Integrand size = 15, antiderivative size = 65 \[ \int (a+b x)^2 (c+d x)^7 \, dx=\frac {(b c-a d)^2 (c+d x)^8}{8 d^3}-\frac {2 b (b c-a d) (c+d x)^9}{9 d^3}+\frac {b^2 (c+d x)^{10}}{10 d^3} \] Output:
1/8*(-a*d+b*c)^2*(d*x+c)^8/d^3-2/9*b*(-a*d+b*c)*(d*x+c)^9/d^3+1/10*b^2*(d* x+c)^10/d^3
Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(65)=130\).
Time = 0.02 (sec) , antiderivative size = 261, normalized size of antiderivative = 4.02 \[ \int (a+b x)^2 (c+d x)^7 \, dx=a^2 c^7 x+\frac {1}{2} a c^6 (2 b c+7 a d) x^2+\frac {1}{3} c^5 \left (b^2 c^2+14 a b c d+21 a^2 d^2\right ) x^3+\frac {7}{4} c^4 d \left (b^2 c^2+6 a b c d+5 a^2 d^2\right ) x^4+\frac {7}{5} c^3 d^2 \left (3 b^2 c^2+10 a b c d+5 a^2 d^2\right ) x^5+\frac {7}{6} c^2 d^3 \left (5 b^2 c^2+10 a b c d+3 a^2 d^2\right ) x^6+c d^4 \left (5 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{8} d^5 \left (21 b^2 c^2+14 a b c d+a^2 d^2\right ) x^8+\frac {1}{9} b d^6 (7 b c+2 a d) x^9+\frac {1}{10} b^2 d^7 x^{10} \] Input:
Integrate[(a + b*x)^2*(c + d*x)^7,x]
Output:
a^2*c^7*x + (a*c^6*(2*b*c + 7*a*d)*x^2)/2 + (c^5*(b^2*c^2 + 14*a*b*c*d + 2 1*a^2*d^2)*x^3)/3 + (7*c^4*d*(b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*x^4)/4 + (7 *c^3*d^2*(3*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*x^5)/5 + (7*c^2*d^3*(5*b^2*c ^2 + 10*a*b*c*d + 3*a^2*d^2)*x^6)/6 + c*d^4*(5*b^2*c^2 + 6*a*b*c*d + a^2*d ^2)*x^7 + (d^5*(21*b^2*c^2 + 14*a*b*c*d + a^2*d^2)*x^8)/8 + (b*d^6*(7*b*c + 2*a*d)*x^9)/9 + (b^2*d^7*x^10)/10
Time = 0.29 (sec) , antiderivative size = 65, 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 = {49, 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.
| \(\displaystyle \int (a+b x)^2 (c+d x)^7 \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {2 b (c+d x)^8 (b c-a d)}{d^2}+\frac {(c+d x)^7 (a d-b c)^2}{d^2}+\frac {b^2 (c+d x)^9}{d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b (c+d x)^9 (b c-a d)}{9 d^3}+\frac {(c+d x)^8 (b c-a d)^2}{8 d^3}+\frac {b^2 (c+d x)^{10}}{10 d^3}\) |
Input:
Int[(a + b*x)^2*(c + d*x)^7,x]
Output:
((b*c - a*d)^2*(c + d*x)^8)/(8*d^3) - (2*b*(b*c - a*d)*(c + d*x)^9)/(9*d^3 ) + (b^2*(c + d*x)^10)/(10*d^3)
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(268\) vs. \(2(59)=118\).
