} \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^(-3/2*m - 3/2) + a)^(2/3)*x^m, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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