Optimal. Leaf size=49 \[ \frac{x^2 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+2;p+3;-\frac{c x}{b}\right )}{p+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02233, antiderivative size = 83, normalized size of antiderivative = 1.69, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {640, 624} \[ \frac{\left (b x+c x^2\right )^{p+1} \left (-\frac{c x}{b}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{2 c (p+1)}+\frac{\left (b x+c x^2\right )^{p+1}}{2 c (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 640
Rule 624
Rubi steps
\begin{align*} \int x \left (b x+c x^2\right )^p \, dx &=\frac{\left (b x+c x^2\right )^{1+p}}{2 c (1+p)}-\frac{b \int \left (b x+c x^2\right )^p \, dx}{2 c}\\ &=\frac{\left (b x+c x^2\right )^{1+p}}{2 c (1+p)}+\frac{\left (-\frac{c x}{b}\right )^{-1-p} \left (b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{b+c x}{b}\right )}{2 c (1+p)}\\ \end{align*}
Mathematica [A] time = 0.007747, size = 47, normalized size = 0.96 \[ \frac{x^2 (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+2;p+3;-\frac{c x}{b}\right )}{p+2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.413, size = 0, normalized size = 0. \begin{align*} \int x \left ( c{x}^{2}+bx \right ) ^{p}\, dx \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{\left (c x^{2} + b x\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (x \left (b + c x\right )\right )^{p}\, dx \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{\left (c x^{2} + b x\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]