Time = 0.08 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.14
| method | result | size |
| norman | \(\frac {b^{2} d^{7} x^{10}}{10}+\left (\frac {2}{9} a b \,d^{7}+\frac {7}{9} b^{2} c \,d^{6}\right ) x^{9}+\left (\frac {1}{8} a^{2} d^{7}+\frac {7}{4} a b c \,d^{6}+\frac {21}{8} b^{2} c^{2} d^{5}\right ) x^{8}+\left (a^{2} c \,d^{6}+6 a b \,c^{2} d^{5}+5 b^{2} c^{3} d^{4}\right ) x^{7}+\left (\frac {7}{2} a^{2} c^{2} d^{5}+\frac {35}{3} a b \,c^{3} d^{4}+\frac {35}{6} b^{2} c^{4} d^{3}\right ) x^{6}+\left (7 a^{2} c^{3} d^{4}+14 a b \,c^{4} d^{3}+\frac {21}{5} b^{2} c^{5} d^{2}\right ) x^{5}+\left (\frac {35}{4} a^{2} c^{4} d^{3}+\frac {21}{2} a b \,c^{5} d^{2}+\frac {7}{4} b^{2} c^{6} d \right ) x^{4}+\left (7 a^{2} c^{5} d^{2}+\frac {14}{3} a b \,c^{6} d +\frac {1}{3} b^{2} c^{7}\right ) x^{3}+\left (\frac {7}{2} a^{2} c^{6} d +a b \,c^{7}\right ) x^{2}+a^{2} c^{7} x\) | \(269\) |
| default | \(\frac {b^{2} d^{7} x^{10}}{10}+\frac {\left (2 a b \,d^{7}+7 b^{2} c \,d^{6}\right ) x^{9}}{9}+\frac {\left (a^{2} d^{7}+14 a b c \,d^{6}+21 b^{2} c^{2} d^{5}\right ) x^{8}}{8}+\frac {\left (7 a^{2} c \,d^{6}+42 a b \,c^{2} d^{5}+35 b^{2} c^{3} d^{4}\right ) x^{7}}{7}+\frac {\left (21 a^{2} c^{2} d^{5}+70 a b \,c^{3} d^{4}+35 b^{2} c^{4} d^{3}\right ) x^{6}}{6}+\frac {\left (35 a^{2} c^{3} d^{4}+70 a b \,c^{4} d^{3}+21 b^{2} c^{5} d^{2}\right ) x^{5}}{5}+\frac {\left (35 a^{2} c^{4} d^{3}+42 a b \,c^{5} d^{2}+7 b^{2} c^{6} d \right ) x^{4}}{4}+\frac {\left (21 a^{2} c^{5} d^{2}+14 a b \,c^{6} d +b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{2} c^{6} d +2 a b \,c^{7}\right ) x^{2}}{2}+a^{2} c^{7} x\) | \(277\) |
| gosper | \(\frac {1}{10} b^{2} d^{7} x^{10}+\frac {2}{9} x^{9} a b \,d^{7}+\frac {7}{9} x^{9} b^{2} c \,d^{6}+\frac {1}{8} x^{8} a^{2} d^{7}+\frac {7}{4} x^{8} a b c \,d^{6}+\frac {21}{8} x^{8} b^{2} c^{2} d^{5}+a^{2} c \,d^{6} x^{7}+6 a b \,c^{2} d^{5} x^{7}+5 b^{2} c^{3} d^{4} x^{7}+\frac {7}{2} x^{6} a^{2} c^{2} d^{5}+\frac {35}{3} x^{6} a b \,c^{3} d^{4}+\frac {35}{6} x^{6} b^{2} c^{4} d^{3}+7 x^{5} a^{2} c^{3} d^{4}+14 x^{5} a b \,c^{4} d^{3}+\frac {21}{5} x^{5} b^{2} c^{5} d^{2}+\frac {35}{4} x^{4} a^{2} c^{4} d^{3}+\frac {21}{2} x^{4} a b \,c^{5} d^{2}+\frac {7}{4} x^{4} b^{2} c^{6} d +7 x^{3} a^{2} c^{5} d^{2}+\frac {14}{3} x^{3} a b \,c^{6} d +\frac {1}{3} x^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{2} c^{6} d +x^{2} a b \,c^{7}+a^{2} c^{7} x\) | \(295\) |
