3.27 \(\int \frac{\sqrt [3]{a+b x^3}}{c+d x} \, dx\)

Optimal. Leaf size=435 \[ \frac{x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{c \sqrt [3]{\frac{b x^3}{a}+1}}+\frac{\sqrt [3]{b c^3-a d^3} \log \left (c^3+d^3 x^3\right )}{3 d^2}-\frac{\sqrt [3]{b c^3-a d^3} \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{2 d^2}-\frac{\sqrt [3]{b c^3-a d^3} \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 d^2}-\frac{\sqrt [3]{b c^3-a d^3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{\sqrt [3]{b c^3-a d^3} \tan ^{-1}\left (\frac{1-\frac{2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{\sqrt [3]{b} c \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d^2}+\frac{\sqrt [3]{b} c \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{\sqrt [3]{a+b x^3}}{d} \]

[Out]

(a + b*x^3)^(1/3)/d + (x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c*(1
+ (b*x^3)/a)^(1/3)) + (b^(1/3)*c*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) - ((b*c^
3 - a*d^3)^(1/3)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*d^2) + ((b*
c^3 - a*d^3)^(1/3)*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + ((b*c^
3 - a*d^3)^(1/3)*Log[c^3 + d^3*x^3])/(3*d^2) + (b^(1/3)*c*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/(2*d^2) - ((b*c^
3 - a*d^3)^(1/3)*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(2*d^2) - ((b*c^3 - a*d^3)^(1/3)*Log[(b
*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)])/(2*d^2)

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Rubi [F]  time = 0.0815726, 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 = {} \[ \int \frac{\sqrt [3]{a+b x^3}}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*x^3)^(1/3)/(c + d*x),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{c+d x} \, dx &=\int \frac{\sqrt [3]{a+b x^3}}{c+d x} \, dx\\ \end{align*}

Mathematica [F]  time = 0.332267, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{a+b x^3}}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*x^3)^(1/3)/(c + d*x),x]

[Out]

Integrate[(a + b*x^3)^(1/3)/(c + d*x), x]

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c}\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/(d*x+c),x)

[Out]

int((b*x^3+a)^(1/3)/(d*x+c),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x^{3}}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/(d*x+c),x)

[Out]

Integral((a + b*x**3)**(1/3)/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)/(d*x + c), x)