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

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

[Out]

-1/2*ln(d*x+c)/c^2/d-1/4*ln(d*x-(d^3*x^3+2*c^3)^(1/3))/c^2/d+3/4*ln(d*(d*x+2*c)-d*(d^3*x^3+2*c^3)^(1/3))/c^2/d
-1/6*arctan(1/3*(1+2*d*x/(d^3*x^3+2*c^3)^(1/3))*3^(1/2))/c^2/d*3^(1/2)+1/2*arctan(1/3*(1+2*(d*x+2*c)/(d^3*x^3+
2*c^3)^(1/3))*3^(1/2))*3^(1/2)/c^2/d

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Rubi [F]  time = 0.11, 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 = {} \[ \int \frac {1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]

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*}

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Mathematica [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]

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

[Out]

Timed out

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

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)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {2}{3}}}\, dx \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (2\,c^3+d^3\,x^3\right )}^{2/3}\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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