Optimal. Leaf size=45 \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0147312, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {190, 43} \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{1+\sqrt [5]{x}} \, dx &=5 \operatorname{Subst}\left (\int \frac{x^4}{1+x} \, dx,x,\sqrt [5]{x}\right )\\ &=5 \operatorname{Subst}\left (\int \left (-1+x-x^2+x^3+\frac{1}{1+x}\right ) \, dx,x,\sqrt [5]{x}\right )\\ &=-5 \sqrt [5]{x}+\frac{5 x^{2/5}}{2}-\frac{5 x^{3/5}}{3}+\frac{5 x^{4/5}}{4}+5 \log \left (1+\sqrt [5]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0163861, size = 45, normalized size = 1. \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.044, size = 79, normalized size = 1.8 \begin{align*} \ln \left ( 1+x \right ) +{\frac{5}{2}{x}^{{\frac{2}{5}}}}+4\,\ln \left ( 1+\sqrt [5]{x} \right ) -\ln \left ( -\sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) -\ln \left ( \sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) +{\frac{5}{4}{x}^{{\frac{4}{5}}}}-5\,\sqrt [5]{x}-{\frac{5}{3}{x}^{{\frac{3}{5}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04067, size = 57, normalized size = 1.27 \begin{align*} \frac{5}{4} \,{\left (x^{\frac{1}{5}} + 1\right )}^{4} - \frac{20}{3} \,{\left (x^{\frac{1}{5}} + 1\right )}^{3} + 15 \,{\left (x^{\frac{1}{5}} + 1\right )}^{2} - 20 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) - 20 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57499, size = 100, normalized size = 2.22 \begin{align*} \frac{5}{4} \, x^{\frac{4}{5}} - \frac{5}{3} \, x^{\frac{3}{5}} + \frac{5}{2} \, x^{\frac{2}{5}} - 5 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.9917, size = 41, normalized size = 0.91 \begin{align*} \frac{5 x^{\frac{4}{5}}}{4} - \frac{5 x^{\frac{3}{5}}}{3} + \frac{5 x^{\frac{2}{5}}}{2} - 5 \sqrt [5]{x} + 5 \log{\left (\sqrt [5]{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2002, size = 39, normalized size = 0.87 \begin{align*} \frac{5}{4} \, x^{\frac{4}{5}} - \frac{5}{3} \, x^{\frac{3}{5}} + \frac{5}{2} \, x^{\frac{2}{5}} - 5 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]