3.28 \(\int \frac{\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=818 \[ -\frac{d^3 \sqrt [3]{b x^3+a} F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right ) x^4}{2 c^5 \sqrt [3]{\frac{b x^3}{a}+1}}-\frac{d \sqrt [3]{b x^3+a} x^2}{c^3+d^3 x^3}+\frac{\sqrt [3]{b x^3+a} F_1\left (\frac{1}{3};-\frac{1}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right ) x}{c^2 \sqrt [3]{\frac{b x^3}{a}+1}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{b x^3+a}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{2 a d \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c \left (b c^3-a d^3\right )^{2/3}}+\frac{\left (3 b c^3-2 a d^3\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac{b c^2 \tan ^{-1}\left (\frac{1-\frac{2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt{3}}\right )}{\sqrt{3} d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac{a d \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{2/3}}-\frac{\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac{b c^2 \log \left (c^3+d^3 x^3\right )}{6 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{b x^3+a}\right )}{2 d^2}+\frac{a d \log \left (\frac{\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{3 c \left (b c^3-a d^3\right )^{2/3}}+\frac{\left (3 b c^3-2 a d^3\right ) \log \left (\frac{\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{6 c d^2 \left (b c^3-a d^3\right )^{2/3}}+\frac{b c^2 \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{2 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac{c^2 \sqrt [3]{b x^3+a}}{d \left (c^3+d^3 x^3\right )} \]

[Out]

-((c^2*(a + b*x^3)^(1/3))/(d*(c^3 + d^3*x^3))) - (d*x^2*(a + b*x^3)^(1/3))/(c^3 + d^3*x^3) + (x*(a + b*x^3)^(1
/3)*AppellF1[1/3, -1/3, 2, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c^2*(1 + (b*x^3)/a)^(1/3)) - (d^3*x^4*(a + b
*x^3)^(1/3)*AppellF1[4/3, -1/3, 2, 7/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(2*c^5*(1 + (b*x^3)/a)^(1/3)) - (b^(1
/3)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + (2*a*d*ArcTan[(1 + (2*(b*c^3 - a*d^
3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c*(b*c^3 - a*d^3)^(2/3)) + ((3*b*c^3 - 2*a*d^3)*ArcTan
[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c*d^2*(b*c^3 - a*d^3)^(2/3)) - (
b*c^2*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)^(2/3))
- (b*c^2*Log[c^3 + d^3*x^3])/(6*d^2*(b*c^3 - a*d^3)^(2/3)) - (a*d*Log[c^3 + d^3*x^3])/(9*c*(b*c^3 - a*d^3)^(2
/3)) - ((3*b*c^3 - 2*a*d^3)*Log[c^3 + d^3*x^3])/(18*c*d^2*(b*c^3 - a*d^3)^(2/3)) - (b^(1/3)*Log[b^(1/3)*x - (a
+ b*x^3)^(1/3)])/(2*d^2) + (a*d*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(3*c*(b*c^3 - a*d^3)^(2
/3)) + ((3*b*c^3 - 2*a*d^3)*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(6*c*d^2*(b*c^3 - a*d^3)^(2/
3)) + (b*c^2*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)])/(2*d^2*(b*c^3 - a*d^3)^(2/3))

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Rubi [F]  time = 0.0794583, 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 = {} \[ \int \frac{\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*x^3)^(1/3)/(c + d*x)^2,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx &=\int \frac{\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx\\ \end{align*}

Mathematica [F]  time = 0.209719, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2}}\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)^2,x)

[Out]

int((b*x^3+a)^(1/3)/(d*x+c)^2,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}}}{{\left (d x + c\right )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)/(d*x + c)^2, 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)^2,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}}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x**3)**(1/3)/(c + d*x)**2, 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}}}{{\left (d x + c\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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