| risch | \(\frac {1}{10} b^{2} d^{7} x^{10}+\frac {2}{9} x^{9} a b \,d^{7}+\frac {7}{9} x^{9} b^{2} c \,d^{6}+\frac {1}{8} x^{8} a^{2} d^{7}+\frac {7}{4} x^{8} a b c \,d^{6}+\frac {21}{8} x^{8} b^{2} c^{2} d^{5}+a^{2} c \,d^{6} x^{7}+6 a b \,c^{2} d^{5} x^{7}+5 b^{2} c^{3} d^{4} x^{7}+\frac {7}{2} x^{6} a^{2} c^{2} d^{5}+\frac {35}{3} x^{6} a b \,c^{3} d^{4}+\frac {35}{6} x^{6} b^{2} c^{4} d^{3}+7 x^{5} a^{2} c^{3} d^{4}+14 x^{5} a b \,c^{4} d^{3}+\frac {21}{5} x^{5} b^{2} c^{5} d^{2}+\frac {35}{4} x^{4} a^{2} c^{4} d^{3}+\frac {21}{2} x^{4} a b \,c^{5} d^{2}+\frac {7}{4} x^{4} b^{2} c^{6} d +7 x^{3} a^{2} c^{5} d^{2}+\frac {14}{3} x^{3} a b \,c^{6} d +\frac {1}{3} x^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{2} c^{6} d +x^{2} a b \,c^{7}+a^{2} c^{7} x\) | \(295\) |
| parallelrisch | \(\frac {1}{10} b^{2} d^{7} x^{10}+\frac {2}{9} x^{9} a b \,d^{7}+\frac {7}{9} x^{9} b^{2} c \,d^{6}+\frac {1}{8} x^{8} a^{2} d^{7}+\frac {7}{4} x^{8} a b c \,d^{6}+\frac {21}{8} x^{8} b^{2} c^{2} d^{5}+a^{2} c \,d^{6} x^{7}+6 a b \,c^{2} d^{5} x^{7}+5 b^{2} c^{3} d^{4} x^{7}+\frac {7}{2} x^{6} a^{2} c^{2} d^{5}+\frac {35}{3} x^{6} a b \,c^{3} d^{4}+\frac {35}{6} x^{6} b^{2} c^{4} d^{3}+7 x^{5} a^{2} c^{3} d^{4}+14 x^{5} a b \,c^{4} d^{3}+\frac {21}{5} x^{5} b^{2} c^{5} d^{2}+\frac {35}{4} x^{4} a^{2} c^{4} d^{3}+\frac {21}{2} x^{4} a b \,c^{5} d^{2}+\frac {7}{4} x^{4} b^{2} c^{6} d +7 x^{3} a^{2} c^{5} d^{2}+\frac {14}{3} x^{3} a b \,c^{6} d +\frac {1}{3} x^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{2} c^{6} d +x^{2} a b \,c^{7}+a^{2} c^{7} x\) | \(295\) |
| orering | \(\frac {x \left (36 b^{2} d^{7} x^{9}+80 a b \,d^{7} x^{8}+280 b^{2} c \,d^{6} x^{8}+45 a^{2} d^{7} x^{7}+630 a b c \,d^{6} x^{7}+945 b^{2} c^{2} d^{5} x^{7}+360 a^{2} c \,d^{6} x^{6}+2160 a b \,c^{2} d^{5} x^{6}+1800 b^{2} c^{3} d^{4} x^{6}+1260 a^{2} c^{2} d^{5} x^{5}+4200 a b \,c^{3} d^{4} x^{5}+2100 b^{2} c^{4} d^{3} x^{5}+2520 a^{2} c^{3} d^{4} x^{4}+5040 a b \,c^{4} d^{3} x^{4}+1512 b^{2} c^{5} d^{2} x^{4}+3150 a^{2} c^{4} d^{3} x^{3}+3780 a b \,c^{5} d^{2} x^{3}+630 b^{2} c^{6} d \,x^{3}+2520 a^{2} c^{5} d^{2} x^{2}+1680 a b \,c^{6} d \,x^{2}+120 b^{2} c^{7} x^{2}+1260 a^{2} c^{6} d x +360 a b \,c^{7} x +360 a^{2} c^{7}\right )}{360}\) | \(296\) |
