Optimal. Leaf size=237 \[ -\frac{169 b^3 \sqrt [3]{a+b x^3}}{140 a^4 d x}-\frac{37 b^2 \sqrt [3]{a+b x^3}}{140 a^3 d x^4}+\frac{b^{10/3} \log \left (a d-b d x^3\right )}{3\ 2^{2/3} a^4 d}-\frac{b^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^4 d}-\frac{\sqrt [3]{2} b^{10/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a^4 d}-\frac{11 b \sqrt [3]{a+b x^3}}{70 a^2 d x^7}-\frac{\sqrt [3]{a+b x^3}}{10 a d x^{10}} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 29.209, antiderivative size = 423, normalized size of antiderivative = 1.78, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {511, 510} \[ -\frac{-54 b x^3 \left (a-b x^3\right )^2 \left (2 a+3 b x^3\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+27 b x^3 \left (a-b x^3\right )^3 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )-36 a^2 b^2 x^6 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+99 a^2 b^2 x^6 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+90 a^2 b^2 x^6-28 a^3 b x^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+117 a^3 b x^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+64 a^3 b x^3+28 a^4-162 b^4 x^{12} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )-297 b^4 x^{12} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )-54 a b^3 x^9 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+81 a b^3 x^9 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+216 a b^3 x^9+162 b^4 x^{12}}{280 a^4 d x^{10} \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^{11} \left (a d-b d x^3\right )} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{\sqrt [3]{1+\frac{b x^3}{a}}}{x^{11} \left (a d-b d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{28 a^4+64 a^3 b x^3+90 a^2 b^2 x^6+216 a b^3 x^9+162 b^4 x^{12}-28 a^3 b x^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )-36 a^2 b^2 x^6 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )-54 a b^3 x^9 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )-162 b^4 x^{12} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )+117 a^3 b x^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )+99 a^2 b^2 x^6 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )+81 a b^3 x^9 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )-297 b^4 x^{12} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )-54 b x^3 \left (a-b x^3\right )^2 \left (2 a+3 b x^3\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )+27 b x^3 \left (a-b x^3\right )^3 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )}{280 a^4 d x^{10} \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 5.15406, size = 130, normalized size = 0.55 \[ \frac{-59 a^2 b^2 x^6-36 a^3 b x^3-14 a^4+\frac{140 b^4 x^{12} \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right )}{\left (1-\frac{b x^3}{a}\right )^{2/3}}-206 a b^3 x^9-169 b^4 x^{12}}{140 a^4 d x^{10} \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{11} \left ( -bd{x}^{3}+ad \right ) }\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{11}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{11}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]