Optimal. Leaf size=761 \[ -\frac {d x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},2;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^3 \sqrt [3]{a+b x^3}}+\frac {d^4 x^5 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},2;\frac {8}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{5 c^6 \sqrt [3]{a+b x^3}}-\frac {c d^3 x \left (a+b x^3\right )^{2/3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}+\frac {a d^3 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{4/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c \left (b c^3-a d^3\right )^{4/3}}-\frac {a d^3 \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{4/3}}-\frac {\left (3 b c^3-2 a d^3\right ) \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{6 c \left (b c^3-a d^3\right )^{4/3}}+\frac {2 a d^3 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{4/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{4/3}}+\frac {b c^2 \log \left (c^3+d^3 x^3\right )}{6 \left (b c^3-a d^3\right )^{4/3}}-\frac {b c^2 \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{4/3}}-\frac {b c^2 \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{4/3}}+\frac {c^2 d^2 \left (a+b x^3\right )^{2/3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.08, 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 = {} \[ \int \frac {1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx &=\int \frac {1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{1/3}\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{a + b x^{3}} \left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________