Optimal. Leaf size=26 \[ \frac {\left (b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 629
Rule 2034
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 = 0.13, 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]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 36, normalized size = 1.38 \[ \frac {{\left (c x^{2 \, n} + b x^{n}\right )} {\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{n p + n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.83, size = 26, normalized size = 1.00 \[ \frac {{\left (c x^{2 \, n} + b x^{n}\right )}^{p + 1}}{n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.11, size = 155, normalized size = 5.96 \[ \frac {\left (c \,x^{n}+b \right ) x^{n} {\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right )\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (c \,x^{n}+b \right )\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )^{3}+2 \ln \left (x^{n}\right )+2 \ln \left (c \,x^{n}+b \right )\right ) p}{2}}}{\left (p +1\right ) n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.77, size = 40, normalized size = 1.54 \[ \frac {{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.13, size = 34, normalized size = 1.31 \[ \frac {x^n\,\left (b+c\,x^n\right )\,{\left (b\,x^n+c\,x^{2\,n}\right )}^p}{n\,\left (p+1\right )} \]
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]
________________________________________________________________________________________