\(\int (a+b x)^2 (c+d x)^n \, dx\) [1853]

Optimal result

Integrand size = 15, antiderivative size = 78 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {(b c-a d)^2 (c+d x)^{1+n}}{d^3 (1+n)}-\frac {2 b (b c-a d) (c+d x)^{2+n}}{d^3 (2+n)}+\frac {b^2 (c+d x)^{3+n}}{d^3 (3+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {(b c-a d)^2 (c+d x)^{n+1}}{d^3 (n+1)}-\frac {2 b (b c-a d) (c+d x)^{n+2}}{d^3 (n+2)}+\frac {b^2 (c+d x)^{n+3}}{d^3 (n+3)} \]

[In]

Int[(a + b*x)^2*(c + d*x)^n,x]

[Out]

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

Rule 45

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {(c+d x)^{1+n} \left (\frac {(b c-a d)^2}{1+n}-\frac {2 b (b c-a d) (c+d x)}{2+n}+\frac {b^2 (c+d x)^2}{3+n}\right )}{d^3} \]

[In]

Integrate[(a + b*x)^2*(c + d*x)^n,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(78)=156\).

Time = 0.40 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.04

method	result	size
gosper	\(\frac {\left (d x +c \right )^{1+n} \left (b^{2} d^{2} n^{2} x^{2}+2 a b \,d^{2} n^{2} x +3 b^{2} d^{2} n \,x^{2}+a^{2} d^{2} n^{2}+8 a b \,d^{2} n x -2 b^{2} c d n x +2 d^{2} x^{2} b^{2}+5 a^{2} d^{2} n -2 a b c d n +6 x a b \,d^{2}-2 x \,b^{2} c d +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right )}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\)	\(159\)
norman	\(\frac {b^{2} x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{3+n}+\frac {c \left (a^{2} d^{2} n^{2}+5 a^{2} d^{2} n -2 a b c d n +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (a^{2} d^{2} n^{2}+2 a b c d \,n^{2}+5 a^{2} d^{2} n +6 a b c d n -2 b^{2} c^{2} n +6 a^{2} d^{2}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (2 a d n +b c n +6 a d \right ) b \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+5 n +6\right )}\)	\(224\)
risch	\(\frac {\left (b^{2} d^{3} n^{2} x^{3}+2 a b \,d^{3} n^{2} x^{2}+b^{2} c \,d^{2} n^{2} x^{2}+3 b^{2} d^{3} n \,x^{3}+a^{2} d^{3} n^{2} x +2 a b c \,d^{2} n^{2} x +8 a b \,d^{3} n \,x^{2}+b^{2} c \,d^{2} n \,x^{2}+2 b^{2} x^{3} d^{3}+a^{2} c \,d^{2} n^{2}+5 a^{2} d^{3} n x +6 a b c \,d^{2} n x +6 a b \,d^{3} x^{2}-2 b^{2} c^{2} d n x +5 a^{2} c \,d^{2} n +6 a^{2} d^{3} x -2 a b \,c^{2} d n +6 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 b^{2} c^{3}\right ) \left (d x +c \right )^{n}}{\left (2+n \right ) \left (3+n \right ) \left (1+n \right ) d^{3}}\)	\(242\)
parallelrisch	\(\frac {x^{3} \left (d x +c \right )^{n} b^{2} c \,d^{3} n^{2}+3 x^{3} \left (d x +c \right )^{n} b^{2} c \,d^{3} n +2 x^{2} \left (d x +c \right )^{n} a b c \,d^{3} n^{2}+x^{2} \left (d x +c \right )^{n} b^{2} c^{2} d^{2} n^{2}+2 x^{3} \left (d x +c \right )^{n} b^{2} c \,d^{3}+8 x^{2} \left (d x +c \right )^{n} a b c \,d^{3} n +x^{2} \left (d x +c \right )^{n} b^{2} c^{2} d^{2} n +x \left (d x +c \right )^{n} a^{2} c \,d^{3} n^{2}+2 x \left (d x +c \right )^{n} a b \,c^{2} d^{2} n^{2}+6 x^{2} \left (d x +c \right )^{n} a b c \,d^{3}+5 x \left (d x +c \right )^{n} a^{2} c \,d^{3} n +6 x \left (d x +c \right )^{n} a b \,c^{2} d^{2} n -2 x \left (d x +c \right )^{n} b^{2} c^{3} d n +\left (d x +c \right )^{n} a^{2} c^{2} d^{2} n^{2}+6 x \left (d x +c \right )^{n} a^{2} c \,d^{3}+5 \left (d x +c \right )^{n} a^{2} c^{2} d^{2} n -2 \left (d x +c \right )^{n} a b \,c^{3} d n +6 \left (d x +c \right )^{n} a^{2} c^{2} d^{2}-6 \left (d x +c \right )^{n} a b \,c^{3} d +2 \left (d x +c \right )^{n} b^{2} c^{4}}{\left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{3} c}\)	\(401\)

[In]

int((b*x+a)^2*(d*x+c)^n,x,method=_RETURNVERBOSE)

[Out]

1/d^3*(d*x+c)^(1+n)/(n^3+6*n^2+11*n+6)*(b^2*d^2*n^2*x^2+2*a*b*d^2*n^2*x+3*b^2*d^2*n*x^2+a^2*d^2*n^2+8*a*b*d^2*
n*x-2*b^2*c*d*n*x+2*b^2*d^2*x^2+5*a^2*d^2*n-2*a*b*c*d*n+6*a*b*d^2*x-2*b^2*c*d*x+6*a^2*d^2-6*a*b*c*d+2*b^2*c^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (78) = 156\).

