Optimal. Leaf size=318 \[ -\frac {\sqrt [3]{a+b x^3} \left (35 a^2 d^2-40 a b c d+2 b^2 c^2\right )}{140 a c^3 x^4}+\frac {\sqrt [3]{a+b x^3} \left (140 a^3 d^3-175 a^2 b c d^2+20 a b^2 c^2 d+6 b^3 c^3\right )}{140 a^2 c^4 x}+\frac {d^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{13/3}}-\frac {d^2 (b c-a d)^{4/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{13/3}}-\frac {d^2 (b c-a d)^{4/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{13/3}}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{70 c^2 x^7}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}} \]
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Rubi [C] time = 1.80, antiderivative size = 260, normalized size of antiderivative = 0.82, 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} \[ \frac {-18 c x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (-\frac {1}{3},2,2;\frac {2}{3},1;\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 c x^3 \left (a+b x^3\right ) \left (11 c^2+2 c d x^3-9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-\left (14 c^2-12 c d x^3+9 d^2 x^6\right ) \left (c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+5 b c x^3\right )-2 x^6 (b c-a d)^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )}{140 c^5 x^{10} \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx &=\frac {\left (a \sqrt [3]{a+b x^3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {6 c (b c-a d) x^3 \left (a+b x^3\right ) \left (11 c^2+2 c d x^3-9 d^2 x^6\right ) \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-\left (14 c^2-12 c d x^3+9 d^2 x^6\right ) \left (c \left (a+b x^3\right ) \left (5 b c x^3+a \left (c-4 d x^3\right )\right )-2 (b c-a 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 )\right )-18 c (b c-a d) x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, _3F_2\left (-\frac {1}{3},2,2;\frac {2}{3},1;\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{140 c^5 x^{10} \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 2.70, size = 270, normalized size = 0.85 \[ -\frac {a \left (\frac {b x^3}{a}+1\right ) \left (18 c x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (-\frac {1}{3},2,2;\frac {2}{3},1;\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 c x^3 \left (a+b x^3\right ) \left (-11 c^2-2 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+\left (14 c^2-12 c d x^3+9 d^2 x^6\right ) \left (c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+5 b c x^3\right )-2 x^6 (b c-a d)^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )\right )}{140 c^5 x^{10} \left (a+b x^3\right )^{5/3}} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{\left (d \,x^{3}+c \right ) x^{11}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{11}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{4/3}}{x^{11}\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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