Optimal. Leaf size=320 \[ \frac {\left (a+b x^3\right )^{2/3} \left (-44 a^2 d^2+11 a b c d+6 b^2 c^2\right )}{220 a^2 c^3 x^5}-\frac {\left (a+b x^3\right )^{2/3} \left (-220 a^3 d^3+88 a^2 b c d^2+33 a b^2 c^2 d+18 b^3 c^3\right )}{440 a^3 c^4 x^2}-\frac {d^3 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{14/3}}+\frac {d^3 (b c-a d)^{2/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{14/3}}-\frac {d^3 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{14/3}}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{88 a c^2 x^8}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}} \]
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Rubi [C] time = 2.63, antiderivative size = 819, normalized size of antiderivative = 2.56, 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} \begin {gather*} -\frac {-81 b c d^3 x^{12}-162 a d^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{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 {1}{3},1;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-81 a c d^3 x^9+54 b c^2 d^2 x^9+108 a c d^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-108 b c^2 d^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+54 a c^2 d^2 x^6-45 b c^3 d x^6-90 a c^2 d^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+90 b c^3 d \, _2F_1\left (\frac {1}{3},1;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+40 b c^4 x^3-45 a c^3 d x^3-80 b c^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+80 a c^3 d \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+198 b c^4 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-198 a c^3 d \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-27 (b c-a d) \left (5 c-6 d x^3\right ) \left (d x^3+c\right )^2 \, _3F_2\left (\frac {1}{3},2,2;1,\frac {4}{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 {1}{3},2,2,2;1,1,\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+40 a c^4}{440 c^5 x^{11} \sqrt [3]{b x^3+a}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {40 a c^4+40 b c^4 x^3-45 a c^3 d x^3-45 b c^3 d x^6+54 a c^2 d^2 x^6+54 b c^2 d^2 x^9-81 a c d^3 x^9-81 b c d^3 x^{12}-80 b c^4 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+80 a c^3 d x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+90 b c^3 d x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-90 a c^2 d^2 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-108 b c^2 d^2 x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+108 a c d^3 x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{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 {1}{3},1;\frac {4}{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 {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+198 b c^4 x^3 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-198 a c^3 d x^3 \, _2F_1\left (\frac {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-27 (b c-a d) x^3 \left (5 c-6 d x^3\right ) \left (c+d x^3\right )^2 \, _3F_2\left (\frac {1}{3},2,2;1,\frac {4}{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 {1}{3},2,2,2;1,1,\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{440 c^5 x^{11} \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 4.99, size = 819, normalized size = 2.56 \begin {gather*} -\frac {-81 b c d^3 x^{12}-162 a d^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{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 {1}{3},1;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-81 a c d^3 x^9+54 b c^2 d^2 x^9+108 a c d^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-108 b c^2 d^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+54 a c^2 d^2 x^6-45 b c^3 d x^6-90 a c^2 d^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+90 b c^3 d \, _2F_1\left (\frac {1}{3},1;\frac {4}{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 {1}{3},2;\frac {4}{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 {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+40 b c^4 x^3-45 a c^3 d x^3-80 b c^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+80 a c^3 d \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+198 b c^4 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-198 a c^3 d \, _2F_1\left (\frac {1}{3},2;\frac {4}{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 )^2 \left (6 d x^3-5 c\right ) \, _3F_2\left (\frac {1}{3},2,2;1,\frac {4}{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 {1}{3},2,2,2;1,1,\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+40 a c^4}{440 c^5 x^{11} \sqrt [3]{b x^3+a}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 3.35, size = 501, normalized size = 1.57 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-40 a^3 c^3+55 a^3 c^2 d x^3-88 a^3 c d^2 x^6+220 a^3 d^3 x^9-10 a^2 b c^3 x^3+22 a^2 b c^2 d x^6-88 a^2 b c d^2 x^9+12 a b^2 c^3 x^6-33 a b^2 c^2 d x^9-18 b^3 c^3 x^9\right )}{440 a^3 c^4 x^{11}}-\frac {i \left (\sqrt {3} d^3 (b c-a d)^{2/3}-i d^3 (b c-a d)^{2/3}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 c^{14/3}}+\frac {\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} d^3 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{14/3}}+\frac {\left (d^3 (b c-a d)^{2/3}+i \sqrt {3} d^3 (b c-a d)^{2/3}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 c^{14/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right ) x^{12}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^{12}\,\left (d\,x^3+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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