\(\int (b x)^m (c-b c x)^n \, dx\) [751]

Optimal result

Integrand size = 15, antiderivative size = 40 \[ \int (b x)^m (c-b c x)^n \, dx=-\frac {(c-b c x)^{1+n} \operatorname {Hypergeometric2F1}(-m,1+n,2+n,1-b x)}{b c (1+n)} \]

[Out]

-(-b*c*x+c)^(1+n)*hypergeom([-m, 1+n],[2+n],-b*x+1)/b/c/(1+n)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {67} \[ \int (b x)^m (c-b c x)^n \, dx=-\frac {(c-b c x)^{n+1} \operatorname {Hypergeometric2F1}(-m,n+1,n+2,1-b x)}{b c (n+1)} \]

[In]

Int[(b*x)^m*(c - b*c*x)^n,x]

[Out]

-(((c - b*c*x)^(1 + n)*Hypergeometric2F1[-m, 1 + n, 2 + n, 1 - b*x])/(b*c*(1 + n)))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10 \[ \int (b x)^m (c-b c x)^n \, dx=\frac {x (b x)^m (1-b x)^{-n} (c-b c x)^n \operatorname {Hypergeometric2F1}(1+m,-n,2+m,b x)}{1+m} \]

[In]

Integrate[(b*x)^m*(c - b*c*x)^n,x]

[Out]

(x*(b*x)^m*(c - b*c*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, b*x])/((1 + m)*(1 - b*x)^n)

Maple [F]

\[\int \left (b x \right )^{m} \left (-b c x +c \right )^{n}d x\]

[In]

int((b*x)^m*(-b*c*x+c)^n,x)

[Out]

int((b*x)^m*(-b*c*x+c)^n,x)

Fricas [F]

\[ \int (b x)^m (c-b c x)^n \, dx=\int { {\left (-b c x + c\right )}^{n} \left (b x\right )^{m} \,d x } \]

[In]

integrate((b*x)^m*(-b*c*x+c)^n,x, algorithm="fricas")

[Out]

integral((-b*c*x + c)^n*(b*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.57 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int (b x)^m (c-b c x)^n \, dx=\frac {b^{m} c^{n} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {b x e^{2 i \pi }} \right )}}{\Gamma \left (m + 2\right )} \]

[In]

integrate((b*x)**m*(-b*c*x+c)**n,x)

[Out]

b**m*c**n*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), b*x*exp_polar(2*I*pi))/gamma(m + 2)

Maxima [F]

\[ \int (b x)^m (c-b c x)^n \, dx=\int { {\left (-b c x + c\right )}^{n} \left (b x\right )^{m} \,d x } \]

[In]

integrate((b*x)^m*(-b*c*x+c)^n,x, algorithm="maxima")

[Out]

integrate((-b*c*x + c)^n*(b*x)^m, x)

Giac [F]

\[ \int (b x)^m (c-b c x)^n \, dx=\int { {\left (-b c x + c\right )}^{n} \left (b x\right )^{m} \,d x } \]

[In]

integrate((b*x)^m*(-b*c*x+c)^n,x, algorithm="giac")

[Out]

integrate((-b*c*x + c)^n*(b*x)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (b x)^m (c-b c x)^n \, dx=\int {\left (b\,x\right )}^m\,{\left (c-b\,c\,x\right )}^n \,d x \]

[In]

int((b*x)^m*(c - b*c*x)^n,x)

[Out]

int((b*x)^m*(c - b*c*x)^n, x)