Optimal. Leaf size=258 \[ \frac {\sqrt [3]{a+b x^3} \left (-28 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{28 a^2 c^3 x}+\frac {d^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^{10/3}}-\frac {d^2 \sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{10/3}}-\frac {d^2 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{10/3}}-\frac {\sqrt [3]{a+b x^3} (b c-7 a d)}{28 a c^2 x^4}-\frac {\sqrt [3]{a+b x^3}}{7 c x^7} \]
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Rubi [C] time = 0.94, antiderivative size = 451, normalized size of antiderivative = 1.75, 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} \[ -\frac {-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (\frac {2}{3},2,2;1,\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-2 x^3 \left (2 c^2-3 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-12 b c^2 d x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+15 b c^3 x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-15 a c^2 d x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+12 a c d^2 x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-12 a c^2 d x^3+8 a c^3+36 a c d^2 x^6-12 b c^2 d x^6+8 b c^3 x^3+36 b c d^2 x^9}{56 c^4 x^7 \left (a+b x^3\right )^{2/3}} \]
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^8 \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^8 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=-\frac {8 a c^3+8 b c^3 x^3-12 a c^2 d x^3-12 b c^2 d x^6+36 a c d^2 x^6+36 b c d^2 x^9-2 (b c-a d) x^3 \left (2 c^2-3 c d x^3+9 d^2 x^6\right ) \, _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 )+15 b c^3 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 )-15 a c^2 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 )-12 b c^2 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 )+12 a c 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 )-27 b c 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 )+27 a 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 )-9 (b c-a d) x^3 \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 )}{56 c^4 x^7 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 2.75, size = 451, normalized size = 1.75 \[ -\frac {-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (\frac {2}{3},2,2;1,\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+15 b c^3 x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-2 x^3 \left (2 c^2-3 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-15 a c^2 d x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-12 b c^2 d x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+12 a c d^2 x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+8 a c^3-12 a c^2 d x^3+36 a c d^2 x^6+8 b c^3 x^3-12 b c^2 d x^6+36 b c d^2 x^9}{56 c^4 x^7 \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (d \,x^{3}+c \right ) x^{8}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^8\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \int \frac {\sqrt [3]{a + b x^{3}}}{x^{8} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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