Time = 0.23 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.04 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {{\left (a^{2} c d^{2} n^{2} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} + {\left (b^{2} d^{3} n^{2} + 3 \, b^{2} d^{3} n + 2 \, b^{2} d^{3}\right )} x^{3} + {\left (6 \, a b d^{3} + {\left (b^{2} c d^{2} + 2 \, a b d^{3}\right )} n^{2} + {\left (b^{2} c d^{2} + 8 \, a b d^{3}\right )} n\right )} x^{2} - {\left (2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} n + {\left (6 \, a^{2} d^{3} + {\left (2 \, a b c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (2 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \]

[In]

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

[Out]

(a^2*c*d^2*n^2 + 2*b^2*c^3 - 6*a*b*c^2*d + 6*a^2*c*d^2 + (b^2*d^3*n^2 + 3*b^2*d^3*n + 2*b^2*d^3)*x^3 + (6*a*b*
d^3 + (b^2*c*d^2 + 2*a*b*d^3)*n^2 + (b^2*c*d^2 + 8*a*b*d^3)*n)*x^2 - (2*a*b*c^2*d - 5*a^2*c*d^2)*n + (6*a^2*d^
3 + (2*a*b*c*d^2 + a^2*d^3)*n^2 - (2*b^2*c^2*d - 6*a*b*c*d^2 - 5*a^2*d^3)*n)*x)*(d*x + c)^n/(d^3*n^3 + 6*d^3*n
^2 + 11*d^3*n + 6*d^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1506 vs. \(2 (66) = 132\).

Time = 0.58 (sec) , antiderivative size = 1506, normalized size of antiderivative = 19.31 \[ \int (a+b x)^2 (c+d x)^n \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((c**n*(a**2*x + a*b*x**2 + b**2*x**3/3), Eq(d, 0)), (-a**2*d**2/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x
**2) - 2*a*b*c*d/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x**2) - 4*a*b*d**2*x/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x*
*2) + 2*b**2*c**2*log(c/d + x)/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x**2) + 3*b**2*c**2/(2*c**2*d**3 + 4*c*d**4*
x + 2*d**5*x**2) + 4*b**2*c*d*x*log(c/d + x)/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x**2) + 4*b**2*c*d*x/(2*c**2*d
**3 + 4*c*d**4*x + 2*d**5*x**2) + 2*b**2*d**2*x**2*log(c/d + x)/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x**2), Eq(n
, -3)), (-a**2*d**2/(c*d**3 + d**4*x) + 2*a*b*c*d*log(c/d + x)/(c*d**3 + d**4*x) + 2*a*b*c*d/(c*d**3 + d**4*x)
+ 2*a*b*d**2*x*log(c/d + x)/(c*d**3 + d**4*x) - 2*b**2*c**2*log(c/d + x)/(c*d**3 + d**4*x) - 2*b**2*c**2/(c*d
**3 + d**4*x) - 2*b**2*c*d*x*log(c/d + x)/(c*d**3 + d**4*x) + b**2*d**2*x**2/(c*d**3 + d**4*x), Eq(n, -2)), (a
**2*log(c/d + x)/d - 2*a*b*c*log(c/d + x)/d**2 + 2*a*b*x/d + b**2*c**2*log(c/d + x)/d**3 - b**2*c*x/d**2 + b**
2*x**2/(2*d), Eq(n, -1)), (a**2*c*d**2*n**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 5*a*
*2*c*d**2*n*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a**2*c*d**2*(c + d*x)**n/(d**3*n**
3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + a**2*d**3*n**2*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6
*d**3) + 5*a**2*d**3*n*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a**2*d**3*x*(c + d*x)
**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) - 2*a*b*c**2*d*n*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11
*d**3*n + 6*d**3) - 6*a*b*c**2*d*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*a*b*c*d**2*n*
*2*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a*b*c*d**2*n*x*(c + d*x)**n/(d**3*n**3 +
6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*a*b*d**3*n**2*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6
*d**3) + 8*a*b*d**3*n*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a*b*d**3*x**2*(c +
d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*b**2*c**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 1
1*d**3*n + 6*d**3) - 2*b**2*c**2*d*n*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b**2*c*d*
*2*n**2*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b**2*c*d**2*n*x**2*(c + d*x)**n/(d*
*3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b**2*d**3*n**2*x**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d
**3*n + 6*d**3) + 3*b**2*d**3*n*x**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*b**2*d**3
*x**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3), True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.77 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {2 \, {\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a b}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d x + c\right )}^{n + 1} a^{2}}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} b^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (78) = 156\).