Input:
int((b*x+a)^2*(d*x+c)^7,x,method=_RETURNVERBOSE)
Output:
1/10*b^2*d^7*x^10+(2/9*a*b*d^7+7/9*b^2*c*d^6)*x^9+(1/8*a^2*d^7+7/4*a*b*c*d ^6+21/8*b^2*c^2*d^5)*x^8+(a^2*c*d^6+6*a*b*c^2*d^5+5*b^2*c^3*d^4)*x^7+(7/2* a^2*c^2*d^5+35/3*a*b*c^3*d^4+35/6*b^2*c^4*d^3)*x^6+(7*a^2*c^3*d^4+14*a*b*c ^4*d^3+21/5*b^2*c^5*d^2)*x^5+(35/4*a^2*c^4*d^3+21/2*a*b*c^5*d^2+7/4*b^2*c^ 6*d)*x^4+(7*a^2*c^5*d^2+14/3*a*b*c^6*d+1/3*b^2*c^7)*x^3+(7/2*a^2*c^6*d+a*b *c^7)*x^2+a^2*c^7*x
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (59) = 118\).
Time = 0.09 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.20 \[ \int (a+b x)^2 (c+d x)^7 \, dx=\frac {1}{10} \, b^{2} d^{7} x^{10} + a^{2} c^{7} x + \frac {1}{9} \, {\left (7 \, b^{2} c d^{6} + 2 \, a b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{2} c^{2} d^{5} + 14 \, a b c d^{6} + a^{2} d^{7}\right )} x^{8} + {\left (5 \, b^{2} c^{3} d^{4} + 6 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b^{2} c^{4} d^{3} + 10 \, a b c^{3} d^{4} + 3 \, a^{2} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (3 \, b^{2} c^{5} d^{2} + 10 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (b^{2} c^{6} d + 6 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{7} + 14 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{7} + 7 \, a^{2} c^{6} d\right )} x^{2} \] Input:
integrate((b*x+a)^2*(d*x+c)^7,x, algorithm="fricas")
Output:
1/10*b^2*d^7*x^10 + a^2*c^7*x + 1/9*(7*b^2*c*d^6 + 2*a*b*d^7)*x^9 + 1/8*(2 1*b^2*c^2*d^5 + 14*a*b*c*d^6 + a^2*d^7)*x^8 + (5*b^2*c^3*d^4 + 6*a*b*c^2*d ^5 + a^2*c*d^6)*x^7 + 7/6*(5*b^2*c^4*d^3 + 10*a*b*c^3*d^4 + 3*a^2*c^2*d^5) *x^6 + 7/5*(3*b^2*c^5*d^2 + 10*a*b*c^4*d^3 + 5*a^2*c^3*d^4)*x^5 + 7/4*(b^2 *c^6*d + 6*a*b*c^5*d^2 + 5*a^2*c^4*d^3)*x^4 + 1/3*(b^2*c^7 + 14*a*b*c^6*d + 21*a^2*c^5*d^2)*x^3 + 1/2*(2*a*b*c^7 + 7*a^2*c^6*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (56) = 112\).
