Optimal. Leaf size=440 \[ \frac {\sqrt [3]{a+b x^3} \left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right )}{9 a c^3}+\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{2/3} c^3}-\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} c^3}-\frac {\log (x) \left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right )}{18 a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^3}-\frac {d^{2/3} (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^3}+\frac {d \sqrt [3]{a+b x^3} (b c-a d)}{c^3}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6} \]
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Rubi [A] time = 0.62, antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {446, 103, 149, 156, 50, 57, 617, 204, 31, 58} \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right )}{9 a c^3}+\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{2/3} c^3}-\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} c^3}-\frac {\log (x) \left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right )}{18 a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^3}-\frac {d^{2/3} (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 c^3}+\frac {d^{2/3} (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} c^3}-\frac {\left (a+b x^3\right )^{4/3} (b c-6 a d)}{18 a c^2 x^3}+\frac {d \sqrt [3]{a+b x^3} (b c-a d)}{c^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 58
Rule 103
Rule 149
Rule 156
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{4/3}}{x^7 \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{x^3 (c+d x)} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{4/3} \left (\frac {1}{3} (-b c+6 a d)-\frac {b d x}{3}\right )}{x^2 (c+d x)} \, dx,x,x^3\right )}{6 a c}\\ &=-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x} \left (-\frac {2}{9} \left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right )-\frac {2}{9} b d (2 b c-3 a d) x\right )}{x (c+d x)} \, dx,x,x^3\right )}{6 a c^2}\\ &=-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}+\frac {\left (d^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 c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x} \, dx,x,x^3\right )}{27 a c^3}\\ &=\frac {d (b c-a d) \sqrt [3]{a+b x^3}}{c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{9 a c^3}-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\left (d (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 c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{27 c^3}\\ &=\frac {d (b c-a d) \sqrt [3]{a+b x^3}}{c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{9 a c^3}-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log (x)}{18 a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^3}-\frac {\left (d^{2/3} (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 c^3}-\frac {\left (\sqrt [3]{d} (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 c^3}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{2/3} c^3}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 \sqrt [3]{a} c^3}\\ &=\frac {d (b c-a d) \sqrt [3]{a+b x^3}}{c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{9 a c^3}-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log (x)}{18 a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{2/3} c^3}-\frac {d^{2/3} (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 c^3}-\frac {\left (d^{2/3} (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 )}{c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{9 a^{2/3} c^3}\\ &=\frac {d (b c-a d) \sqrt [3]{a+b x^3}}{c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{9 a c^3}-\frac {(b c-6 a d) \left (a+b x^3\right )^{4/3}}{18 a c^2 x^3}-\frac {\left (a+b x^3\right )^{7/3}}{6 a c x^6}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{2/3} c^3}+\frac {d^{2/3} (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} c^3}-\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log (x)}{18 a^{2/3} c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^3}+\frac {\left (2 b^2 c^2-12 a b c d+9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{2/3} c^3}-\frac {d^{2/3} (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 c^3}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 429, normalized size = 0.98 \begin {gather*} \frac {x^6 \left (4 \left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \left (-\frac {1}{2} a^{4/3} \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 )+3 a \sqrt [3]{a+b x^3}+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )-9 a^2 d^{2/3} \left (3 d^{4/3} \left (a+b x^3\right )^{4/3}-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 {\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt {3}}\right )\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )\right )\right )-18 a c^2 \left (a+b x^3\right )^{7/3}-6 c x^3 \left (a+b x^3\right )^{7/3} (b c-6 a d)}{108 a^2 c^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.07, size = 445, normalized size = 1.01 \begin {gather*} \frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{a}\right )}{27 a^{2/3} c^3}+\frac {\left (-9 a^2 d^2+12 a b c d-2 b^2 c^2\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{54 a^{2/3} c^3}-\frac {\left (9 a^2 d^2-12 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} c^3}-\frac {d^{2/3} (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 c^3}+\frac {d^{2/3} (b c-a d)^{4/3} \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^3}+\frac {d^{2/3} (b c-a d)^{4/3} \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^3}+\frac {\sqrt [3]{a+b x^3} \left (-3 a c+6 a d x^3-7 b c x^3\right )}{18 c^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.02, size = 503, normalized size = 1.14 \begin {gather*} \frac {18 \, \sqrt {3} {\left (a^{2} b c - a^{3} d\right )} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} x^{6} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b c d^{2} - a d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \sqrt {3} {\left (b c d - a d^{2}\right )}}{3 \, {\left (b c d - a d^{2}\right )}}\right ) - 2 \, \sqrt {3} {\left (2 \, a b^{2} c^{2} - 12 \, a^{2} b c d + 9 \, a^{3} d^{2}\right )} {\left (a^{2}\right )}^{\frac {1}{6}} x^{6} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\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 )}}{3 \, a^{2}}\right ) - {\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}} x^{6} \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 (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 9 \, {\left (a^{2} b c - a^{3} d\right )} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} d^{2} - {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c d^{2} - a d^{3}\right )}^{\frac {2}{3}}\right ) - 18 \, {\left (a^{2} b c - a^{3} d\right )} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}}\right ) - 3 \, {\left (3 \, a^{3} c^{2} + {\left (7 \, a^{2} b c^{2} - 6 \, a^{3} c d\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, a^{2} c^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 481, normalized size = 1.09 \begin {gather*} \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\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 c^{4} - a c^{3} d\right )}} - \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a 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 \, c^{3}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a 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 \, c^{3}} - \frac {\sqrt {3} {\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\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 )}{27 \, a^{\frac {2}{3}} c^{3}} - \frac {{\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\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 )}{54 \, a^{\frac {2}{3}} c^{3}} + \frac {{\left (2 \, b^{2} c^{2} - 12 \, a b c d + 9 \, a^{2} d^{2}\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{27 \, a^{\frac {2}{3}} c^{3}} - \frac {7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2} c - 4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{2} c - 6 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a b d + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b d}{18 \, b^{2} c^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{\left (d \,x^{3}+c \right ) x^{7}}\, 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 {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.25, size = 2841, normalized size = 6.46
result too large to display
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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