Optimal. Leaf size=318 \[ -\frac{\sqrt [3]{a+b x^3} \left (35 a^2 b c d^2-140 a^3 d^3+15 a b^2 c^2 d+9 b^3 c^3\right )}{140 a^3 c^4 x}+\frac{\sqrt [3]{a+b x^3} \left (-35 a^2 d^2+5 a b c d+3 b^2 c^2\right )}{140 a^2 c^3 x^4}-\frac{d^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^{13/3}}+\frac{d^3 \sqrt [3]{b c-a d} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{13/3}}+\frac{d^3 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{13/3}}-\frac{\sqrt [3]{a+b x^3} (b c-10 a d)}{70 a c^2 x^7}-\frac{\sqrt [3]{a+b x^3}}{10 c x^{10}} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 2.64184, antiderivative size = 905, normalized size of antiderivative = 2.85, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{-324 b c d^3 x^{12}-162 a d^4 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+162 b c d^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-297 a d^4 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+297 b c d^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-324 a c d^3 x^9+108 b c^2 d^2 x^9+54 a c d^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-54 b c^2 d^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-81 a c d^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+81 b c^2 d^2 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+108 a c^2 d^2 x^6-72 b c^3 d x^6-36 a c^2 d^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+36 b c^3 d \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+99 a c^2 d^2 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-99 b c^3 d \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+56 b c^4 x^3-72 a c^3 d x^3-28 b c^4 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+28 a c^3 d \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+117 b c^4 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-117 a c^3 d \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-54 (b c-a d) \left (2 c-3 d x^3\right ) \left (d x^3+c\right )^2 \, _3F_2\left (\frac{2}{3},2,2;1,\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^3 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+56 a c^4}{560 c^5 x^{10} \left (b x^3+a\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 (c+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 (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{56 a c^4+56 b c^4 x^3-72 a c^3 d x^3-72 b c^3 d x^6+108 a c^2 d^2 x^6+108 b c^2 d^2 x^9-324 a c d^3 x^9-324 b c d^3 x^{12}-28 b c^4 x^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+28 a c^3 d x^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+36 b c^3 d x^6 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-36 a c^2 d^2 x^6 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-54 b c^2 d^2 x^9 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+54 a c d^3 x^9 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+162 b c d^3 x^{12} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-162 a d^4 x^{12} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+117 b c^4 x^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-117 a c^3 d x^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-99 b c^3 d x^6 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+99 a c^2 d^2 x^6 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+81 b c^2 d^2 x^9 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-81 a c d^3 x^9 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+297 b c d^3 x^{12} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-297 a d^4 x^{12} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-54 (b c-a d) x^3 \left (2 c-3 d x^3\right ) \left (c+d x^3\right )^2 \, _3F_2\left (\frac{2}{3},2,2;1,\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+27 (b c-a d) x^3 \left (c+d x^3\right )^3 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{560 c^5 x^{10} \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 3.79019, size = 905, normalized size = 2.85 \[ -\frac{-324 b c d^3 x^{12}-162 a d^4 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+162 b c d^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-297 a d^4 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+297 b c d^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-324 a c d^3 x^9+108 b c^2 d^2 x^9+54 a c d^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-54 b c^2 d^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-81 a c d^3 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+81 b c^2 d^2 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+108 a c^2 d^2 x^6-72 b c^3 d x^6-36 a c^2 d^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+36 b c^3 d \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+99 a c^2 d^2 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-99 b c^3 d \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+56 b c^4 x^3-72 a c^3 d x^3-28 b c^4 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+28 a c^3 d \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+117 b c^4 \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-117 a c^3 d \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+54 (b c-a d) \left (d x^3+c\right )^2 \left (3 d x^3-2 c\right ) \text{HypergeometricPFQ}\left (\left \{\frac{2}{3},2,2\right \},\left \{1,\frac{5}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{2}{3},2,2,2\right \},\left \{1,1,\frac{5}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+56 a c^4}{560 c^5 x^{10} \left (b x^3+a\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 ( d{x}^{3}+c \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 (d x^{3} + c\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 (d x^{3} + c\right )} x^{11}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]