Time = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 4.66 \[ \int (a+b x)^2 (c+d x)^7 \, dx=a^{2} c^{7} x + \frac {b^{2} d^{7} x^{10}}{10} + x^{9} \cdot \left (\frac {2 a b d^{7}}{9} + \frac {7 b^{2} c d^{6}}{9}\right ) + x^{8} \left (\frac {a^{2} d^{7}}{8} + \frac {7 a b c d^{6}}{4} + \frac {21 b^{2} c^{2} d^{5}}{8}\right ) + x^{7} \left (a^{2} c d^{6} + 6 a b c^{2} d^{5} + 5 b^{2} c^{3} d^{4}\right ) + x^{6} \cdot \left (\frac {7 a^{2} c^{2} d^{5}}{2} + \frac {35 a b c^{3} d^{4}}{3} + \frac {35 b^{2} c^{4} d^{3}}{6}\right ) + x^{5} \cdot \left (7 a^{2} c^{3} d^{4} + 14 a b c^{4} d^{3} + \frac {21 b^{2} c^{5} d^{2}}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{2} c^{4} d^{3}}{4} + \frac {21 a b c^{5} d^{2}}{2} + \frac {7 b^{2} c^{6} d}{4}\right ) + x^{3} \cdot \left (7 a^{2} c^{5} d^{2} + \frac {14 a b c^{6} d}{3} + \frac {b^{2} c^{7}}{3}\right ) + x^{2} \cdot \left (\frac {7 a^{2} c^{6} d}{2} + a b c^{7}\right ) \] Input:
integrate((b*x+a)**2*(d*x+c)**7,x)
Output:
a**2*c**7*x + b**2*d**7*x**10/10 + x**9*(2*a*b*d**7/9 + 7*b**2*c*d**6/9) + x**8*(a**2*d**7/8 + 7*a*b*c*d**6/4 + 21*b**2*c**2*d**5/8) + x**7*(a**2*c* d**6 + 6*a*b*c**2*d**5 + 5*b**2*c**3*d**4) + x**6*(7*a**2*c**2*d**5/2 + 35 *a*b*c**3*d**4/3 + 35*b**2*c**4*d**3/6) + x**5*(7*a**2*c**3*d**4 + 14*a*b* c**4*d**3 + 21*b**2*c**5*d**2/5) + x**4*(35*a**2*c**4*d**3/4 + 21*a*b*c**5 *d**2/2 + 7*b**2*c**6*d/4) + x**3*(7*a**2*c**5*d**2 + 14*a*b*c**6*d/3 + b* *2*c**7/3) + x**2*(7*a**2*c**6*d/2 + a*b*c**7)
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (59) = 118\).
Time = 0.03 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.20 \[ \int (a+b x)^2 (c+d x)^7 \, dx=\frac {1}{10} \, b^{2} d^{7} x^{10} + a^{2} c^{7} x + \frac {1}{9} \, {\left (7 \, b^{2} c d^{6} + 2 \, a b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{2} c^{2} d^{5} + 14 \, a b c d^{6} + a^{2} d^{7}\right )} x^{8} + {\left (5 \, b^{2} c^{3} d^{4} + 6 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b^{2} c^{4} d^{3} + 10 \, a b c^{3} d^{4} + 3 \, a^{2} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (3 \, b^{2} c^{5} d^{2} + 10 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (b^{2} c^{6} d + 6 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{7} + 14 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{7} + 7 \, a^{2} c^{6} d\right )} x^{2} \] Input:
integrate((b*x+a)^2*(d*x+c)^7,x, algorithm="maxima")
Output:
1/10*b^2*d^7*x^10 + a^2*c^7*x + 1/9*(7*b^2*c*d^6 + 2*a*b*d^7)*x^9 + 1/8*(2 1*b^2*c^2*d^5 + 14*a*b*c*d^6 + a^2*d^7)*x^8 + (5*b^2*c^3*d^4 + 6*a*b*c^2*d ^5 + a^2*c*d^6)*x^7 + 7/6*(5*b^2*c^4*d^3 + 10*a*b*c^3*d^4 + 3*a^2*c^2*d^5) *x^6 + 7/5*(3*b^2*c^5*d^2 + 10*a*b*c^4*d^3 + 5*a^2*c^3*d^4)*x^5 + 7/4*(b^2 *c^6*d + 6*a*b*c^5*d^2 + 5*a^2*c^4*d^3)*x^4 + 1/3*(b^2*c^7 + 14*a*b*c^6*d + 21*a^2*c^5*d^2)*x^3 + 1/2*(2*a*b*c^7 + 7*a^2*c^6*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (59) = 118\).
