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

Optimal. Leaf size=40 \[ -\frac {(c-b c x)^{1+n} \, _2F_1(-m,1+n;2+n;1-b x)}{b c (1+n)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {67} \begin {gather*} -\frac {(c-b c x)^{n+1} \, _2F_1(-m,n+1;n+2;1-b x)}{b c (n+1)} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.05, size = 44, normalized size = 1.10 \begin {gather*} \frac {x (b x)^m (1-b x)^{-n} (c-b c x)^n \, _2F_1(1+m,-n;2+m;b x)}{1+m} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 2.86, size = 36, normalized size = 0.90 \begin {gather*} \frac {b^m c^n x^{1+m} \text {hyper}\left [\left \{-n,1+m\right \},\left \{2+m\right \},b x \text {exp\_polar}\left [2 I \text {Pi}\right ]\right ]}{1+m} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(b*x)^m*(c - b*c*x)^n,x]')

[Out]

b ^ m c ^ n x ^ (1 + m) hyper[{-n, 1 + m}, {2 + m}, b x exp_polar[2 I Pi]] / (1 + m)

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \left (-b c x +c \right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 1.27, size = 37, normalized size = 0.92 \begin {gather*} \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 {gather*}

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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (b\,x\right )}^m\,{\left (c-b\,c\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(c - b*c*x)^n,x)

[Out]

int((b*x)^m*(c - b*c*x)^n, x)

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