3.751 \(\int (b x)^m (c-b c x)^n \, dx\)

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]

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

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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]

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 65

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

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]

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)

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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]

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

[Out]

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

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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]

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

[Out]

integrate((-b*c*x + c)^n*(b*x)^m, x)

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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]

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

[Out]

integral((-b*c*x + c)^n*(b*x)^m, x)

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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]

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

[Out]

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

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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]

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

[Out]

integrate((-b*c*x + c)^n*(b*x)^m, x)