Time = 0.12 (sec) , antiderivative size = 294, normalized size of antiderivative = 4.52 \[ \int (a+b x)^2 (c+d x)^7 \, dx=\frac {1}{10} \, b^{2} d^{7} x^{10} + \frac {7}{9} \, b^{2} c d^{6} x^{9} + \frac {2}{9} \, a b d^{7} x^{9} + \frac {21}{8} \, b^{2} c^{2} d^{5} x^{8} + \frac {7}{4} \, a b c d^{6} x^{8} + \frac {1}{8} \, a^{2} d^{7} x^{8} + 5 \, b^{2} c^{3} d^{4} x^{7} + 6 \, a b c^{2} d^{5} x^{7} + a^{2} c d^{6} x^{7} + \frac {35}{6} \, b^{2} c^{4} d^{3} x^{6} + \frac {35}{3} \, a b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{2} c^{2} d^{5} x^{6} + \frac {21}{5} \, b^{2} c^{5} d^{2} x^{5} + 14 \, a b c^{4} d^{3} x^{5} + 7 \, a^{2} c^{3} d^{4} x^{5} + \frac {7}{4} \, b^{2} c^{6} d x^{4} + \frac {21}{2} \, a b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{2} c^{4} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{7} x^{3} + \frac {14}{3} \, a b c^{6} d x^{3} + 7 \, a^{2} c^{5} d^{2} x^{3} + a b c^{7} x^{2} + \frac {7}{2} \, a^{2} c^{6} d x^{2} + a^{2} c^{7} x \] Input:
integrate((b*x+a)^2*(d*x+c)^7,x, algorithm="giac")
Output:
1/10*b^2*d^7*x^10 + 7/9*b^2*c*d^6*x^9 + 2/9*a*b*d^7*x^9 + 21/8*b^2*c^2*d^5 *x^8 + 7/4*a*b*c*d^6*x^8 + 1/8*a^2*d^7*x^8 + 5*b^2*c^3*d^4*x^7 + 6*a*b*c^2 *d^5*x^7 + a^2*c*d^6*x^7 + 35/6*b^2*c^4*d^3*x^6 + 35/3*a*b*c^3*d^4*x^6 + 7 /2*a^2*c^2*d^5*x^6 + 21/5*b^2*c^5*d^2*x^5 + 14*a*b*c^4*d^3*x^5 + 7*a^2*c^3 *d^4*x^5 + 7/4*b^2*c^6*d*x^4 + 21/2*a*b*c^5*d^2*x^4 + 35/4*a^2*c^4*d^3*x^4 + 1/3*b^2*c^7*x^3 + 14/3*a*b*c^6*d*x^3 + 7*a^2*c^5*d^2*x^3 + a*b*c^7*x^2 + 7/2*a^2*c^6*d*x^2 + a^2*c^7*x
Time = 0.15 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.83 \[ \int (a+b x)^2 (c+d x)^7 \, dx=x^3\,\left (7\,a^2\,c^5\,d^2+\frac {14\,a\,b\,c^6\,d}{3}+\frac {b^2\,c^7}{3}\right )+x^8\,\left (\frac {a^2\,d^7}{8}+\frac {7\,a\,b\,c\,d^6}{4}+\frac {21\,b^2\,c^2\,d^5}{8}\right )+a^2\,c^7\,x+\frac {b^2\,d^7\,x^{10}}{10}+\frac {a\,c^6\,x^2\,\left (7\,a\,d+2\,b\,c\right )}{2}+\frac {b\,d^6\,x^9\,\left (2\,a\,d+7\,b\,c\right )}{9}+\frac {7\,c^4\,d\,x^4\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{4}+c\,d^4\,x^7\,\left (a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )+\frac {7\,c^3\,d^2\,x^5\,\left (5\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{5}+\frac {7\,c^2\,d^3\,x^6\,\left (3\,a^2\,d^2+10\,a\,b\,c\,d+5\,b^2\,c^2\right )}{6} \] Input:
