Optimal. Leaf size=299 \[ -\frac {(3 a d+2 b c) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{5/3} c^2}+\frac {(3 a d+2 b c) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} c^2}+\frac {\log (x) (3 a d+2 b c)}{6 a^{5/3} c^2}-\frac {d^{5/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{2/3}}+\frac {d^{5/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{2/3}}-\frac {d^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2 (b c-a d)^{2/3}}-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3} \]
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Rubi [A] time = 0.32, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {446, 103, 156, 57, 617, 204, 31, 58} \begin {gather*} -\frac {(3 a d+2 b c) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{5/3} c^2}+\frac {(3 a d+2 b c) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} c^2}+\frac {\log (x) (3 a d+2 b c)}{6 a^{5/3} c^2}-\frac {d^{5/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{2/3}}+\frac {d^{5/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{2/3}}-\frac {d^{5/3} \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} c^2 (b c-a d)^{2/3}}-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 58
Rule 103
Rule 156
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{3} (2 b c+3 a d)+\frac {2 b d x}{3}}{x (a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3}+\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 c^2}-\frac {(2 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{9 a c^2}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3}+\frac {(2 b c+3 a d) \log (x)}{6 a^{5/3} c^2}-\frac {d^{5/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{2/3}}+\frac {d^{5/3} \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 c^2 (b c-a d)^{2/3}}+\frac {d^{4/3} \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 c^2 \sqrt [3]{b c-a d}}+\frac {(2 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{5/3} c^2}+\frac {(2 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{4/3} c^2}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3}+\frac {(2 b c+3 a d) \log (x)}{6 a^{5/3} c^2}-\frac {d^{5/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{2/3}}-\frac {(2 b c+3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{5/3} c^2}+\frac {d^{5/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{2/3}}+\frac {d^{5/3} \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 )}{c^2 (b c-a d)^{2/3}}-\frac {(2 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 a^{5/3} c^2}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3}+\frac {(2 b c+3 a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} c^2}-\frac {d^{5/3} \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} c^2 (b c-a d)^{2/3}}+\frac {(2 b c+3 a d) \log (x)}{6 a^{5/3} c^2}-\frac {d^{5/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{2/3}}-\frac {(2 b c+3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{5/3} c^2}+\frac {d^{5/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 303, normalized size = 1.01 \begin {gather*} \frac {\frac {(3 a d+2 b c) \left (\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )\right )}{a^{2/3} c}+\frac {3 a d^{5/3} \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 {\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt {3}}\right )\right )}{c (b c-a d)^{2/3}}-\frac {6 \sqrt [3]{a+b x^3}}{x^3}}{18 a c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.73, size = 388, normalized size = 1.30 \begin {gather*} \frac {(-3 a d-2 b c) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{a}\right )}{9 a^{5/3} c^2}+\frac {(3 a d+2 b c) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{18 a^{5/3} c^2}+\frac {(3 a d+2 b c) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} c^2}+\frac {d^{5/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 c^2 (b c-a d)^{2/3}}-\frac {d^{5/3} \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 c^2 (b c-a d)^{2/3}}-\frac {d^{5/3} \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} c^2 (b c-a d)^{2/3}}-\frac {\sqrt [3]{a+b x^3}}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.25, size = 562, normalized size = 1.88 \begin {gather*} -\frac {6 \, \sqrt {3} a^{3} d \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} x^{3} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) + 3 \, a^{3} d \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} d^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c d - a d^{2}\right )} \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {2}{3}}\right ) - 6 \, a^{3} d \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c - a d\right )} \left (\frac {d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, \sqrt {3} {\left (2 \, a b c + 3 \, a^{2} d\right )} x^{3} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2}}\right ) - \left (-a^{2}\right )^{\frac {2}{3}} {\left (2 \, b c + 3 \, a d\right )} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a^{2}\right )^{\frac {2}{3}} {\left (2 \, b c + 3 \, a d\right )} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} c}{18 \, a^{3} c^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 377, normalized size = 1.26 \begin {gather*} -\frac {d^{2} \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 c^{3} - a c^{2} d\right )}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} d \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^{3} - \sqrt {3} a c^{2} d} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} d \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^{3} - a c^{2} d\right )}} + \frac {\sqrt {3} {\left (2 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {5}{3}} c^{2}} + \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {5}{3}} c^{2}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {5}{3}} c^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, a c x^{3}} \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 {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right ) x^{4}}\, dx \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.14, size = 1959, normalized size = 6.55
result too large to display
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 {1}{x^{4} \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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