Optimal. Leaf size=257 \[ \frac {\left (a+b x^3\right )^{2/3} \left (-20 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{40 a^2 c^3 x^2}+\frac {d^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{11/3}}-\frac {d^2 (b c-a d)^{2/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{11/3}}+\frac {d^2 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{11/3}}-\frac {\left (a+b x^3\right )^{2/3} (b c-4 a d)}{20 a c^2 x^5}-\frac {\left (a+b x^3\right )^{2/3}}{8 c x^8} \]
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Rubi [C] time = 0.97, 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} \begin {gather*} -\frac {-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _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 )-2 x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) (b c-a 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 )-6 b c^2 d x^6 \, _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 )+21 b c^3 x^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 )-21 a c^2 d x^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 )+27 a d^3 x^9 \, _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 )-27 b c d^2 x^9 \, _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 )+6 a c d^2 x^6 \, _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 )-6 a c^2 d x^3+5 a c^3+9 a c d^2 x^6-6 b c^2 d x^6+5 b c^3 x^3+9 b c d^2 x^9}{40 c^4 x^8 \sqrt [3]{a+b x^3}} \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^9 \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^9 \left (c+d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {5 a c^3+5 b c^3 x^3-6 a c^2 d x^3-6 b c^2 d x^6+9 a c d^2 x^6+9 b c d^2 x^9-2 (b c-a d) x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) \, _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 )+21 b c^3 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 )-21 a c^2 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 )-6 b c^2 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 )+6 a c 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 )-27 b c d^2 x^9 \, _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 a d^3 x^9 \, _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 )-9 (b c-a d) x^3 \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 )}{40 c^4 x^8 \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 2.20, size = 451, normalized size = 1.75 \begin {gather*} -\frac {-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _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 )+21 b c^3 x^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 )-2 x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) (b c-a 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 )-21 a c^2 d x^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 )-6 b c^2 d x^6 \, _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 )+27 a d^3 x^9 \, _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 )-27 b c d^2 x^9 \, _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 )+6 a c d^2 x^6 \, _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 )+5 a c^3-6 a c^2 d x^3+9 a c d^2 x^6+5 b c^3 x^3-6 b c^2 d x^6+9 b c d^2 x^9}{40 c^4 x^8 \sqrt [3]{a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 2.91, size = 444, normalized size = 1.73 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-5 a^2 c^2+8 a^2 c d x^3-20 a^2 d^2 x^6-2 a b c^2 x^3+8 a b c d x^6+3 b^2 c^2 x^6\right )}{40 a^2 c^3 x^8}+\frac {\left (d^2 (b c-a d)^{2/3}+i \sqrt {3} d^2 (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^{11/3}}-\frac {\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} d^2 (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^{11/3}}-\frac {i \left (\sqrt {3} d^2 (b c-a d)^{2/3}-i d^2 (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^{11/3}} \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^{9}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right ) x^{9}}\, 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^{9}}\,{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^9\,\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 {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{9} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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