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

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

[Out]

1/2*(-x^3+1)^(2/3)/b-1/2*(a^3+b^3)*x^2*AppellF1(2/3,1/3,1,5/3,x^3,-b^3*x^3/a^3)/a^2/b^2+1/2*a*x^2*hypergeom([1
/3, 2/3],[5/3],x^3)/b^2-1/3*(a^3+b^3)^(2/3)*ln(b^3*x^3+a^3)/b^3+1/2*(a^3+b^3)^(2/3)*ln(-(a^3+b^3)^(1/3)*x/a-(-
x^3+1)^(1/3))/b^3-1/2*a^2*ln(x+(-x^3+1)^(1/3))/b^3+1/2*(a^3+b^3)^(2/3)*ln((a^3+b^3)^(1/3)-b*(-x^3+1)^(1/3))/b^
3+1/3*a^2*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))/b^3*3^(1/2)-1/3*(a^3+b^3)^(2/3)*arctan(1/3*(1-2*(a^3+b^3)
^(1/3)*x/a/(-x^3+1)^(1/3))*3^(1/2))/b^3*3^(1/2)+1/3*(a^3+b^3)^(2/3)*arctan(1/3*(1+2*b*(-x^3+1)^(1/3)/(a^3+b^3)
^(1/3))*3^(1/2))/b^3*3^(1/2)

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Rubi [F]  time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{b x + a}\,{d x} \]

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)

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{b x +a}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{b x + a}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-x^3\right )}^{2/3}}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^3)^(2/3)/(a + b*x),x)

[Out]

int((1 - x^3)^(2/3)/(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{a + b x}\, dx \]

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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