Integrand size = 19, antiderivative size = 263 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^4 \text {b1} x+\frac {1}{2} a^3 (8 b \text {b1}+a \text {c1}) x^2+\frac {4}{3} a^2 \left (6 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+a \left (8 b^3 \text {b1}+6 a b \text {b1} c+6 a b^2 \text {c1}+a^2 c \text {c1}\right ) x^4+\frac {2}{5} \left (8 b^4 \text {b1}+24 a b^2 \text {b1} c+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+12 a^2 b c \text {c1}\right ) x^5+\frac {1}{3} \left (16 b^3 \text {b1} c+12 a b \text {b1} c^2+8 b^4 \text {c1}+24 a b^2 c \text {c1}+3 a^2 c^2 \text {c1}\right ) x^6+\frac {4}{7} c \left (6 b^2 \text {b1} c+a \text {b1} c^2+8 b^3 \text {c1}+6 a b c \text {c1}\right ) x^7+\frac {1}{2} c^2 \left (2 b \text {b1} c+6 b^2 \text {c1}+a c \text {c1}\right ) x^8+\frac {1}{9} c^3 (\text {b1} c+8 b \text {c1}) x^9+\frac {1}{10} c^4 \text {c1} x^{10} \] Output:
a^4*b1*x+1/2*a^3*(a*c1+8*b*b1)*x^2+4/3*a^2*(2*a*b*c1+a*b1*c+6*b^2*b1)*x^3+ a*(a^2*c*c1+6*a*b^2*c1+6*a*b*b1*c+8*b^3*b1)*x^4+2/5*(12*a^2*b*c*c1+3*a^2*b 1*c^2+16*a*b^3*c1+24*a*b^2*b1*c+8*b^4*b1)*x^5+1/3*(3*a^2*c^2*c1+24*a*b^2*c *c1+12*a*b*b1*c^2+8*b^4*c1+16*b^3*b1*c)*x^6+4/7*c*(6*a*b*c*c1+a*b1*c^2+8*b ^3*c1+6*b^2*b1*c)*x^7+1/2*c^2*(a*c*c1+6*b^2*c1+2*b*b1*c)*x^8+1/9*c^3*(8*b* c1+b1*c)*x^9+1/10*c^4*c1*x^10
Time = 0.04 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^4 \text {b1} x+\frac {1}{2} a^3 (8 b \text {b1}+a \text {c1}) x^2+\frac {4}{3} a^2 \left (6 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+a \left (8 b^3 \text {b1}+6 a b \text {b1} c+6 a b^2 \text {c1}+a^2 c \text {c1}\right ) x^4+\frac {2}{5} \left (8 b^4 \text {b1}+24 a b^2 \text {b1} c+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+12 a^2 b c \text {c1}\right ) x^5+\frac {1}{3} \left (16 b^3 \text {b1} c+12 a b \text {b1} c^2+8 b^4 \text {c1}+24 a b^2 c \text {c1}+3 a^2 c^2 \text {c1}\right ) x^6+\frac {4}{7} c \left (6 b^2 \text {b1} c+a \text {b1} c^2+8 b^3 \text {c1}+6 a b c \text {c1}\right ) x^7+\frac {1}{2} c^2 \left (2 b \text {b1} c+6 b^2 \text {c1}+a c \text {c1}\right ) x^8+\frac {1}{9} c^3 (\text {b1} c+8 b \text {c1}) x^9+\frac {1}{10} c^4 \text {c1} x^{10} \] Input:
Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2)^4,x]
Output:
a^4*b1*x + (a^3*(8*b*b1 + a*c1)*x^2)/2 + (4*a^2*(6*b^2*b1 + a*b1*c + 2*a*b *c1)*x^3)/3 + a*(8*b^3*b1 + 6*a*b*b1*c + 6*a*b^2*c1 + a^2*c*c1)*x^4 + (2*( 8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^2 + 16*a*b^3*c1 + 12*a^2*b*c*c1)*x^5 )/5 + ((16*b^3*b1*c + 12*a*b*b1*c^2 + 8*b^4*c1 + 24*a*b^2*c*c1 + 3*a^2*c^2 *c1)*x^6)/3 + (4*c*(6*b^2*b1*c + a*b1*c^2 + 8*b^3*c1 + 6*a*b*c*c1)*x^7)/7 + (c^2*(2*b*b1*c + 6*b^2*c1 + a*c*c1)*x^8)/2 + (c^3*(b1*c + 8*b*c1)*x^9)/9 + (c^4*c1*x^10)/10
Time = 0.52 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1140, 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.
