3.2276 \(\int \frac{1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=148 \[ \frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}}-\frac{10 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3}}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac{\sqrt [3]{a+b x^{3/2}}}{3 a x^3} \]

[Out]

-(a + b*x^(3/2))^(1/3)/(3*a*x^3) + (5*b*(a + b*x^(3/2))^(1/3))/(9*a^2*x^(3/2)) -
(10*b^2*ArcTan[(a^(1/3) + 2*(a + b*x^(3/2))^(1/3))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[
3]*a^(8/3)) - (5*b^2*Log[x])/(18*a^(8/3)) + (5*b^2*Log[a^(1/3) - (a + b*x^(3/2))
^(1/3)])/(9*a^(8/3))

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Rubi [A]  time = 0.209792, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}}-\frac{10 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3}}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac{\sqrt [3]{a+b x^{3/2}}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^(3/2))^(1/3)/(3*a*x^3) + (5*b*(a + b*x^(3/2))^(1/3))/(9*a^2*x^(3/2)) -
(10*b^2*ArcTan[(a^(1/3) + 2*(a + b*x^(3/2))^(1/3))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[
3]*a^(8/3)) - (5*b^2*Log[x])/(18*a^(8/3)) + (5*b^2*Log[a^(1/3) - (a + b*x^(3/2))
^(1/3)])/(9*a^(8/3))

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Rubi in Sympy [A]  time = 16.0176, size = 143, normalized size = 0.97 \[ - \frac{\sqrt [3]{a + b x^{\frac{3}{2}}}}{3 a x^{3}} + \frac{5 b \sqrt [3]{a + b x^{\frac{3}{2}}}}{9 a^{2} x^{\frac{3}{2}}} - \frac{5 b^{2} \log{\left (x^{\frac{3}{2}} \right )}}{27 a^{\frac{8}{3}}} + \frac{5 b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{\frac{3}{2}}} \right )}}{9 a^{\frac{8}{3}}} - \frac{10 \sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{\frac{3}{2}}}}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**(3/2))**(1/3)/(3*a*x**3) + 5*b*(a + b*x**(3/2))**(1/3)/(9*a**2*x**(3/
2)) - 5*b**2*log(x**(3/2))/(27*a**(8/3)) + 5*b**2*log(a**(1/3) - (a + b*x**(3/2)
)**(1/3))/(9*a**(8/3)) - 10*sqrt(3)*b**2*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**
(3/2))**(1/3)/3)/a**(1/3))/(27*a**(8/3))

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Mathematica [C]  time = 0.0589437, size = 91, normalized size = 0.61 \[ \frac{-3 a^2-5 b^2 x^3 \left (\frac{a}{b x^{3/2}}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^{3/2}}\right )+2 a b x^{3/2}+5 b^2 x^3}{9 a^2 x^3 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^2 + 2*a*b*x^(3/2) + 5*b^2*x^3 - 5*b^2*(1 + a/(b*x^(3/2)))^(2/3)*x^3*Hyperg
eometric2F1[2/3, 2/3, 5/3, -(a/(b*x^(3/2)))])/(9*a^2*x^3*(a + b*x^(3/2))^(2/3))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^4/(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(1/((b*x^(3/2) + a)^(2/3)*x^4),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(1/((b*x^(3/2) + a)^(2/3)*x^4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 109.3, size = 42, normalized size = 0.28 \[ - \frac{2 \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{\frac{3}{2}}}} \right )}}{3 b^{\frac{2}{3}} x^{4} \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*gamma(8/3)*hyper((2/3, 8/3), (11/3,), a*exp_polar(I*pi)/(b*x**(3/2)))/(3*b**(
2/3)*x**4*gamma(11/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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