Optimal. Leaf size=288 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{c} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.143002, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {482, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{c} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 482
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac{b \int \frac{x}{a+b x^3} \, dx}{b c-a d}-\frac{d \int \frac{x}{c+d x^3} \, dx}{b c-a d}\\ &=-\frac{b^{2/3} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} (b c-a d)}+\frac{b^{2/3} \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 \sqrt [3]{a} (b c-a d)}+\frac{d^{2/3} \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 \sqrt [3]{c} (b c-a d)}-\frac{d^{2/3} \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 \sqrt [3]{c} (b c-a d)}\\ &=-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}+\frac{\sqrt [3]{b} \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 \sqrt [3]{a} (b c-a d)}+\frac{b^{2/3} \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 (b c-a d)}-\frac{\sqrt [3]{d} \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 \sqrt [3]{c} (b c-a d)}-\frac{d^{2/3} \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 (b c-a d)}\\ &=-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{c} (b c-a d)}\\ &=-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0991425, size = 224, normalized size = 0.78 \[ \frac{-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{c}}-\frac{2 \sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{c}}-\frac{2 \sqrt{3} \sqrt [3]{d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{\sqrt [3]{c}}}{6 a d-6 b c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 222, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{1}{6\,ad-6\,bc}\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{\sqrt{3}}{3\,ad-3\,bc}\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{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{6\,ad-6\,bc}\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{\sqrt{3}}{3\,ad-3\,bc}\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}}}}}} \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 = 1.62779, size = 497, normalized size = 1.73 \begin{align*} \frac{2 \, \sqrt{3} \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 ) - 2 \, \sqrt{3} \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 ) + \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 ) + \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 ) - 2 \, \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (-\frac{d}{c}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c - a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.1138, size = 515, normalized size = 1.79 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{4} d^{3} - 81 a^{3} b c d^{2} + 81 a^{2} b^{2} c^{2} d - 27 a b^{3} c^{3}\right ) - b, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} c d^{3} - 81 a^{2} b c^{2} d^{2} + 81 a b^{2} c^{3} d - 27 b^{3} c^{4}\right ) + d, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15834, size = 392, normalized size = 1.36 \begin{align*} -\frac{b \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 b c - a^{2} d\right )}} + \frac{d \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^{2} - a c d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \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 b^{2} c - \sqrt{3} a^{2} b d} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \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^{2} d - \sqrt{3} a c d^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \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^{2} c - a^{2} b d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \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} d - a c d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]