Integrand size = 15, antiderivative size = 78 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {(b c-a d)^2 (c+d x)^{1+n}}{d^3 (1+n)}-\frac {2 b (b c-a d) (c+d x)^{2+n}}{d^3 (2+n)}+\frac {b^2 (c+d x)^{3+n}}{d^3 (3+n)} \]
(-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)
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {(c+d x)^{1+n} \left (\frac {(b c-a d)^2}{1+n}-\frac {2 b (b c-a d) (c+d x)}{2+n}+\frac {b^2 (c+d x)^2}{3+n}\right )}{d^3} \]
((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
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a+b x)^2 (c+d x)^n \, dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {(a d-b c)^2 (c+d x)^n}{d^2}-\frac {2 b (b c-a d) (c+d x)^{n+1}}{d^2}+\frac {b^2 (c+d x)^{n+2}}{d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(b c-a d)^2 (c+d x)^{n+1}}{d^3 (n+1)}-\frac {2 b (b c-a d) (c+d x)^{n+2}}{d^3 (n+2)}+\frac {b^2 (c+d x)^{n+3}}{d^3 (n+3)}\) |
((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))
3.19.53.3.1 Defintions of rubi rules used
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] && 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])
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 | \(\frac {x^{3} \left (d x +c \right )^{n} b^{2} c \,d^{3} n^{2}+3 x^{3} \left (d x +c \right )^{n} b^{2} c \,d^{3} n +2 x^{2} \left (d x +c \right )^{n} a b c \,d^{3} n^{2}+x^{2} \left (d x +c \right )^{n} b^{2} c^{2} d^{2} n^{2}+2 x^{3} \left (d x +c \right )^{n} b^{2} c \,d^{3}+8 x^{2} \left (d x +c \right )^{n} a b c \,d^{3} n +x^{2} \left (d x +c \right )^{n} b^{2} c^{2} d^{2} n +x \left (d x +c \right )^{n} a^{2} c \,d^{3} n^{2}+2 x \left (d x +c \right )^{n} a b \,c^{2} d^{2} n^{2}+6 x^{2} \left (d x +c \right )^{n} a b c \,d^{3}+5 x \left (d x +c \right )^{n} a^{2} c \,d^{3} n +6 x \left (d x +c \right )^{n} a b \,c^{2} d^{2} n -2 x \left (d x +c \right )^{n} b^{2} c^{3} d n +\left (d x +c \right )^{n} a^{2} c^{2} d^{2} n^{2}+6 x \left (d x +c \right )^{n} a^{2} c \,d^{3}+5 \left (d x +c \right )^{n} a^{2} c^{2} d^{2} n -2 \left (d x +c \right )^{n} a b \,c^{3} d n +6 \left (d x +c \right )^{n} a^{2} c^{2} d^{2}-6 \left (d x +c \right )^{n} a b \,c^{3} d +2 \left (d x +c \right )^{n} b^{2} c^{4}}{\left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{3} c}\) | \(401\) |
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)
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 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {{\left (a^{2} c d^{2} n^{2} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} + {\left (b^{2} d^{3} n^{2} + 3 \, b^{2} d^{3} n + 2 \, b^{2} d^{3}\right )} x^{3} + {\left (6 \, a b d^{3} + {\left (b^{2} c d^{2} + 2 \, a b d^{3}\right )} n^{2} + {\left (b^{2} c d^{2} + 8 \, a b d^{3}\right )} n\right )} x^{2} - {\left (2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} n + {\left (6 \, a^{2} d^{3} + {\left (2 \, a b c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (2 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \]
(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)
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 \[ \int (a+b x)^2 (c+d x)^n \, dx=\text {Too large to display} \]
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*...
