Optimal. Leaf size=85 \[ \frac{b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0261973, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1355, 14} \[ \frac{b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1355
Rule 14
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \frac{a b+b^2 x^n}{x} \, dx}{a b+b^2 x^n}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \left (\frac{a b}{x}+b^2 x^{-1+n}\right ) \, dx}{a b+b^2 x^n}\\ &=\frac{b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \log (x)}{a+b x^n}\\ \end{align*}
Mathematica [A] time = 0.015956, size = 37, normalized size = 0.44 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (a n \log (x)+b x^n\right )}{n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 54, normalized size = 0.6 \begin{align*}{\frac{a\ln \left ( x \right ) }{a+b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{n}}{ \left ( a+b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00288, size = 18, normalized size = 0.21 \begin{align*} a \log \left (x\right ) + \frac{b x^{n}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62657, size = 32, normalized size = 0.38 \begin{align*} \frac{a n \log \left (x\right ) + b x^{n}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\left (a + b x^{n}\right )^{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]