3.1.20 \(\int \frac {1}{(c+d x) (2 c^3+d^3 x^3)^{2/3}} \, dx\) [20]

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

[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 [A]
time = 0.05, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2179} \begin {gather*} -\frac {\text {ArcTan}\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} \text {ArcTan}\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}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2179

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(2/3)), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[(-d)*(ArcTan[(
1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(2*Sqrt[3]*q^2*c^2)), x] + (Simp[Sqrt[3]*d*(ArcTan[(1 + 2*q*((2*c + d*
x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2*q^2*c^2)), x] - Simp[d*(Log[c + d*x]/(2*q^2*c^2)), x] - Simp[d*(Log[q*x
- (a + b*x^3)^(1/3)]/(4*q^2*c^2)), x] + Simp[3*d*(Log[q*(2*c + d*x) - d*(a + b*x^3)^(1/3)]/(4*q^2*c^2)), x])]
/; FreeQ[{a, b, c, d}, x] && EqQ[2*b*c^3 - a*d^3, 0]

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 = 7.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \end {gather*}

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 = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c + d x\right ) \left (2 c^{3} + d^{3} x^{3}\right )^{\frac {2}{3}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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