Optimal. Leaf size=93 \[ \frac{3 a^3}{2 b^4 x^{2/3}}-\frac{3 a^4}{b^5 \sqrt [3]{x}}-\frac{a^2}{b^3 x}+\frac{3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{a^5 \log (x)}{b^6}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{3}{5 b x^{5/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0514041, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 44} \[ \frac{3 a^3}{2 b^4 x^{2/3}}-\frac{3 a^4}{b^5 \sqrt [3]{x}}-\frac{a^2}{b^3 x}+\frac{3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{a^5 \log (x)}{b^6}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{3}{5 b x^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^3} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right ) x^{8/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^6 (b+a x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b x^6}-\frac{a}{b^2 x^5}+\frac{a^2}{b^3 x^4}-\frac{a^3}{b^4 x^3}+\frac{a^4}{b^5 x^2}-\frac{a^5}{b^6 x}+\frac{a^6}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3}{5 b x^{5/3}}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{a^2}{b^3 x}+\frac{3 a^3}{2 b^4 x^{2/3}}-\frac{3 a^4}{b^5 \sqrt [3]{x}}+\frac{3 a^5 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac{a^5 \log (x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0782991, size = 84, normalized size = 0.9 \[ -\frac{\frac{b \left (20 a^2 b^2 x^{2/3}-30 a^3 b x+60 a^4 x^{4/3}-15 a b^3 \sqrt [3]{x}+12 b^4\right )}{x^{5/3}}-60 a^5 \log \left (a \sqrt [3]{x}+b\right )+20 a^5 \log (x)}{20 b^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 78, normalized size = 0.8 \begin{align*} -{\frac{3}{5\,b}{x}^{-{\frac{5}{3}}}}+{\frac{3\,a}{4\,{b}^{2}}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{3\,{a}^{3}}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{a}^{4}}{{b}^{5}\sqrt [3]{x}}}+3\,{\frac{{a}^{5}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}-{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.97827, size = 128, normalized size = 1.38 \begin{align*} \frac{3 \, a^{5} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5}}{5 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a}{4 \, b^{6}} - \frac{10 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{2}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{3}}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{4}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54642, size = 211, normalized size = 2.27 \begin{align*} \frac{60 \, a^{5} x^{2} \log \left (a x^{\frac{1}{3}} + b\right ) - 60 \, a^{5} x^{2} \log \left (x^{\frac{1}{3}}\right ) - 20 \, a^{2} b^{3} x - 15 \,{\left (4 \, a^{4} b x - a b^{4}\right )} x^{\frac{2}{3}} + 6 \,{\left (5 \, a^{3} b^{2} x - 2 \, b^{5}\right )} x^{\frac{1}{3}}}{20 \, b^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.9519, size = 116, normalized size = 1.25 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{5 b x^{\frac{5}{3}}} & \text{for}\: a = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{a^{5} \log{\left (x \right )}}{b^{6}} + \frac{3 a^{5} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{6}} - \frac{3 a^{4}}{b^{5} \sqrt [3]{x}} + \frac{3 a^{3}}{2 b^{4} x^{\frac{2}{3}}} - \frac{a^{2}}{b^{3} x} + \frac{3 a}{4 b^{2} x^{\frac{4}{3}}} - \frac{3}{5 b x^{\frac{5}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23034, size = 109, normalized size = 1.17 \begin{align*} \frac{3 \, a^{5} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} - \frac{a^{5} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{\frac{4}{3}} - 30 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 15 \, a b^{4} x^{\frac{1}{3}} + 12 \, b^{5}}{20 \, b^{6} x^{\frac{5}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]