Time = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.77 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {2 \, {\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a b}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d x + c\right )}^{n + 1} a^{2}}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} b^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} \]
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)
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 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {{\left (d x + c\right )}^{n} b^{2} d^{3} n^{2} x^{3} + {\left (d x + c\right )}^{n} b^{2} c d^{2} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} a b d^{3} n^{2} x^{2} + 3 \, {\left (d x + c\right )}^{n} b^{2} d^{3} n x^{3} + 2 \, {\left (d x + c\right )}^{n} a b c d^{2} n^{2} x + {\left (d x + c\right )}^{n} a^{2} d^{3} n^{2} x + {\left (d x + c\right )}^{n} b^{2} c d^{2} n x^{2} + 8 \, {\left (d x + c\right )}^{n} a b d^{3} n x^{2} + 2 \, {\left (d x + c\right )}^{n} b^{2} d^{3} x^{3} + {\left (d x + c\right )}^{n} a^{2} c d^{2} n^{2} - 2 \, {\left (d x + c\right )}^{n} b^{2} c^{2} d n x + 6 \, {\left (d x + c\right )}^{n} a b c d^{2} n x + 5 \, {\left (d x + c\right )}^{n} a^{2} d^{3} n x + 6 \, {\left (d x + c\right )}^{n} a b d^{3} x^{2} - 2 \, {\left (d x + c\right )}^{n} a b c^{2} d n + 5 \, {\left (d x + c\right )}^{n} a^{2} c d^{2} n + 6 \, {\left (d x + c\right )}^{n} a^{2} d^{3} x + 2 \, {\left (d x + c\right )}^{n} b^{2} c^{3} - 6 \, {\left (d x + c\right )}^{n} a b c^{2} d + 6 \, {\left (d x + c\right )}^{n} a^{2} c d^{2}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \]
((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)
Time = 0.58 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.90 \[ \int (a+b x)^2 (c+d x)^n \, dx={\left (c+d\,x\right )}^n\,\left (\frac {c\,\left (a^2\,d^2\,n^2+5\,a^2\,d^2\,n+6\,a^2\,d^2-2\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+2\,b^2\,c^2\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b^2\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {x\,\left (a^2\,d^3\,n^2+5\,a^2\,d^3\,n+6\,a^2\,d^3+2\,a\,b\,c\,d^2\,n^2+6\,a\,b\,c\,d^2\,n-2\,b^2\,c^2\,d\,n\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,x^2\,\left (n+1\right )\,\left (6\,a\,d+2\,a\,d\,n+b\,c\,n\right )}{d\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \]
(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)))
Time = 0.01 (sec) , antiderivative size = 241, normalized size of antiderivative = 3.09 \[ \int (a+b x)^2 (c+d x)^n \, dx=\frac {\left (d x +c \right )^{n} \left (b^{2} d^{3} n^{2} x^{3}+2 a b \,d^{3} n^{2} x^{2}+b^{2} c \,d^{2} n^{2} x^{2}+3 b^{2} d^{3} n \,x^{3}+a^{2} d^{3} n^{2} x +2 a b c \,d^{2} n^{2} x +8 a b \,d^{3} n \,x^{2}+b^{2} c \,d^{2} n \,x^{2}+2 b^{2} d^{3} x^{3}+a^{2} c \,d^{2} n^{2}+5 a^{2} d^{3} n x +6 a b c \,d^{2} n x +6 a b \,d^{3} x^{2}-2 b^{2} c^{2} d n x +5 a^{2} c \,d^{2} n +6 a^{2} d^{3} x -2 a b \,c^{2} d n +6 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 b^{2} c^{3}\right )}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )} \]
((c + d*x)**n*(a**2*c*d**2*n**2 + 5*a**2*c*d**2*n + 6*a**2*c*d**2 + a**2*d **3*n**2*x + 5*a**2*d**3*n*x + 6*a**2*d**3*x - 2*a*b*c**2*d*n - 6*a*b*c**2 *d + 2*a*b*c*d**2*n**2*x + 6*a*b*c*d**2*n*x + 2*a*b*d**3*n**2*x**2 + 8*a*b *d**3*n*x**2 + 6*a*b*d**3*x**2 + 2*b**2*c**3 - 2*b**2*c**2*d*n*x + b**2*c* d**2*n**2*x**2 + b**2*c*d**2*n*x**2 + b**2*d**3*n**2*x**3 + 3*b**2*d**3*n* x**3 + 2*b**2*d**3*x**3))/(d**3*(n**3 + 6*n**2 + 11*n + 6))