3.20 \(\int \frac{1}{(c+d x) (2 c^3+d^3 x^3)^{2/3}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{\log \left (d x-\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}-\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c^2 d}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c^2 d}-\frac{\log (c+d x)}{2 c^2 d} \]

[Out]

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

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Rubi [F]  time = 0.105546, 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 = {} \[ \int \frac{1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.0625508, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c} \left ({d}^{3}{x}^{3}+2\,{c}^{3} \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x)

[Out]

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x, algorithm="maxima")

[Out]

integrate(1/((d^3*x^3 + 2*c^3)^(2/3)*(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(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((c + d*x)*(2*c**3 + d**3*x**3)**(2/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x, algorithm="giac")

[Out]

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