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

Optimal. Leaf size=107 \[ -\frac{\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a}}-\frac{b \log (x)}{3 \sqrt [3]{a}} \]

[Out]

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

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Rubi [A]  time = 0.147619, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a}}-\frac{b \log (x)}{3 \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.97169, size = 100, normalized size = 0.93 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{3 x^{3}} - \frac{b \log{\left (x^{3} \right )}}{9 \sqrt [3]{a}} + \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{3 \sqrt [3]{a}} + \frac{2 \sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 \sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0421171, size = 67, normalized size = 0.63 \[ \frac{-2 b x^3 \sqrt [3]{\frac{a}{b x^3}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^3}\right )-a-b x^3}{3 x^3 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271424, size = 173, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) - 2 \, \sqrt{3} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right ) - 6 \, b x^{3} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}}\right )}}{27 \, a^{\frac{1}{3}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*sqrt(3)*(sqrt(3)*b*x^3*log((b*x^3 + a)^(2/3)*a^(1/3) + (b*x^3 + a)^(1/3)*a
^(2/3) + a) - 2*sqrt(3)*b*x^3*log((b*x^3 + a)^(1/3)*a^(2/3) - a) - 6*b*x^3*arcta
n(1/3*(2*sqrt(3)*(b*x^3 + a)^(1/3)*a^(2/3) + sqrt(3)*a)/a) + 3*sqrt(3)*(b*x^3 +
a)^(2/3)*a^(1/3))/(a^(1/3)*x^3)

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Sympy [A]  time = 4.83783, size = 39, normalized size = 0.36 \[ - \frac{b^{\frac{2}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 x \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(2/3)*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), a*exp_polar(I*pi)/(b*x**3))/(3*x
*gamma(4/3))

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GIAC/XCAS [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 + a)^(2/3)/x^4,x, algorithm="giac")

[Out]

Timed out