Optimal. Leaf size=458 \[ \frac {b x \left (a+b x^3\right )^{2/3} \left (-5 a^2 d^2-7 a b c d+18 b^2 c^2\right )}{18 c^2 d^3}+\frac {(b c-a d)^{5/3} \left (5 a^2 d^2+12 a b c d+27 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^4}-\frac {(b c-a d)^{5/3} \left (5 a^2 d^2+12 a b c d+27 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^4}+\frac {(b c-a d)^{5/3} \left (5 a^2 d^2+12 a b c d+27 b^2 c^2\right ) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^4}+\frac {b^{8/3} (9 b c-11 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 d^4}-\frac {b^{8/3} (9 b c-11 a d) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} d^4}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d) (5 a d+9 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right )^{8/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
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Rubi [C] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 0.14, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac {a^3 x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {11}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{11/3}}{\left (c+d x^3\right )^3} \, dx &=\frac {\left (a^3 \left (a+b x^3\right )^{2/3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{11/3}}{\left (c+d x^3\right )^3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {a^3 x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {11}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 1.84, size = 908, normalized size = 1.98 \[ \frac {1}{108} \left (\frac {10 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac {(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^4}{c^{8/3} \sqrt [3]{b c-a d}}+\frac {4 b \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac {(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^3}{c^{5/3} d \sqrt [3]{b c-a d}}+\frac {16 b^2 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac {(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^2}{c^{2/3} d^2 \sqrt [3]{b c-a d}}+\frac {99 b^3 x^4 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) a}{c d^2 \sqrt [3]{b x^3+a}}-\frac {18 b^3 \sqrt [3]{c} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac {(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a}{d^3 \sqrt [3]{b c-a d}}+\frac {6 x \left (b x^3+a\right )^{2/3} \left (6 b^3+\frac {5 (b c-a d)^2 (3 b c+a d)}{c^2 \left (d x^3+c\right )}-\frac {3 (b c-a d)^3}{c \left (d x^3+c\right )^2}\right )}{d^3}-\frac {81 b^4 x^4 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{d^3 \sqrt [3]{b x^3+a}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 79.23, size = 1246, normalized size = 2.72 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {11}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {11}{3}}}{\left (d \,x^{3}+c \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {11}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{11/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]
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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