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 = 0.0291204, 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 \[ -\frac{(c-b c x)^{n+1} \, _2F_1(-m,n+1;n+2;1-b x)}{b c (n+1)} \]

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
)))

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Rubi in Sympy [A]  time = 5.57311, size = 29, normalized size = 0.72 \[ - \frac{\left (- b c x + c\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} - m, n + 1 \\ n + 2 \end{matrix}\middle |{- b x + 1} \right )}}{b c \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x)**m*(-b*c*x+c)**n,x)

[Out]

-(-b*c*x + c)**(n + 1)*hyper((-m, n + 1), (n + 2,), -b*x + 1)/(b*c*(n + 1))

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Mathematica [A]  time = 0.0385477, 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 = 0.174, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{m} \left ( -bcx+c \right ) ^{n}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*c*x + c)^n*(b*x)^m,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. \[{\rm integral}\left ({\left (-b c x + c\right )}^{n} \left (b x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*c*x + c)^n*(b*x)^m, x)

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Sympy [A]  time = 8.63452, size = 37, normalized size = 0.92 \[ \frac{b^{m} c^{n} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{b x e^{2 i \pi }} \right )}}{\Gamma \left (m + 2\right )} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-b c x + c\right )}^{n} \left (b x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*c*x + c)^n*(b*x)^m, x)