3.1229 \(\int (a+b x)^2 (a c-b c x)^n \, dx\)

Optimal. Leaf size=83 \[ -\frac{4 a^2 (a c-b c x)^{n+1}}{b c (n+1)}+\frac{4 a (a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac{(a c-b c x)^{n+3}}{b c^3 (n+3)} \]

[Out]

(-4*a^2*(a*c - b*c*x)^(1 + n))/(b*c*(1 + n)) + (4*a*(a*c - b*c*x)^(2 + n))/(b*c^2*(2 + n)) - (a*c - b*c*x)^(3
+ n)/(b*c^3*(3 + n))

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Rubi [A]  time = 0.0285228, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ -\frac{4 a^2 (a c-b c x)^{n+1}}{b c (n+1)}+\frac{4 a (a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac{(a c-b c x)^{n+3}}{b c^3 (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(a*c - b*c*x)^n,x]

[Out]

(-4*a^2*(a*c - b*c*x)^(1 + n))/(b*c*(1 + n)) + (4*a*(a*c - b*c*x)^(2 + n))/(b*c^2*(2 + n)) - (a*c - b*c*x)^(3
+ n)/(b*c^3*(3 + 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)^2 (a c-b c x)^n \, dx &=\int \left (4 a^2 (a c-b c x)^n-\frac{4 a (a c-b c x)^{1+n}}{c}+\frac{(a c-b c x)^{2+n}}{c^2}\right ) \, dx\\ &=-\frac{4 a^2 (a c-b c x)^{1+n}}{b c (1+n)}+\frac{4 a (a c-b c x)^{2+n}}{b c^2 (2+n)}-\frac{(a c-b c x)^{3+n}}{b c^3 (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.0382107, size = 77, normalized size = 0.93 \[ \frac{(b x-a) \left (a^2 \left (n^2+7 n+14\right )+2 a b \left (n^2+5 n+4\right ) x+b^2 \left (n^2+3 n+2\right ) x^2\right ) (c (a-b x))^n}{b (n+1) (n+2) (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(a*c - b*c*x)^n,x]

[Out]

((c*(a - b*x))^n*(-a + b*x)*(a^2*(14 + 7*n + n^2) + 2*a*b*(4 + 5*n + n^2)*x + b^2*(2 + 3*n + n^2)*x^2))/(b*(1
+ n)*(2 + n)*(3 + n))

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Maple [A]  time = 0.006, size = 103, normalized size = 1.2 \begin{align*} -{\frac{ \left ({b}^{2}{n}^{2}{x}^{2}+2\,ab{n}^{2}x+3\,{b}^{2}n{x}^{2}+{a}^{2}{n}^{2}+10\,abnx+2\,{b}^{2}{x}^{2}+7\,{a}^{2}n+8\,abx+14\,{a}^{2} \right ) \left ( -bx+a \right ) \left ( -bcx+ac \right ) ^{n}}{b \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(-b*c*x+a*c)^n,x)

[Out]

-(-b*x+a)*(b^2*n^2*x^2+2*a*b*n^2*x+3*b^2*n*x^2+a^2*n^2+10*a*b*n*x+2*b^2*x^2+7*a^2*n+8*a*b*x+14*a^2)*(-b*c*x+a*
c)^n/b/(n^3+6*n^2+11*n+6)

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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)^2*(-b*c*x+a*c)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.24628, size = 261, normalized size = 3.14 \begin{align*} -\frac{{\left (a^{3} n^{2} + 7 \, a^{3} n -{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} + 14 \, a^{3} -{\left (a b^{2} n^{2} + 7 \, a b^{2} n + 6 \, a b^{2}\right )} x^{2} +{\left (a^{2} b n^{2} + 3 \, a^{2} b n - 6 \, a^{2} b\right )} x\right )}{\left (-b c x + a c\right )}^{n}}{b n^{3} + 6 \, b n^{2} + 11 \, b n + 6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(-b*c*x+a*c)^n,x, algorithm="fricas")

[Out]

-(a^3*n^2 + 7*a^3*n - (b^3*n^2 + 3*b^3*n + 2*b^3)*x^3 + 14*a^3 - (a*b^2*n^2 + 7*a*b^2*n + 6*a*b^2)*x^2 + (a^2*
b*n^2 + 3*a^2*b*n - 6*a^2*b)*x)*(-b*c*x + a*c)^n/(b*n^3 + 6*b*n^2 + 11*b*n + 6*b)

