Optimal. Leaf size=36 \[ \frac {x^{-((1-n) (p+1))} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2036} \[ \frac {x^{-(1-n) (p+1)} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2036
Rubi steps
\begin {align*} \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx &=\frac {x^{-(1-n) (1+p)} \left (b x+c x^{1+n}\right )^{1+p}}{n (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 108, normalized size = 3.00 \[ \frac {x^{-p} \left (x \left (b+c x^n\right )\right )^p \left (\frac {c x^n}{b}+1\right )^{-p} \left (b (p+2) x^{n (p+1)} \, _2F_1\left (-p,p+1;p+2;-\frac {c x^n}{b}\right )+2 c (p+1) x^{n (p+2)} \, _2F_1\left (-p,p+2;p+3;-\frac {c x^n}{b}\right )\right )}{n (p+1) (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 42, normalized size = 1.17 \[ \frac {{\left (b x + c x^{n + 1}\right )} {\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} p + n - 1}}{n p + n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (2 \, c x^{n} + b\right )} {\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} {\left (p + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (2 c \,x^{n}+b \right ) x^{\left (n -1\right ) \left (p +1\right )} \left (b x +c \,x^{n +1}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.07, size = 39, normalized size = 1.08 \[ \frac {{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (n p \log \relax (x) + p \log \left (c x^{n} + b\right )\right )}}{n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int x^{\left (n-1\right )\,\left (p+1\right )}\,{\left (b\,x+c\,x^{n+1}\right )}^p\,\left (b+2\,c\,x^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________