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 = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 14
Rule 1355
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 = 0.02, 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]
________________________________________________________________________________________
fricas [A] time = 0.89, size = 15, normalized size = 0.18 \[ \frac {a n \log \relax (x) + b x^{n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 54, normalized size = 0.64 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a \ln \relax (x )}{b \,x^{n}+a}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, b \,x^{n}}{\left (b \,x^{n}+a \right ) n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.80, size = 13, normalized size = 0.15 \[ a \log \relax (x) + \frac {b x^{n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (a + b x^{n}\right )^{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________