Optimal. Leaf size=211 \[ \frac {c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac {c (b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3}}+\frac {c \sqrt [3]{a+b x^3} (b c-a d)}{d^3}-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d} \]
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Rubi [A] time = 0.24, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {446, 80, 50, 58, 617, 204, 31} \[ -\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {c \sqrt [3]{a+b x^3} (b c-a d)}{d^3}+\frac {c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac {c (b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{10/3}}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 58
Rule 80
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x (a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3\right )^{7/3}}{7 b d}-\frac {c \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d}+\frac {(c (b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac {c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d}-\frac {\left (c (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^3}\\ &=\frac {c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d}+\frac {c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {\left (c (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^{10/3}}-\frac {\left (c (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^{11/3}}\\ &=\frac {c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d}+\frac {c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c (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^{10/3}}-\frac {\left (c (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^{10/3}}\\ &=\frac {c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d}+\frac {c (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^{10/3}}+\frac {c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac {c (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^{10/3}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 255, normalized size = 1.21 \[ \frac {c (b c-a d) \left (\sqrt [3]{b c-a d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt {3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt {3}}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{6 d^{10/3}}-\frac {c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 298, normalized size = 1.41 \[ \frac {28 \, \sqrt {3} {\left (b^{2} c^{2} - a b c d\right )} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 14 \, {\left (b^{2} c^{2} - a b c d\right )} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 28 \, {\left (b^{2} c^{2} - a b c d\right )} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, b^{2} d^{2} x^{6} + 28 \, b^{2} c^{2} - 35 \, a b c d + 4 \, a^{2} d^{2} - {\left (7 \, b^{2} c d - 8 \, a b d^{2}\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{84 \, b d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 348, normalized size = 1.65 \[ \frac {{\left (b^{10} c^{3} d^{4} - 2 \, a b^{9} c^{2} d^{5} + a^{2} b^{8} c d^{6}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{9} c d^{7} - a b^{8} d^{8}\right )}} - \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c^{2} - a c d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d^{4}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c^{2} - a c d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, d^{4}} + \frac {28 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{8} c^{2} d^{4} - 7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{7} c d^{5} - 28 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{7} c d^{5} + 4 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{6} d^{6}}{28 \, b^{7} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} x^{5}}{d \,x^{3}+c}\, dx \]
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 = 5.06, size = 348, normalized size = 1.65 \[ \frac {{\left (b\,x^3+a\right )}^{7/3}}{7\,b\,d}-{\left (b\,x^3+a\right )}^{4/3}\,\left (\frac {a}{4\,b\,d}+\frac {b^2\,c-a\,b\,d}{4\,b^2\,d^2}\right )-\frac {c\,\ln \left (\frac {3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (a^2\,c\,d^2-2\,a\,b\,c^2\,d+b^2\,c^3\right )}{d}-\frac {c\,{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{10/3}}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{10/3}}-\frac {c\,\ln \left (\frac {3\,c\,{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d}-\frac {3\,c\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{7/3}}{d^{4/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{10/3}}+\frac {c\,\ln \left (\frac {3\,c\,{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d}+\frac {3\,c\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{7/3}}{d^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{10/3}}+\frac {{\left (b\,x^3+a\right )}^{1/3}\,\left (b^2\,c-a\,b\,d\right )\,\left (\frac {a}{b\,d}+\frac {b^2\,c-a\,b\,d}{b^2\,d^2}\right )}{b\,d} \]
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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