Optimal. Leaf size=103 \[ -\frac{12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{1}{b^3 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0684681, antiderivative size = 103, 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} \[ -\frac{12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{1}{b^3 x} \]
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 )^3 x^3} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^3 x^2} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^4 (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^4}-\frac{3 a}{b^4 x^3}+\frac{6 a^2}{b^5 x^2}-\frac{10 a^3}{b^6 x}+\frac{a^4}{b^4 (b+a x)^3}+\frac{4 a^4}{b^5 (b+a x)^2}+\frac{10 a^4}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^3}{2 b^4 \left (b+a \sqrt [3]{x}\right )^2}-\frac{12 a^3}{b^5 \left (b+a \sqrt [3]{x}\right )}-\frac{1}{b^3 x}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{30 a^3 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.142595, size = 93, normalized size = 0.9 \[ -\frac{\frac{b \left (20 a^2 b^2 x^{2/3}+90 a^3 b x+60 a^4 x^{4/3}-5 a b^3 \sqrt [3]{x}+2 b^4\right )}{x \left (a \sqrt [3]{x}+b\right )^2}-60 a^3 \log \left (a \sqrt [3]{x}+b\right )+20 a^3 \log (x)}{2 b^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 90, normalized size = 0.9 \begin{align*} -{\frac{3\,{a}^{3}}{2\,{b}^{4}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-12\,{\frac{{a}^{3}}{{b}^{5} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{1}{{b}^{3}x}}+{\frac{9\,a}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}-18\,{\frac{{a}^{2}}{{b}^{5}\sqrt [3]{x}}}+30\,{\frac{{a}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}-10\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.982002, size = 128, normalized size = 1.24 \begin{align*} \frac{30 \, a^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a}{2 \, b^{6}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{2}}{b^{6}} + \frac{15 \, a^{4}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{6}} - \frac{3 \, a^{5}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.57148, size = 421, normalized size = 4.09 \begin{align*} -\frac{20 \, a^{6} b^{3} x^{2} + 31 \, a^{3} b^{6} x + 2 \, b^{9} - 60 \,{\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 60 \,{\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (x^{\frac{1}{3}}\right ) + 3 \,{\left (20 \, a^{8} b x^{2} + 35 \, a^{5} b^{4} x + 12 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} - 3 \,{\left (10 \, a^{7} b^{2} x^{2} + 16 \, a^{4} b^{5} x + 3 \, a b^{8}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{6} b^{6} x^{3} + 2 \, a^{3} b^{9} x^{2} + b^{12} x\right )}} \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 = 1.17221, size = 122, normalized size = 1.18 \begin{align*} \frac{30 \, a^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} - \frac{10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{\frac{4}{3}} + 90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 5 \, a b^{4} x^{\frac{1}{3}} + 2 \, b^{5}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{6} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]