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

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

[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 = 0.0770858, 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 \[ -\frac{4 a^2 (a c-b c x)^{n+1}}{b c (n+1)}-\frac{(a c-b c x)^{n+3}}{b c^3 (n+3)}+\frac{4 a (a c-b c x)^{n+2}}{b c^2 (n+2)} \]

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))

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Rubi in Sympy [A]  time = 18.998, size = 66, normalized size = 0.8 \[ - \frac{4 a^{2} \left (a c - b c x\right )^{n + 1}}{b c \left (n + 1\right )} + \frac{4 a \left (a c - b c x\right )^{n + 2}}{b c^{2} \left (n + 2\right )} - \frac{\left (a c - b c x\right )^{n + 3}}{b c^{3} \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(-b*c*x+a*c)**n,x)

[Out]

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

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Mathematica [A]  time = 0.0582491, 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 = 0.01, size = 103, normalized size = 1.2 \[ -{\frac{ \left ({b}^{2}{n}^{2}{x}^{2}+2\,ab{n}^{2}x+3\,{b}^{2}n{x}^{2}+{a}^{2}{n}^{2}+10\,abnx+2\,{b}^{2}{x}^{2}+7\,{a}^{2}n+8\,abx+14\,{a}^{2} \right ) \left ( -bx+a \right ) \left ( -bcx+ac \right ) ^{n}}{b \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.240046, size = 173, normalized size = 2.08 \[ -\frac{{\left (a^{3} n^{2} + 7 \, a^{3} n -{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} + 14 \, a^{3} -{\left (a b^{2} n^{2} + 7 \, a b^{2} n + 6 \, a b^{2}\right )} x^{2} +{\left (a^{2} b n^{2} + 3 \, a^{2} b n - 6 \, a^{2} b\right )} x\right )}{\left (-b c x + a c\right )}^{n}}{b n^{3} + 6 \, b n^{2} + 11 \, b n + 6 \, b} \]

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 = 3.21102, size = 819, normalized size = 9.87 \[ \text{result too large to display} \]

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**2/(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)
+ 4*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), 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/XCAS [A]  time = 0.222816, size = 378, normalized size = 4.55 \[ \frac{b^{3} n^{2} x^{3} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} + a b^{2} n^{2} x^{2} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} + 3 \, b^{3} n x^{3} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} - a^{2} b n^{2} x e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} + 7 \, a b^{2} n x^{2} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} + 2 \, b^{3} x^{3} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} - a^{3} n^{2} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} - 3 \, a^{2} b n x e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} + 6 \, a b^{2} x^{2} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} - 7 \, a^{3} n e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} + 6 \, a^{2} b x e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )} - 14 \, a^{3} e^{\left (n{\rm ln}\left (-b c x + a c\right )\right )}}{b n^{3} + 6 \, b n^{2} + 11 \, b n + 6 \, b} \]

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