Optimal. Leaf size=301 \[ \frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac {b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} (b c-a d)}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{5/3} (b c-a d)}-\frac {1}{2 a c x^2} \]
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Rubi [A] time = 0.25, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {480, 522, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac {b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} (b c-a d)}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{5/3} (b c-a d)}-\frac {1}{2 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 480
Rule 522
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac {1}{2 a c x^2}+\frac {\int \frac {-2 (b c+a d)-2 b d x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{2 a c}\\ &=-\frac {1}{2 a c x^2}-\frac {b^2 \int \frac {1}{a+b x^3} \, dx}{a (b c-a d)}+\frac {d^2 \int \frac {1}{c+d x^3} \, dx}{c (b c-a d)}\\ &=-\frac {1}{2 a c x^2}-\frac {b^2 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{5/3} (b c-a d)}-\frac {b^2 \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{5/3} (b c-a d)}+\frac {d^2 \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{5/3} (b c-a d)}+\frac {d^2 \int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{5/3} (b c-a d)}\\ &=-\frac {1}{2 a c x^2}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}+\frac {b^{5/3} \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{5/3} (b c-a d)}-\frac {b^2 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{4/3} (b c-a d)}-\frac {d^{5/3} \int \frac {-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{5/3} (b c-a d)}+\frac {d^2 \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^{4/3} (b c-a d)}\\ &=-\frac {1}{2 a c x^2}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}+\frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)}-\frac {b^{5/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{5/3} (b c-a d)}+\frac {d^{5/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{5/3} (b c-a d)}\\ &=-\frac {1}{2 a c x^2}+\frac {b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} (b c-a d)}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{5/3} (b c-a d)}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}+\frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 259, normalized size = 0.86 \begin {gather*} \frac {\frac {2 b^{5/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {b^{5/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}-\frac {2 \sqrt {3} b^{5/3} x^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {3 b}{a}-\frac {2 d^{5/3} x^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3}}+\frac {d^{5/3} x^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3}}+\frac {2 \sqrt {3} d^{5/3} x^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{5/3}}-\frac {3 d}{c}}{6 x^2 (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 2.62, size = 301, normalized size = 1.00 \begin {gather*} -\frac {2 \, \sqrt {3} b c x^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, \sqrt {3} a d x^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - b c x^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - a d x^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} + c d x \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, b c x^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, a d x^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x - c \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) + 3 \, b c - 3 \, a d}{6 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 309, normalized size = 1.03 \begin {gather*} \frac {b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{2} b c - \sqrt {3} a^{3} d} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{3} - \sqrt {3} a c^{2} d} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} b \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} b c - a^{3} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {1}{2 \, a c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 257, normalized size = 0.85 \begin {gather*} \frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {b \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {\sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {2}{3}} c}-\frac {d \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {2}{3}} c}+\frac {d \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {2}{3}} c}-\frac {1}{2 a c \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 328, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {b \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {d \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} - \frac {b \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {d \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} - \frac {1}{2 \, a c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.83, size = 1829, normalized size = 6.08
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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