3.19 \(\int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx\)

Optimal. Leaf size=186 \[ -\frac{\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d} \]

[Out]

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

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Rubi [A]  time = 0.203521, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2149, 239, 2151} \[ -\frac{\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2149

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[1/(2*c), Int[1/(a + b*x^3)^(1/3), x
], x] + Dist[1/(2*c), Int[(c - d*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[2*b
*c^3 - a*d^3, 0]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 2151

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*f*ArcTan
[(1 + (2*Rt[b, 3]*(2*c + d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Rt[b, 3]*d), x] + (Simp[(f*Log[c + d*x])/(Rt[
b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx &=\frac{\int \frac{1}{\sqrt [3]{2 c^3+d^3 x^3}} \, dx}{2 c}+\frac{\int \frac{c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx}{2 c}\\ &=\frac{\tan ^{-1}\left (\frac{1+\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d}-\frac{\log \left (-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}\\ \end{align*}

Mathematica [F]  time = 0.0633519, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}+2\,{c}^{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)^(1/3),x)

[Out]

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

[Out]

integrate(1/((d^3*x^3 + 2*c^3)^(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(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),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 ) \sqrt [3]{2 c^{3} + d^{3} x^{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)**(1/3),x)

[Out]

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

[Out]

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