Optimal. Leaf size=318 \[ \frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{d^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{7/3} (b c-a d)}-\frac{1}{4 a c x^4} \]
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Rubi [A] time = 0.379855, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {480, 583, 584, 292, 31, 634, 617, 204, 628} \[ \frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{d^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{7/3} (b c-a d)}-\frac{1}{4 a c x^4} \]
Antiderivative was successfully verified.
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Rule 480
Rule 583
Rule 584
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac{1}{4 a c x^4}+\frac{\int \frac{-4 (b c+a d)-4 b d x^3}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{4 a c}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}-\frac{\int \frac{x \left (-4 \left (b^2 c^2+a b c d+a^2 d^2\right )-4 b d (b c+a d) x^3\right )}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{4 a^2 c^2}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}-\frac{\int \left (-\frac{4 b^3 c^2 x}{(b c-a d) \left (a+b x^3\right )}-\frac{4 a^2 d^3 x}{(-b c+a d) \left (c+d x^3\right )}\right ) \, dx}{4 a^2 c^2}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}+\frac{b^3 \int \frac{x}{a+b x^3} \, dx}{a^2 (b c-a d)}-\frac{d^3 \int \frac{x}{c+d x^3} \, dx}{c^2 (b c-a d)}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}-\frac{b^{8/3} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{7/3} (b c-a d)}+\frac{b^{8/3} \int \frac{\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^{7/3} (b c-a d)}+\frac{d^{8/3} \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{7/3} (b c-a d)}-\frac{d^{8/3} \int \frac{\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^{7/3} (b c-a d)}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{b^{7/3} \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^{7/3} (b c-a d)}+\frac{b^{8/3} \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^2 (b c-a d)}-\frac{d^{7/3} \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^{7/3} (b c-a d)}-\frac{d^{8/3} \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^2 (b c-a d)}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac{d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac{b^{7/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{7/3} (b c-a d)}-\frac{d^{7/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{7/3} (b c-a d)}\\ &=-\frac{1}{4 a c x^4}+\frac{b c+a d}{a^2 c^2 x}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} (b c-a d)}+\frac{d^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{7/3} (b c-a d)}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac{d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.182195, size = 282, normalized size = 0.89 \[ \frac{-\frac{12 b^2 x^3}{a^2}+\frac{4 b^{7/3} x^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3}}-\frac{2 b^{7/3} x^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3}}+\frac{4 \sqrt{3} b^{7/3} x^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{7/3}}+\frac{3 b}{a}+\frac{12 d^2 x^3}{c^2}-\frac{4 d^{7/3} x^4 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{7/3}}+\frac{2 d^{7/3} x^4 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{7/3}}-\frac{4 \sqrt{3} d^{7/3} x^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{7/3}}-\frac{3 d}{c}}{12 x^4 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 291, normalized size = 0.9 \begin{align*} -{\frac{{d}^{2}}{3\,{c}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{d}^{2}}{6\,{c}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{d}^{2}\sqrt{3}}{3\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{b}^{2}}{3\,{a}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}}{6\,{a}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{4\,ac{x}^{4}}}+{\frac{d}{a{c}^{2}x}}+{\frac{b}{{a}^{2}cx}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.76087, size = 697, normalized size = 2.19 \begin{align*} \frac{4 \, \sqrt{3} b^{2} c^{2} x^{4} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) - 4 \, \sqrt{3} a^{2} d^{2} x^{4} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{d}{c}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 2 \, b^{2} c^{2} x^{4} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 2 \, a^{2} d^{2} x^{4} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (-\frac{d}{c}\right )^{\frac{2}{3}} - c \left (-\frac{d}{c}\right )^{\frac{1}{3}}\right ) - 4 \, b^{2} c^{2} x^{4} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 4 \, a^{2} d^{2} x^{4} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (-\frac{d}{c}\right )^{\frac{2}{3}}\right ) - 3 \, a b c^{2} + 3 \, a^{2} c d + 12 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12395, size = 443, normalized size = 1.39 \begin{align*} -\frac{b^{3} \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \left (-\frac{c}{d}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{4} - a c^{3} d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{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^{3} b c - \sqrt{3} a^{4} d} + \frac{\left (-c d^{2}\right )^{\frac{2}{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^{4} - \sqrt{3} a c^{3} d} + \frac{\left (-a b^{2}\right )^{\frac{2}{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^{3} b c - a^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{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^{4} - a c^{3} d\right )}} + \frac{4 \, b c x^{3} + 4 \, a d x^{3} - a c}{4 \, a^{2} c^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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