Optimal. Leaf size=43 \[ \frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{n+1}{2 n}}}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0134285, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {1343, 191} \[ \frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{n+1}{2 n}}}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1343
Rule 191
Rubi steps
\begin{align*} \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{1+n}{2 n}} \, dx &=\left (\left (2 a b+2 b^2 x^n\right )^{\frac{1+n}{n}} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{1+n}{2 n}}\right ) \int \left (2 a b+2 b^2 x^n\right )^{-\frac{1+n}{n}} \, dx\\ &=\frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{1+n}{2 n}}}{a}\\ \end{align*}
Mathematica [A] time = 0.0609589, size = 32, normalized size = 0.74 \[ \frac{x \left (a+b x^n\right ) \left (\left (a+b x^n\right )^2\right )^{-\frac{n+1}{2 n}}}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 51, normalized size = 1.2 \begin{align*}{ \left ( x+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}} \right ) \left ({{\rm e}^{{\frac{ \left ( 1+n \right ) \ln \left ({a}^{2}+2\,ab{{\rm e}^{n\ln \left ( x \right ) }}+{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \right ) }{2\,n}}}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{n + 1}{2 \, n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64416, size = 93, normalized size = 2.16 \begin{align*} \frac{b x x^{n} + a x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{n + 1}{2 \, n}} a} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{n + 1}{2 \, n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]