3.1230 \(\int (a+b x) (a c-b c x)^n \, dx\)

Optimal. Leaf size=53 \[ \frac{(a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac{2 a (a c-b c x)^{n+1}}{b c (n+1)} \]

[Out]

(-2*a*(a*c - b*c*x)^(1 + n))/(b*c*(1 + n)) + (a*c - b*c*x)^(2 + n)/(b*c^2*(2 + n))

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Rubi [A]  time = 0.0170643, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{(a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac{2 a (a c-b c x)^{n+1}}{b c (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x)^n,x]

[Out]

(-2*a*(a*c - b*c*x)^(1 + n))/(b*c*(1 + n)) + (a*c - b*c*x)^(2 + n)/(b*c^2*(2 + n))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (a c-b c x)^n \, dx &=\int \left (2 a (a c-b c x)^n-\frac{(a c-b c x)^{1+n}}{c}\right ) \, dx\\ &=-\frac{2 a (a c-b c x)^{1+n}}{b c (1+n)}+\frac{(a c-b c x)^{2+n}}{b c^2 (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.0210716, size = 43, normalized size = 0.81 \[ \frac{(b x-a) (a (n+3)+b (n+1) x) (c (a-b x))^n}{b (n+1) (n+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c - b*c*x)^n,x]

[Out]

((c*(a - b*x))^n*(-a + b*x)*(a*(3 + n) + b*(1 + n)*x))/(b*(1 + n)*(2 + n))

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Maple [A]  time = 0.002, size = 47, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -bcx+ac \right ) ^{n} \left ( bnx+an+bx+3\,a \right ) \left ( -bx+a \right ) }{b \left ({n}^{2}+3\,n+2 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b*c*x+a*c)^n*(b*n*x+a*n+b*x+3*a)*(-b*x+a)/b/(n^2+3*n+2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.13825, size = 117, normalized size = 2.21 \begin{align*} -\frac{{\left (a^{2} n - 2 \, a b x -{\left (b^{2} n + b^{2}\right )} x^{2} + 3 \, a^{2}\right )}{\left (-b c x + a c\right )}^{n}}{b n^{2} + 3 \, b n + 2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*n - 2*a*b*x - (b^2*n + b^2)*x^2 + 3*a^2)*(-b*c*x + a*c)^n/(b*n^2 + 3*b*n + 2*b)

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Sympy [A]  time = 0.615571, size = 245, normalized size = 4.62 \begin{align*} \begin{cases} a x \left (a c\right )^{n} & \text{for}\: b = 0 \\- \frac{a \log{\left (- \frac{a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} - \frac{2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac{b x \log{\left (- \frac{a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} & \text{for}\: n = -2 \\- \frac{2 a \log{\left (- \frac{a}{b} + x \right )}}{b c} - \frac{x}{c} & \text{for}\: n = -1 \\- \frac{a^{2} n \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} - \frac{3 a^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac{2 a b x \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac{b^{2} n x^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac{b^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x*(a*c)**n, Eq(b, 0)), (-a*log(-a/b + x)/(-a*b*c**2 + b**2*c**2*x) - 2*a/(-a*b*c**2 + b**2*c**2*x
) + b*x*log(-a/b + x)/(-a*b*c**2 + b**2*c**2*x), Eq(n, -2)), (-2*a*log(-a/b + x)/(b*c) - x/c, Eq(n, -1)), (-a*
*2*n*(a*c - b*c*x)**n/(b*n**2 + 3*b*n + 2*b) - 3*a**2*(a*c - b*c*x)**n/(b*n**2 + 3*b*n + 2*b) + 2*a*b*x*(a*c -
b*c*x)**n/(b*n**2 + 3*b*n + 2*b) + b**2*n*x**2*(a*c - b*c*x)**n/(b*n**2 + 3*b*n + 2*b) + b**2*x**2*(a*c - b*c
*x)**n/(b*n**2 + 3*b*n + 2*b), True))

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Giac [A]  time = 1.05295, size = 139, normalized size = 2.62 \begin{align*} \frac{{\left (-b c x + a c\right )}^{n} b^{2} n x^{2} +{\left (-b c x + a c\right )}^{n} b^{2} x^{2} -{\left (-b c x + a c\right )}^{n} a^{2} n + 2 \,{\left (-b c x + a c\right )}^{n} a b x - 3 \,{\left (-b c x + a c\right )}^{n} a^{2}}{b n^{2} + 3 \, b n + 2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-b*c*x + a*c)^n*b^2*n*x^2 + (-b*c*x + a*c)^n*b^2*x^2 - (-b*c*x + a*c)^n*a^2*n + 2*(-b*c*x + a*c)^n*a*b*x - 3
*(-b*c*x + a*c)^n*a^2)/(b*n^2 + 3*b*n + 2*b)