Optimal. Leaf size=165 \[ \frac {c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac {c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}+\frac {c \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{4/3} (b c-a d)^{2/3}}+\frac {\sqrt [3]{a+b x^3}}{b d} \]
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Rubi [A] time = 0.16, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 80, 58, 617, 204, 31} \begin {gather*} \frac {c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac {c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}+\frac {c \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^{4/3} (b c-a d)^{2/3}}+\frac {\sqrt [3]{a+b x^3}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 58
Rule 80
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac {\sqrt [3]{a+b x^3}}{b d}-\frac {c \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d}\\ &=\frac {\sqrt [3]{a+b x^3}}{b d}+\frac {c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac {c \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^{4/3} (b c-a d)^{2/3}}-\frac {c \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^{5/3} \sqrt [3]{b c-a d}}\\ &=\frac {\sqrt [3]{a+b x^3}}{b d}+\frac {c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac {c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}-\frac {c \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^{4/3} (b c-a d)^{2/3}}\\ &=\frac {\sqrt [3]{a+b x^3}}{b d}+\frac {c \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^{4/3} (b c-a d)^{2/3}}+\frac {c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac {c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 202, normalized size = 1.22 \begin {gather*} \frac {b c \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 )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3} (b c-a d)^{2/3}-2 b c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt {3} b c \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 b d^{4/3} (b c-a d)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 221, normalized size = 1.34 \begin {gather*} -\frac {c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 d^{4/3} (b c-a d)^{2/3}}+\frac {c \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 )}{6 d^{4/3} (b c-a d)^{2/3}}+\frac {c \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{b c-a d}}\right )}{\sqrt {3} d^{4/3} (b c-a d)^{2/3}}+\frac {\sqrt [3]{a+b x^3}}{b d} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 1060, normalized size = 6.42
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 253, normalized size = 1.53 \begin {gather*} -\frac {\frac {6 \, {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} b c \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^{2} - \sqrt {3} a d^{3}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} b c \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 )}{b c d^{2} - a d^{3}} - \frac {2 \, b c \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 )}{b c d - a d^{2}} - \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{d}}{6 \, b} \end {gather*}
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 \begin {gather*} \int \frac {x^{5}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 232, normalized size = 1.41 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{1/3}}{b\,d}-\frac {c\,\ln \left (3\,c\,d\,{\left (b\,x^3+a\right )}^{1/3}-\frac {c\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{4/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )}{3\,d^{4/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {\ln \left (3\,c\,d\,{\left (b\,x^3+a\right )}^{1/3}+\frac {\left (9\,a\,d^3-9\,b\,c\,d^2\right )\,\left (c-\sqrt {3}\,c\,1{}\mathrm {i}\right )}{6\,d^{4/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (c-\sqrt {3}\,c\,1{}\mathrm {i}\right )}{6\,d^{4/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {\ln \left (3\,c\,d\,{\left (b\,x^3+a\right )}^{1/3}+\frac {\left (9\,a\,d^3-9\,b\,c\,d^2\right )\,\left (c+\sqrt {3}\,c\,1{}\mathrm {i}\right )}{6\,d^{4/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (c+\sqrt {3}\,c\,1{}\mathrm {i}\right )}{6\,d^{4/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \end {gather*}
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 \begin {gather*} \int \frac {x^{5}}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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