Optimal. Leaf size=40 \[ -\frac{(c-b c x)^{n+1} \, _2F_1(-m,n+1;n+2;1-b x)}{b c (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0091687, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {65} \[ -\frac{(c-b c x)^{n+1} \, _2F_1(-m,n+1;n+2;1-b x)}{b c (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rubi steps
\begin{align*} \int (b x)^m (c-b c x)^n \, dx &=-\frac{(c-b c x)^{1+n} \, _2F_1(-m,1+n;2+n;1-b x)}{b c (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0102818, size = 44, normalized size = 1.1 \[ \frac{x (b x)^m (1-b x)^{-n} (c-b c x)^n \, _2F_1(m+1,-n;m+2;b x)}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{m} \left ( -bcx+c \right ) ^{n}\, 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 (-b c x + c\right )}^{n} \left (b x\right )^{m}\,{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 (-b c x + c\right )}^{n} \left (b x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 2.1403, size = 37, normalized size = 0.92 \begin{align*} \frac{b^{m} c^{n} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{b x e^{2 i \pi }} \right )}}{\Gamma \left (m + 2\right )} \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 (-b c x + c\right )}^{n} \left (b x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]