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

Optimal. Leaf size=760 \[ -\frac{d^3 x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{2 c^5 \left (a+b x^3\right )^{2/3}}+\frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{2}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{c^2 \left (a+b x^3\right )^{2/3}}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}+\frac{c^2 d^2 \sqrt [3]{a+b x^3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}-\frac{a d^4 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}+\frac{a d^4 \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}-\frac{d \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}-\frac{b c^2 d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{5/3}}+\frac{d \left (3 b c^3-a d^3\right ) \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}+\frac{b c^2 d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{\left (b c^3-a d^3\right )^{5/3}}+\frac{2 a d^4 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c \left (b c^3-a d^3\right )^{5/3}}+\frac{2 d \left (3 b c^3-a d^3\right ) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c \left (b c^3-a d^3\right )^{5/3}}-\frac{2 b c^2 d \tan ^{-1}\left (\frac{1-\frac{2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt{3}}\right )}{\sqrt{3} \left (b c^3-a d^3\right )^{5/3}} \]

[Out]

(c^2*d^2*(a + b*x^3)^(1/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) + (d^4*x^2*(a + b*x^3)^(1/3))/((b*c^3 - a*d^3)*(
c^3 + d^3*x^3)) + (x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 2/3, 2, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c^2*(a
+ b*x^3)^(2/3)) - (d^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/
(2*c^5*(a + b*x^3)^(2/3)) + (2*a*d^4*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/
(3*Sqrt[3]*c*(b*c^3 - a*d^3)^(5/3)) + (2*d*(3*b*c^3 - a*d^3)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b
*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c*(b*c^3 - a*d^3)^(5/3)) - (2*b*c^2*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)^(5/3)) - (b*c^2*d*Log[c^3 + d^3*x^3])/(3*(b*c^3 - a*d
^3)^(5/3)) - (a*d^4*Log[c^3 + d^3*x^3])/(9*c*(b*c^3 - a*d^3)^(5/3)) - (d*(3*b*c^3 - a*d^3)*Log[c^3 + d^3*x^3])
/(9*c*(b*c^3 - a*d^3)^(5/3)) + (a*d^4*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(3*c*(b*c^3 - a*d^
3)^(5/3)) + (d*(3*b*c^3 - a*d^3)*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(3*c*(b*c^3 - a*d^3)^(5
/3)) + (b*c^2*d*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)])/(b*c^3 - a*d^3)^(5/3)

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Rubi [F]  time = 0.0848816, 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 = {} \[ \int \frac{1}{(c+d x)^2 \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.323316, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^2 \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

[Out]

int(1/(d*x+c)^2/(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 )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*x**3)**(2/3)*(c + d*x)**2), 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 )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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