int((a + b*x)^2*(c + d*x)^7,x)
Output:
x^3*((b^2*c^7)/3 + 7*a^2*c^5*d^2 + (14*a*b*c^6*d)/3) + x^8*((a^2*d^7)/8 + (21*b^2*c^2*d^5)/8 + (7*a*b*c*d^6)/4) + a^2*c^7*x + (b^2*d^7*x^10)/10 + (a *c^6*x^2*(7*a*d + 2*b*c))/2 + (b*d^6*x^9*(2*a*d + 7*b*c))/9 + (7*c^4*d*x^4 *(5*a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/4 + c*d^4*x^7*(a^2*d^2 + 5*b^2*c^2 + 6 *a*b*c*d) + (7*c^3*d^2*x^5*(5*a^2*d^2 + 3*b^2*c^2 + 10*a*b*c*d))/5 + (7*c^ 2*d^3*x^6*(3*a^2*d^2 + 5*b^2*c^2 + 10*a*b*c*d))/6
Time = 0.18 (sec) , antiderivative size = 295, normalized size of antiderivative = 4.54 \[ \int (a+b x)^2 (c+d x)^7 \, dx=\frac {x \left (36 b^{2} d^{7} x^{9}+80 a b \,d^{7} x^{8}+280 b^{2} c \,d^{6} x^{8}+45 a^{2} d^{7} x^{7}+630 a b c \,d^{6} x^{7}+945 b^{2} c^{2} d^{5} x^{7}+360 a^{2} c \,d^{6} x^{6}+2160 a b \,c^{2} d^{5} x^{6}+1800 b^{2} c^{3} d^{4} x^{6}+1260 a^{2} c^{2} d^{5} x^{5}+4200 a b \,c^{3} d^{4} x^{5}+2100 b^{2} c^{4} d^{3} x^{5}+2520 a^{2} c^{3} d^{4} x^{4}+5040 a b \,c^{4} d^{3} x^{4}+1512 b^{2} c^{5} d^{2} x^{4}+3150 a^{2} c^{4} d^{3} x^{3}+3780 a b \,c^{5} d^{2} x^{3}+630 b^{2} c^{6} d \,x^{3}+2520 a^{2} c^{5} d^{2} x^{2}+1680 a b \,c^{6} d \,x^{2}+120 b^{2} c^{7} x^{2}+1260 a^{2} c^{6} d x +360 a b \,c^{7} x +360 a^{2} c^{7}\right )}{360} \] Input:
int((b*x+a)^2*(d*x+c)^7,x)
Output:
(x*(360*a**2*c**7 + 1260*a**2*c**6*d*x + 2520*a**2*c**5*d**2*x**2 + 3150*a **2*c**4*d**3*x**3 + 2520*a**2*c**3*d**4*x**4 + 1260*a**2*c**2*d**5*x**5 + 360*a**2*c*d**6*x**6 + 45*a**2*d**7*x**7 + 360*a*b*c**7*x + 1680*a*b*c**6 *d*x**2 + 3780*a*b*c**5*d**2*x**3 + 5040*a*b*c**4*d**3*x**4 + 4200*a*b*c** 3*d**4*x**5 + 2160*a*b*c**2*d**5*x**6 + 630*a*b*c*d**6*x**7 + 80*a*b*d**7* x**8 + 120*b**2*c**7*x**2 + 630*b**2*c**6*d*x**3 + 1512*b**2*c**5*d**2*x** 4 + 2100*b**2*c**4*d**3*x**5 + 1800*b**2*c**3*d**4*x**6 + 945*b**2*c**2*d* *5*x**7 + 280*b**2*c*d**6*x**8 + 36*b**2*d**7*x**9))/360