Optimal. Leaf size=251 \[ -\frac {\left (a+b x^3\right )^{7/3} (a d+b c)}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}-\frac {c^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac {c^2 (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}-\frac {c^2 (b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{13/3}}-\frac {c^2 \sqrt [3]{a+b x^3} (b c-a d)}{d^4}+\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3} \]
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Rubi [A] time = 0.36, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {446, 88, 50, 58, 617, 204, 31} \[ -\frac {\left (a+b x^3\right )^{7/3} (a d+b c)}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}+\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3}-\frac {c^2 \sqrt [3]{a+b x^3} (b c-a d)}{d^4}-\frac {c^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac {c^2 (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}-\frac {c^2 (b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{13/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 58
Rule 88
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^8 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 (a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {(-b c-a d) (a+b x)^{4/3}}{b d^2}+\frac {(a+b x)^{7/3}}{b d}+\frac {c^2 (a+b x)^{4/3}}{d^2 (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {(b c+a d) \left (a+b x^3\right )^{7/3}}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}+\frac {c^2 \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{7/3}}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}-\frac {\left (c^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d^3}\\ &=-\frac {c^2 (b c-a d) \sqrt [3]{a+b x^3}}{d^4}+\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{7/3}}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}+\frac {\left (c^2 (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^4}\\ &=-\frac {c^2 (b c-a d) \sqrt [3]{a+b x^3}}{d^4}+\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{7/3}}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}-\frac {c^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac {\left (c^2 (b c-a d)^{4/3}\right ) \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^{13/3}}+\frac {\left (c^2 (b c-a d)^{5/3}\right ) \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^{14/3}}\\ &=-\frac {c^2 (b c-a d) \sqrt [3]{a+b x^3}}{d^4}+\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{7/3}}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}-\frac {c^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac {c^2 (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}+\frac {\left (c^2 (b c-a d)^{4/3}\right ) \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^{13/3}}\\ &=-\frac {c^2 (b c-a d) \sqrt [3]{a+b x^3}}{d^4}+\frac {c^2 \left (a+b x^3\right )^{4/3}}{4 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{7/3}}{7 b^2 d^2}+\frac {\left (a+b x^3\right )^{10/3}}{10 b^2 d}-\frac {c^2 (b c-a d)^{4/3} \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^{13/3}}-\frac {c^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac {c^2 (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 258, normalized size = 1.03 \[ \frac {-\frac {60 d \left (a+b x^3\right )^{7/3} (a d+b c)}{b^2}+\frac {42 d^2 \left (a+b x^3\right )^{10/3}}{b^2}-\frac {70 c^2 (b c-a d) \left (\sqrt [3]{b c-a d} \left (\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{d^{4/3}}+105 c^2 \left (a+b x^3\right )^{4/3}}{420 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 369, normalized size = 1.47 \[ \frac {140 \, \sqrt {3} {\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 70 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 140 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (14 \, b^{3} d^{3} x^{9} - 2 \, {\left (10 \, b^{3} c d^{2} - 11 \, a b^{2} d^{3}\right )} x^{6} - 140 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 20 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (35 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{420 \, b^{2} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 394, normalized size = 1.57 \[ -\frac {{\left (b^{24} c^{4} d^{6} - 2 \, a b^{23} c^{3} d^{7} + a^{2} b^{22} c^{2} d^{8}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{23} c d^{10} - a b^{22} d^{11}\right )}} + \frac {\sqrt {3} {\left (b c^{3} - a c^{2} d\right )} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d^{5}} + \frac {{\left (b c^{3} - a c^{2} d\right )} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, d^{5}} - \frac {140 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{21} c^{3} d^{6} - 35 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{20} c^{2} d^{7} - 140 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{20} c^{2} d^{7} + 20 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{19} c d^{8} - 14 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} b^{18} d^{9} + 20 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a b^{18} d^{9}}{140 \, b^{20} d^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} x^{8}}{d \,x^{3}+c}\, dx \]
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 = 5.12, size = 477, normalized size = 1.90 \[ \left (\frac {a^2}{4\,b^2\,d}+\frac {\left (\frac {2\,a}{b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{b^4\,d^2}\right )\,\left (b^3\,c-a\,b^2\,d\right )}{4\,b^2\,d}\right )\,{\left (b\,x^3+a\right )}^{4/3}-\left (\frac {2\,a}{7\,b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{7\,b^4\,d^2}\right )\,{\left (b\,x^3+a\right )}^{7/3}+\frac {{\left (b\,x^3+a\right )}^{10/3}}{10\,b^2\,d}+\frac {c^2\,\ln \left (\frac {3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (a^2\,c^2\,d^2-2\,a\,b\,c^3\,d+b^2\,c^4\right )}{d^2}-\frac {c^2\,{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{13/3}}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{13/3}}-\frac {\left (\frac {a^2}{b^2\,d}+\frac {\left (\frac {2\,a}{b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{b^4\,d^2}\right )\,\left (b^3\,c-a\,b^2\,d\right )}{b^2\,d}\right )\,{\left (b\,x^3+a\right )}^{1/3}\,\left (b^3\,c-a\,b^2\,d\right )}{b^2\,d}-\frac {c^2\,\ln \left (\frac {3\,c^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{7/3}}{d^{7/3}}+\frac {3\,c^2\,{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{13/3}}+\frac {c^2\,\ln \left (\frac {3\,c^2\,{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d^2}-\frac {9\,c^2\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{7/3}}{d^{7/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{d^{13/3}} \]
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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