Optimal. Leaf size=241 \[ \frac {\sqrt [3]{a+b x^3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}-\frac {\left (a+b x^3\right )^{4/3} (2 a d+b c)}{4 b^3 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {c^3 \log \left (c+d x^3\right )}{6 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3} (b c-a d)^{2/3}}+\frac {c^3 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3} (b c-a d)^{2/3}} \]
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Rubi [A] time = 0.26, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 88, 58, 617, 204, 31} \[ \frac {\sqrt [3]{a+b x^3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}-\frac {\left (a+b x^3\right )^{4/3} (2 a d+b c)}{4 b^3 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {c^3 \log \left (c+d x^3\right )}{6 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3} (b c-a d)^{2/3}}+\frac {c^3 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 58
Rule 88
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {b^2 c^2+a b c d+a^2 d^2}{b^2 d^3 (a+b x)^{2/3}}+\frac {(-b c-2 a d) \sqrt [3]{a+b x}}{b^2 d^2}+\frac {(a+b x)^{4/3}}{b^2 d}-\frac {c^3}{d^3 (a+b x)^{2/3} (c+d x)}\right ) \, dx,x,x^3\right )\\ &=\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \sqrt [3]{a+b x^3}}{b^3 d^3}-\frac {(b c+2 a d) \left (a+b x^3\right )^{4/3}}{4 b^3 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^3}\\ &=\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \sqrt [3]{a+b x^3}}{b^3 d^3}-\frac {(b c+2 a d) \left (a+b x^3\right )^{4/3}}{4 b^3 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {c^3 \log \left (c+d x^3\right )}{6 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{11/3} \sqrt [3]{b c-a d}}\\ &=\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \sqrt [3]{a+b x^3}}{b^3 d^3}-\frac {(b c+2 a d) \left (a+b x^3\right )^{4/3}}{4 b^3 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {c^3 \log \left (c+d x^3\right )}{6 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{d^{10/3} (b c-a d)^{2/3}}\\ &=\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \sqrt [3]{a+b x^3}}{b^3 d^3}-\frac {(b c+2 a d) \left (a+b x^3\right )^{4/3}}{4 b^3 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac {c^3 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3} (b c-a d)^{2/3}}+\frac {c^3 \log \left (c+d x^3\right )}{6 d^{10/3} (b c-a d)^{2/3}}-\frac {c^3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3} (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 251, normalized size = 1.04 \[ \frac {\frac {84 \sqrt [3]{a+b x^3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3}-\frac {21 d \left (a+b x^3\right )^{4/3} (2 a d+b c)}{b^3}+\frac {12 d^2 \left (a+b x^3\right )^{7/3}}{b^3}+\frac {14 c^3 \left (\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}}{84 d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 1322, normalized size = 5.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 372, normalized size = 1.54 \[ \frac {b^{24} c^{3} d^{4} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{25} c d^{7} - a b^{24} d^{8}\right )}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} c^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d^{4} - \sqrt {3} a d^{5}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} c^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d^{4} - a d^{5}\right )}} + \frac {28 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{20} c^{2} d^{4} - 7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{19} c d^{5} + 28 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{19} c d^{5} + 4 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{18} d^{6} - 14 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a b^{18} d^{6} + 28 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b^{18} d^{6}}{28 \, b^{21} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {x^{11}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 331, normalized size = 1.37 \[ \left (\frac {3\,a^2}{b^3\,d}+\frac {\left (\frac {3\,a}{b^3\,d}+\frac {b^4\,c-a\,b^3\,d}{b^6\,d^2}\right )\,\left (b^4\,c-a\,b^3\,d\right )}{b^3\,d}\right )\,{\left (b\,x^3+a\right )}^{1/3}-\left (\frac {3\,a}{4\,b^3\,d}+\frac {b^4\,c-a\,b^3\,d}{4\,b^6\,d^2}\right )\,{\left (b\,x^3+a\right )}^{4/3}+\frac {{\left (b\,x^3+a\right )}^{7/3}}{7\,b^3\,d}+\frac {\ln \left (\frac {3\,c^3\,{\left (b\,x^3+a\right )}^{1/3}}{d}+\frac {3\,c^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{2\,d^{4/3}}\right )\,\left (c^3+\sqrt {3}\,c^3\,1{}\mathrm {i}\right )}{6\,d^{10/3}\,{\left (a\,d-b\,c\right )}^{2/3}}-\frac {c^3\,\ln \left (\frac {3\,c^3\,{\left (b\,x^3+a\right )}^{1/3}}{d}-\frac {3\,c^3\,{\left (a\,d-b\,c\right )}^{1/3}}{d^{4/3}}\right )}{3\,d^{10/3}\,{\left (a\,d-b\,c\right )}^{2/3}}-\frac {c^3\,\ln \left (\frac {3\,c^3\,{\left (b\,x^3+a\right )}^{1/3}}{d}-\frac {3\,c^3\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{d^{4/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,d^{10/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \]
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 \[ \int \frac {x^{11}}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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