Optimal. Leaf size=119 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3}}+\frac{x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0660836, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {193, 321, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3}}+\frac{x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 193
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{a+\frac{b}{x^3}} \, dx &=\int \frac{x^3}{b+a x^3} \, dx\\ &=\frac{x}{a}-\frac{b \int \frac{1}{b+a x^3} \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt [3]{b} \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a}-\frac{\sqrt [3]{b} \int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a}\\ &=\frac{x}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{4/3}}-\frac{b^{2/3} \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a}\\ &=\frac{x}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{4/3}}\\ &=\frac{x}{a}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0160645, size = 108, normalized size = 0.91 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )+6 \sqrt [3]{a} x}{6 a^{4/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 99, normalized size = 0.8 \begin{align*}{\frac{x}{a}}-{\frac{b}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{6\,{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b\sqrt{3}}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \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 = 1.50063, size = 254, normalized size = 2.13 \begin{align*} \frac{2 \, \sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 2 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 6 \, x}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.405454, size = 22, normalized size = 0.18 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{4} + b, \left ( t \mapsto t \log{\left (- 3 t a + x \right )} \right )\right )} + \frac{x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25554, size = 150, normalized size = 1.26 \begin{align*} \frac{\left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} + \frac{x}{a} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2}} - \frac{\left (-a^{2} b\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]