Optimal. Leaf size=26 \[ \frac{\left (b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0789861, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2034, 629} \[ \frac{\left (b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2034
Rule 629
Rubi steps
\begin{align*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (b x^n+c x^{2 n}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int (b+2 c x) \left (b x+c x^2\right )^p \, dx,x,x^n\right )}{n}\\ &=\frac{\left (b x^n+c x^{2 n}\right )^{1+p}}{n (1+p)}\\ \end{align*}
Mathematica [C] time = 0.130146, size = 111, normalized size = 4.27 \[ \frac{x^{-n p} \left (x^n \left (b+c x^n\right )\right )^p \left (\frac{c x^n}{b}+1\right )^{-p} \left (b (p+2) x^{n (p+1)} \, _2F_1\left (-p,p+1;p+2;-\frac{c x^n}{b}\right )+2 c (p+1) x^{n (p+2)} \, _2F_1\left (-p,p+2;p+3;-\frac{c x^n}{b}\right )\right )}{n (p+1) (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.098, size = 155, normalized size = 6. \begin{align*}{\frac{{x}^{n} \left ( b+c{x}^{n} \right ) }{n \left ( 1+p \right ) }{{\rm e}^{-{\frac{p \left ( i\pi \, \left ({\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ) \right ) ^{2}{\it csgn} \left ( i \left ( b+c{x}^{n} \right ) \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( i \left ( b+c{x}^{n} \right ) \right ) -2\,\ln \left ({x}^{n} \right ) -2\,\ln \left ( b+c{x}^{n} \right ) \right ) }{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.32212, size = 54, normalized size = 2.08 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.29047, size = 72, normalized size = 2.77 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{n p + n} \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 [A] time = 1.14458, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )}^{p + 1}}{n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]