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

Optimal. Leaf size=332 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{c \left (a+b x^3\right )^{2/3}}-\frac{d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{2/3}}+\frac{d \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{2/3}}+\frac{d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \left (b c^3-a d^3\right )^{2/3}}-\frac{d \tan ^{-1}\left (\frac{1-\frac{2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt{3}}\right )}{\sqrt{3} \left (b c^3-a d^3\right )^{2/3}} \]

[Out]

(x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c*(a + b*x^3)^(2/3)) + (
d*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*(b*c^3 - a*d^3)^(2/3)) - (
d*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*(b*c^3 - a*d^3)^(2/3)) - (d*Lo
g[c^3 + d^3*x^3])/(3*(b*c^3 - a*d^3)^(2/3)) + (d*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(2*(b*c
^3 - a*d^3)^(2/3)) + (d*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)])/(2*(b*c^3 - a*d^3)^(2/3))

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Rubi [F]  time = 0.0913542, 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 = {} \[ \int \frac{1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.041249, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\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)/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\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)/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

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