Optimal. Leaf size=318 \[ \frac {\sqrt [3]{a+b x^3} \left (-35 a^2 d^2+5 a b c d+3 b^2 c^2\right )}{140 a^2 c^3 x^4}-\frac {\sqrt [3]{a+b x^3} \left (-140 a^3 d^3+35 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{140 a^3 c^4 x}-\frac {d^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^{13/3}}+\frac {d^3 \sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{13/3}}+\frac {d^3 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{13/3}}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{70 a c^2 x^7}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}} \]
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Rubi [C] time = 2.64, 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} \begin {gather*} -\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
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*}
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Mathematica [C] time = 5.24, size = 214, normalized size = 0.67 \begin {gather*} \frac {\frac {\left (a+b x^3\right ) \left (a^3 \left (-14 c^3+20 c^2 d x^3-35 c d^2 x^6+140 d^3 x^9\right )+a^2 b c x^3 \left (-2 c^2+5 c d x^3-35 d^2 x^6\right )+3 a b^2 c^2 x^6 \left (c-5 d x^3\right )-9 b^3 c^3 x^9\right )}{70 a^3 c^4 x^{10}}+\frac {d^3 x^2 \left (\frac {b x^3}{a}+1\right )^{2/3} (a d-b c) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{c^5 \left (\frac {d x^3}{c}+1\right )^{2/3}}}{2 \left (a+b x^3\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 2.99, size = 503, normalized size = 1.58 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-14 a^3 c^3+20 a^3 c^2 d x^3-35 a^3 c d^2 x^6+140 a^3 d^3 x^9-2 a^2 b c^3 x^3+5 a^2 b c^2 d x^6-35 a^2 b c d^2 x^9+3 a b^2 c^3 x^6-15 a b^2 c^2 d x^9-9 b^3 c^3 x^9\right )}{140 a^3 c^4 x^{10}}+\frac {i \left (\sqrt {3} d^3 \sqrt [3]{b c-a d}+i d^3 \sqrt [3]{b c-a d}\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^{13/3}}-\frac {\sqrt {-1-i \sqrt {3}} d^3 \sqrt [3]{b c-a d} \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 )}{\sqrt {6} c^{13/3}}+\frac {\left (d^3 \sqrt [3]{b c-a d}-i \sqrt {3} d^3 \sqrt [3]{b c-a d}\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^{13/3}} \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 {1}{3}}}{{\left (d x^{3} + c\right )} x^{11}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (d \,x^{3}+c \right ) x^{11}}\, 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 {1}{3}}}{{\left (d x^{3} + c\right )} x^{11}}\,{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 )}^{1/3}}{x^{11}\,\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{a + b x^{3}}}{x^{11} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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