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Sympy [A]  time = 1.07409, size = 785, normalized size = 9.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(-b*c*x+a*c)**n,x)

[Out]

Piecewise((a**2*x*(a*c)**n, Eq(b, 0)), (-a**2*log(-a/b + x)/(a**2*b*c**3 - 2*a*b**2*c**3*x + b**3*c**3*x**2) +
2*a*b*x*log(-a/b + x)/(a**2*b*c**3 - 2*a*b**2*c**3*x + b**3*c**3*x**2) - b**2*x**2*log(-a/b + x)/(a**2*b*c**3
- 2*a*b**2*c**3*x + b**3*c**3*x**2) + 2*b**2*x**2/(a**2*b*c**3 - 2*a*b**2*c**3*x + b**3*c**3*x**2), Eq(n, -3)
), (-4*a**2*log(-a/b + x)/(-a*b*c**2 + b**2*c**2*x) - 5*a**2/(-a*b*c**2 + b**2*c**2*x) + 4*a*b*x*log(-a/b + x)
/(-a*b*c**2 + b**2*c**2*x) + b**2*x**2/(-a*b*c**2 + b**2*c**2*x), Eq(n, -2)), (-4*a**2*log(-a/b + x)/(b*c) - 3
*a*x/c - b*x**2/(2*c), Eq(n, -1)), (-a**3*n**2*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) - 7*a**3*n*
(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) - 14*a**3*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6
*b) - a**2*b*n**2*x*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) - 3*a**2*b*n*x*(a*c - b*c*x)**n/(b*n**
3 + 6*b*n**2 + 11*b*n + 6*b) + 6*a**2*b*x*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) + a*b**2*n**2*x*
*2*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) + 7*a*b**2*n*x**2*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 +
11*b*n + 6*b) + 6*a*b**2*x**2*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) + b**3*n**2*x**3*(a*c - b*c
*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b) + 3*b**3*n*x**3*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b)
+ 2*b**3*x**3*(a*c - b*c*x)**n/(b*n**3 + 6*b*n**2 + 11*b*n + 6*b), True))

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Giac [B]  time = 1.07298, size = 346, normalized size = 4.17 \begin{align*} \frac{{\left (-b c x + a c\right )}^{n} b^{3} n^{2} x^{3} +{\left (-b c x + a c\right )}^{n} a b^{2} n^{2} x^{2} + 3 \,{\left (-b c x + a c\right )}^{n} b^{3} n x^{3} -{\left (-b c x + a c\right )}^{n} a^{2} b n^{2} x + 7 \,{\left (-b c x + a c\right )}^{n} a b^{2} n x^{2} + 2 \,{\left (-b c x + a c\right )}^{n} b^{3} x^{3} -{\left (-b c x + a c\right )}^{n} a^{3} n^{2} - 3 \,{\left (-b c x + a c\right )}^{n} a^{2} b n x + 6 \,{\left (-b c x + a c\right )}^{n} a b^{2} x^{2} - 7 \,{\left (-b c x + a c\right )}^{n} a^{3} n + 6 \,{\left (-b c x + a c\right )}^{n} a^{2} b x - 14 \,{\left (-b c x + a c\right )}^{n} a^{3}}{b n^{3} + 6 \, b n^{2} + 11 \, b n + 6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(-b*c*x+a*c)^n,x, algorithm="giac")

[Out]

((-b*c*x + a*c)^n*b^3*n^2*x^3 + (-b*c*x + a*c)^n*a*b^2*n^2*x^2 + 3*(-b*c*x + a*c)^n*b^3*n*x^3 - (-b*c*x + a*c)
^n*a^2*b*n^2*x + 7*(-b*c*x + a*c)^n*a*b^2*n*x^2 + 2*(-b*c*x + a*c)^n*b^3*x^3 - (-b*c*x + a*c)^n*a^3*n^2 - 3*(-
b*c*x + a*c)^n*a^2*b*n*x + 6*(-b*c*x + a*c)^n*a*b^2*x^2 - 7*(-b*c*x + a*c)^n*a^3*n + 6*(-b*c*x + a*c)^n*a^2*b*
x - 14*(-b*c*x + a*c)^n*a^3)/(b*n^3 + 6*b*n^2 + 11*b*n + 6*b)