3.108 \(\int \frac{(1-x^3)^{2/3}}{a+b x} \, dx\)

Optimal. Leaf size=384 \[ -\frac{x^2 \left (a^3+b^3\right ) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-\frac{b^3 x^3}{a^3}\right )}{2 a^2 b^2}-\frac{a^2 \log \left (\sqrt [3]{1-x^3}+x\right )}{2 b^3}-\frac{\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac{\left (a^3+b^3\right )^{2/3} \log \left (-\frac{x \sqrt [3]{a^3+b^3}}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac{\left (a^3+b^3\right )^{2/3} \log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 b^3}+\frac{a^2 \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3} b^3}-\frac{\left (a^3+b^3\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 x \sqrt [3]{a^3+b^3}}{a \sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3} b^3}+\frac{\left (a^3+b^3\right )^{2/3} \tan ^{-1}\left (\frac{\frac{2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^3}+\frac{a x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )}{2 b^2}+\frac{\left (1-x^3\right )^{2/3}}{2 b} \]

[Out]

(1 - x^3)^(2/3)/(2*b) - ((a^3 + b^3)*x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -((b^3*x^3)/a^3)])/(2*a^2*b^2) + (a^2
*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^3) - ((a^3 + b^3)^(2/3)*ArcTan[(1 - (2*(a^3 + b^3)^(1
/3)*x)/(a*(1 - x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*b^3) + ((a^3 + b^3)^(2/3)*ArcTan[(1 + (2*b*(1 - x^3)^(1/3))/(a^
3 + b^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^3) + (a*x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/(2*b^2) - ((a^3 + b^3)
^(2/3)*Log[a^3 + b^3*x^3])/(3*b^3) + ((a^3 + b^3)^(2/3)*Log[-(((a^3 + b^3)^(1/3)*x)/a) - (1 - x^3)^(1/3)])/(2*
b^3) - (a^2*Log[x + (1 - x^3)^(1/3)])/(2*b^3) + ((a^3 + b^3)^(2/3)*Log[(a^3 + b^3)^(1/3) - b*(1 - x^3)^(1/3)])
/(2*b^3)

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Rubi [F]  time = 0.0682516, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (1-x^3\right )^{2/3}}{a+b x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - x^3)^(2/3)/(a + b*x),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (1-x^3\right )^{2/3}}{a+b x} \, dx &=\int \frac{\left (1-x^3\right )^{2/3}}{a+b x} \, dx\\ \end{align*}

Mathematica [F]  time = 0.34462, size = 0, normalized size = 0. \[ \int \frac{\left (1-x^3\right )^{2/3}}{a+b x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a} \left ( -{x}^{3}+1 \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3+1)^(2/3)/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}}}{a + b x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)/(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3+1)^(2/3)/(b*x+a),x, algorithm="giac")

[Out]

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