Optimal. Leaf size=44 \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (\frac{b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)-\frac{3}{2} b x^{2/3}+x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0238145, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 44} \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (\frac{b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)-\frac{3}{2} b x^{2/3}+x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 190
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{1+\frac{b}{\sqrt [3]{x}}} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{1}{x^4 (1+b x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{b}{x^3}+\frac{b^2}{x^2}-\frac{b^3}{x}+\frac{b^4}{1+b x}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=3 b^2 \sqrt [3]{x}-\frac{3}{2} b x^{2/3}+x-3 b^3 \log \left (1+\frac{b}{\sqrt [3]{x}}\right )-b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0175819, size = 35, normalized size = 0.8 \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (b+\sqrt [3]{x}\right )-\frac{3}{2} b x^{2/3}+x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 28, normalized size = 0.6 \begin{align*} x-{\frac{3\,b}{2}{x}^{{\frac{2}{3}}}}+3\,{b}^{2}\sqrt [3]{x}-3\,{b}^{3}\ln \left ( \sqrt [3]{x}+b \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06248, size = 54, normalized size = 1.23 \begin{align*} -b^{3} \log \left (x\right ) - 3 \, b^{3} \log \left (\frac{b}{x^{\frac{1}{3}}} + 1\right ) + \frac{1}{2} \,{\left (\frac{6 \, b^{2}}{x^{\frac{2}{3}}} - \frac{3 \, b}{x^{\frac{1}{3}}} + 2\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52849, size = 82, normalized size = 1.86 \begin{align*} -3 \, b^{3} \log \left (b + x^{\frac{1}{3}}\right ) + 3 \, b^{2} x^{\frac{1}{3}} - \frac{3}{2} \, b x^{\frac{2}{3}} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.172669, size = 34, normalized size = 0.77 \begin{align*} - 3 b^{3} \log{\left (b + \sqrt [3]{x} \right )} + 3 b^{2} \sqrt [3]{x} - \frac{3 b x^{\frac{2}{3}}}{2} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19857, size = 38, normalized size = 0.86 \begin{align*} -3 \, b^{3} \log \left ({\left | b + x^{\frac{1}{3}} \right |}\right ) + 3 \, b^{2} x^{\frac{1}{3}} - \frac{3}{2} \, b x^{\frac{2}{3}} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]