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

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

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Rubi [A]  time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 0.93 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^2 (a c-b c x)^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.25, size = 128, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.15, size = 256, normalized size = 3.08 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 103, normalized size = 1.24 \begin {gather*} -\frac {\left (-b x +a \right ) \left (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}\right ) \left (-b c x +a c \right )^{n}}{\left (n^{3}+6 n^{2}+11 n +6\right ) b} \end {gather*}

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 [B]  time = 1.53, size = 167, normalized size = 2.01 \begin {gather*} \frac {2 \, {\left (b^{2} c^{n} {\left (n + 1\right )} x^{2} - a b c^{n} n x - a^{2} c^{n}\right )} {\left (-b x + a\right )}^{n} a}{{\left (n^{2} + 3 \, n + 2\right )} b} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{n} x^{3} - {\left (n^{2} + n\right )} a b^{2} c^{n} x^{2} - 2 \, a^{2} b c^{n} n x - 2 \, a^{3} c^{n}\right )} {\left (-b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b} - \frac {{\left (-b c x + a c\right )}^{n + 1} a^{2}}{b c {\left (n + 1\right )}} \end {gather*}

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]

2*(b^2*c^n*(n + 1)*x^2 - a*b*c^n*n*x - a^2*c^n)*(-b*x + a)^n*a/((n^2 + 3*n + 2)*b) + ((n^2 + 3*n + 2)*b^3*c^n*
x^3 - (n^2 + n)*a*b^2*c^n*x^2 - 2*a^2*b*c^n*n*x - 2*a^3*c^n)*(-b*x + a)^n/((n^3 + 6*n^2 + 11*n + 6)*b) - (-b*c
*x + a*c)^(n + 1)*a^2/(b*c*(n + 1))

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mupad [B]  time = 0.49, size = 133, normalized size = 1.60 \begin {gather*} -{\left (a\,c-b\,c\,x\right )}^n\,\left (\frac {a^2\,x\,\left (n^2+3\,n-6\right )}{n^3+6\,n^2+11\,n+6}+\frac {a^3\,\left (n^2+7\,n+14\right )}{b\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {b^2\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}-\frac {a\,b\,x^2\,\left (n^2+7\,n+6\right )}{n^3+6\,n^2+11\,n+6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^n*(a + b*x)^2,x)

[Out]

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

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sympy [A]  time = 1.30, size = 819, normalized size = 9.87 \begin {gather*} \begin {cases} a^{2} x \left (a c\right )^{n} & \text {for}\: b = 0 \\- \frac {a^{2} \log {\left (- \frac {a}{b} + x \right )}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac {2 a^{2}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} + \frac {2 a b x \log {\left (- \frac {a}{b} + x \right )}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} + \frac {4 a b x}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac {b^{2} x^{2} \log {\left (- \frac {a}{b} + x \right )}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} & \text {for}\: n = -3 \\- \frac {4 a^{2} \log {\left (- \frac {a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} - \frac {5 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac {4 a b x \log {\left (- \frac {a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} + \frac {b^{2} x^{2}}{- a b c^{2} + b^{2} c^{2} x} & \text {for}\: n = -2 \\- \frac {4 a^{2} \log {\left (- \frac {a}{b} + x \right )}}{b c} - \frac {3 a x}{c} - \frac {b x^{2}}{2 c} & \text {for}\: n = -1 \\- \frac {a^{3} n^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {7 a^{3} n \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {14 a^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {a^{2} b n^{2} x \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {3 a^{2} b n x \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {6 a^{2} b x \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {a b^{2} n^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {7 a b^{2} n x^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {6 a b^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {b^{3} n^{2} x^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {3 b^{3} n x^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {2 b^{3} x^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} & \text {otherwise} \end {cases} \end {gather*}

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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