Optimal. Leaf size=202 \[ \frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.117491, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1343, 200, 31, 634, 617, 204, 628} \[ \frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1343
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac{\left (2 a b+2 b^2 x^3\right ) \int \frac{1}{2 a b+2 b^2 x^3} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (2 a b+2 b^2 x^3\right ) \int \frac{1}{\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}+\sqrt [3]{2} b^{2/3} x} \, dx}{3\ 2^{2/3} a^{2/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 a b+2 b^2 x^3\right ) \int \frac{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}-\sqrt [3]{2} b^{2/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{3\ 2^{2/3} a^{2/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (2 a b+2 b^2 x^3\right ) \int \frac{-2^{2/3} \sqrt [3]{a} b+2\ 2^{2/3} b^{4/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{12 a^{2/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 a b+2 b^2 x^3\right ) \int \frac{1}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 a b+2 b^2 x^3\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{2 a^{2/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}
Mathematica [A] time = 0.0229603, size = 109, normalized size = 0.54 \[ -\frac{\left (a+b x^3\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 95, normalized size = 0.5 \begin{align*} -{\frac{b{x}^{3}+a}{6\,b} \left ( 2\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) -2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) +\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \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 = 1.74454, size = 749, normalized size = 3.71 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) - \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.182363, size = 20, normalized size = 0.1 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (3 t a + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12051, size = 165, normalized size = 0.82 \begin{align*} -\frac{1}{6} \,{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{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 )}{a b} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]