Time = 0.31 (sec) , antiderivative size = 385, normalized size of antiderivative = 4.94 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {{\left (d x + c\right )}^{n} b^{2} d^{3} n^{2} x^{3} + {\left (d x + c\right )}^{n} b^{2} c d^{2} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} a b d^{3} n^{2} x^{2} + 3 \, {\left (d x + c\right )}^{n} b^{2} d^{3} n x^{3} + 2 \, {\left (d x + c\right )}^{n} a b c d^{2} n^{2} x + {\left (d x + c\right )}^{n} a^{2} d^{3} n^{2} x + {\left (d x + c\right )}^{n} b^{2} c d^{2} n x^{2} + 8 \, {\left (d x + c\right )}^{n} a b d^{3} n x^{2} + 2 \, {\left (d x + c\right )}^{n} b^{2} d^{3} x^{3} + {\left (d x + c\right )}^{n} a^{2} c d^{2} n^{2} - 2 \, {\left (d x + c\right )}^{n} b^{2} c^{2} d n x + 6 \, {\left (d x + c\right )}^{n} a b c d^{2} n x + 5 \, {\left (d x + c\right )}^{n} a^{2} d^{3} n x + 6 \, {\left (d x + c\right )}^{n} a b d^{3} x^{2} - 2 \, {\left (d x + c\right )}^{n} a b c^{2} d n + 5 \, {\left (d x + c\right )}^{n} a^{2} c d^{2} n + 6 \, {\left (d x + c\right )}^{n} a^{2} d^{3} x + 2 \, {\left (d x + c\right )}^{n} b^{2} c^{3} - 6 \, {\left (d x + c\right )}^{n} a b c^{2} d + 6 \, {\left (d x + c\right )}^{n} a^{2} c d^{2}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \]

[In]

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

[Out]

((d*x + c)^n*b^2*d^3*n^2*x^3 + (d*x + c)^n*b^2*c*d^2*n^2*x^2 + 2*(d*x + c)^n*a*b*d^3*n^2*x^2 + 3*(d*x + c)^n*b
^2*d^3*n*x^3 + 2*(d*x + c)^n*a*b*c*d^2*n^2*x + (d*x + c)^n*a^2*d^3*n^2*x + (d*x + c)^n*b^2*c*d^2*n*x^2 + 8*(d*
x + c)^n*a*b*d^3*n*x^2 + 2*(d*x + c)^n*b^2*d^3*x^3 + (d*x + c)^n*a^2*c*d^2*n^2 - 2*(d*x + c)^n*b^2*c^2*d*n*x +
6*(d*x + c)^n*a*b*c*d^2*n*x + 5*(d*x + c)^n*a^2*d^3*n*x + 6*(d*x + c)^n*a*b*d^3*x^2 - 2*(d*x + c)^n*a*b*c^2*d
*n + 5*(d*x + c)^n*a^2*c*d^2*n + 6*(d*x + c)^n*a^2*d^3*x + 2*(d*x + c)^n*b^2*c^3 - 6*(d*x + c)^n*a*b*c^2*d + 6
*(d*x + c)^n*a^2*c*d^2)/(d^3*n^3 + 6*d^3*n^2 + 11*d^3*n + 6*d^3)

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.90 \[ \int (a+b x)^2 (c+d x)^n \, dx={\left (c+d\,x\right )}^n\,\left (\frac {c\,\left (a^2\,d^2\,n^2+5\,a^2\,d^2\,n+6\,a^2\,d^2-2\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+2\,b^2\,c^2\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b^2\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {x\,\left (a^2\,d^3\,n^2+5\,a^2\,d^3\,n+6\,a^2\,d^3+2\,a\,b\,c\,d^2\,n^2+6\,a\,b\,c\,d^2\,n-2\,b^2\,c^2\,d\,n\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,x^2\,\left (n+1\right )\,\left (6\,a\,d+2\,a\,d\,n+b\,c\,n\right )}{d\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \]

[In]

int((a + b*x)^2*(c + d*x)^n,x)

[Out]

(c + d*x)^n*((c*(6*a^2*d^2 + 2*b^2*c^2 + 5*a^2*d^2*n + a^2*d^2*n^2 - 6*a*b*c*d - 2*a*b*c*d*n))/(d^3*(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) + (x*(6*a^2*d^3 + 5*a^2*d^3*n + a^2*d^3*n
^2 - 2*b^2*c^2*d*n + 2*a*b*c*d^2*n^2 + 6*a*b*c*d^2*n))/(d^3*(11*n + 6*n^2 + n^3 + 6)) + (b*x^2*(n + 1)*(6*a*d
+ 2*a*d*n + b*c*n))/(d*(11*n + 6*n^2 + n^3 + 6)))