Optimal. Leaf size=492 \[ -\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{10/3} \sqrt [3]{b}}+\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{24\ 2^{2/3} a^{10/3} \sqrt [3]{b}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{12\ 2^{2/3} a^{10/3} \sqrt [3]{b}}+\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{48\ 2^{2/3} a^{10/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3} a^{10/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {9 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{20 a^3 \left (a+b x^3\right )^{2/3}}+\frac {13 x}{40 a^3 \left (a+b x^3\right )^{2/3}}+\frac {x}{10 a^2 \left (a+b x^3\right )^{5/3}} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 0.12, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {430, 429} \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};1,\frac {8}{3};\frac {4}{3};\frac {b x^3}{a},-\frac {b x^3}{a}\right )}{a^3 \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{8/3}} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (a-b x^3\right ) \left (1+\frac {b x^3}{a}\right )^{8/3}} \, dx}{a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};1,\frac {8}{3};\frac {4}{3};\frac {b x^3}{a},-\frac {b x^3}{a}\right )}{a^3 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.18, size = 240, normalized size = 0.49 \[ \frac {x \left (\frac {368 a^3 \left (a+b x^3\right ) F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (b x^3 \left (3 F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )+4 a F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )}+16 a^2-13 b x^3 \left (a+b x^3\right ) \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )+52 a \left (a+b x^3\right )\right )}{160 a^4 \left (a+b x^3\right )^{5/3}} \]
Warning: Unable to verify antiderivative.
[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 \[ \int -\frac {1}{{\left (b x^{3} + a\right )}^{\frac {8}{3}} {\left (b x^{3} - a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{3}+a \right ) \left (b \,x^{3}+a \right )^{\frac {8}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {8}{3}} {\left (b x^{3} - a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{8/3}\,\left (a-b\,x^3\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________