Optimal. Leaf size=321 \[ -\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.455683, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {480, 583, 522, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 480
Rule 583
Rule 522
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac{1}{5 a c x^5}+\frac{\int \frac{-5 (b c+a d)-5 b d x^3}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{5 a c}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{2 a^2 c^2 x^2}-\frac{\int \frac{-10 \left (b^2 c^2+a b c d+a^2 d^2\right )-10 b d (b c+a d) x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{10 a^2 c^2}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^3 \int \frac{1}{a+b x^3} \, dx}{a^2 (b c-a d)}-\frac{d^3 \int \frac{1}{c+d x^3} \, dx}{c^2 (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^3 \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{8/3} (b c-a d)}+\frac{b^3 \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^{8/3} (b c-a d)}-\frac{d^3 \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{8/3} (b c-a d)}-\frac{d^3 \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^{8/3} (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}-\frac{b^{8/3} \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^{8/3} (b c-a d)}+\frac{b^3 \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{7/3} (b c-a d)}+\frac{d^{8/3} \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^{8/3} (b c-a d)}-\frac{d^3 \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^{7/3} (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}-\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}+\frac{b^{8/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{8/3} (b c-a d)}-\frac{d^{8/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{8/3} (b c-a d)}\\ &=-\frac{1}{5 a c x^5}+\frac{b c+a d}{2 a^2 c^2 x^2}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}-\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.179111, size = 282, normalized size = 0.88 \[ \frac{-\frac{15 b^2 x^3}{a^2}-\frac{10 b^{8/3} x^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}+\frac{5 b^{8/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}+\frac{10 \sqrt{3} b^{8/3} x^5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{8/3}}+\frac{6 b}{a}+\frac{15 d^2 x^3}{c^2}+\frac{10 d^{8/3} x^5 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{8/3}}-\frac{5 d^{8/3} x^5 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{8/3}}-\frac{10 \sqrt{3} d^{8/3} x^5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{8/3}}-\frac{6 d}{c}}{30 x^5 (a d-b c)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 293, 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 ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\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 ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\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 ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}}{3\,{a}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\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 ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\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 ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{5\,ac{x}^{5}}}+{\frac{d}{2\,a{c}^{2}{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21296, size = 811, normalized size = 2.53 \begin{align*} -\frac{10 \, \sqrt{3} b^{2} c^{2} x^{5} \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 ) + 10 \, \sqrt{3} a^{2} d^{2} x^{5} \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 ) - 5 \, b^{2} c^{2} x^{5} \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 ) - 5 \, a^{2} d^{2} x^{5} \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 ) + 10 \, b^{2} c^{2} x^{5} \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 ) + 10 \, a^{2} d^{2} x^{5} \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 ) + 6 \, a b c^{2} - 6 \, a^{2} c d - 15 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}}{30 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16629, size = 454, normalized size = 1.41 \begin{align*} -\frac{b^{3} \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^{3} b c - a^{4} d\right )}} + \frac{d^{3} \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^{4} - a c^{3} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} \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{1}{3}} d^{2} \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{1}{3}} b^{2} \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{1}{3}} d^{2} \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{5 \, b c x^{3} + 5 \, a d x^{3} - 2 \, a c}{10 \, a^{2} c^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]