| \(\displaystyle \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (a^4 \text {b1}+a^3 x (a \text {c1}+8 b \text {b1})+4 a^2 x^2 \left (2 a b \text {c1}+a \text {b1} c+6 b^2 \text {b1}\right )+4 a x^3 \left (a^2 c \text {c1}+6 a b^2 \text {c1}+6 a b \text {b1} c+8 b^3 \text {b1}\right )+2 x^5 \left (3 a^2 c^2 \text {c1}+24 a b^2 c \text {c1}+12 a b \text {b1} c^2+8 b^4 \text {c1}+16 b^3 \text {b1} c\right )+2 x^4 \left (12 a^2 b c \text {c1}+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+24 a b^2 \text {b1} c+8 b^4 \text {b1}\right )+4 c^2 x^7 \left (a c \text {c1}+6 b^2 \text {c1}+2 b \text {b1} c\right )+4 c x^6 \left (6 a b c \text {c1}+a \text {b1} c^2+8 b^3 \text {c1}+6 b^2 \text {b1} c\right )+c^3 x^8 (8 b \text {c1}+\text {b1} c)+c^4 \text {c1} x^9\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \text {b1} x+\frac {1}{2} a^3 x^2 (a \text {c1}+8 b \text {b1})+\frac {4}{3} a^2 x^3 \left (2 a b \text {c1}+a \text {b1} c+6 b^2 \text {b1}\right )+a x^4 \left (a^2 c \text {c1}+6 a b^2 \text {c1}+6 a b \text {b1} c+8 b^3 \text {b1}\right )+\frac {1}{3} x^6 \left (3 a^2 c^2 \text {c1}+24 a b^2 c \text {c1}+12 a b \text {b1} c^2+8 b^4 \text {c1}+16 b^3 \text {b1} c\right )+\frac {2}{5} x^5 \left (12 a^2 b c \text {c1}+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+24 a b^2 \text {b1} c+8 b^4 \text {b1}\right )+\frac {1}{2} c^2 x^8 \left (a c \text {c1}+6 b^2 \text {c1}+2 b \text {b1} c\right )+\frac {4}{7} c x^7 \left (6 a b c \text {c1}+a \text {b1} c^2+8 b^3 \text {c1}+6 b^2 \text {b1} c\right )+\frac {1}{9} c^3 x^9 (8 b \text {c1}+\text {b1} c)+\frac {1}{10} c^4 \text {c1} x^{10}\) |
Input:
Int[(b1 + c1*x)*(a + 2*b*x + c*x^2)^4,x]
Output:
a^4*b1*x + (a^3*(8*b*b1 + a*c1)*x^2)/2 + (4*a^2*(6*b^2*b1 + a*b1*c + 2*a*b *c1)*x^3)/3 + a*(8*b^3*b1 + 6*a*b*b1*c + 6*a*b^2*c1 + a^2*c*c1)*x^4 + (2*( 8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^2 + 16*a*b^3*c1 + 12*a^2*b*c*c1)*x^5 )/5 + ((16*b^3*b1*c + 12*a*b*b1*c^2 + 8*b^4*c1 + 24*a*b^2*c*c1 + 3*a^2*c^2 *c1)*x^6)/3 + (4*c*(6*b^2*b1*c + a*b1*c^2 + 8*b^3*c1 + 6*a*b*c*c1)*x^7)/7 + (c^2*(2*b*b1*c + 6*b^2*c1 + a*c*c1)*x^8)/2 + (c^3*(b1*c + 8*b*c1)*x^9)/9 + (c^4*c1*x^10)/10
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.18 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00
| method | result | size |
| norman | \(\frac {c^{4} \operatorname {c1} \,x^{10}}{10}+\left (\frac {8}{9} \operatorname {c1} b \,c^{3}+\frac {1}{9} \operatorname {b1} \,c^{4}\right ) x^{9}+\left (\frac {1}{2} a \,c^{3} \operatorname {c1} +3 b^{2} c^{2} \operatorname {c1} +\operatorname {b1} b \,c^{3}\right ) x^{8}+\left (\frac {24}{7} a b \,c^{2} \operatorname {c1} +\frac {4}{7} a \operatorname {b1} \,c^{3}+\frac {32}{7} b^{3} c \operatorname {c1} +\frac {24}{7} b^{2} \operatorname {b1} \,c^{2}\right ) x^{7}+\left (a^{2} c^{2} \operatorname {c1} +8 a \,b^{2} c \operatorname {c1} +4 a b \operatorname {b1} \,c^{2}+\frac {8}{3} b^{4} \operatorname {c1} +\frac {16}{3} b^{3} \operatorname {b1} c \right ) x^{6}+\left (\frac {24}{5} a^{2} b c \operatorname {c1} +\frac {6}{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} a \,b^{3} \operatorname {c1} +\frac {48}{5} a \,b^{2} \operatorname {b1} c +\frac {16}{5} b^{4} \operatorname {b1} \right ) x^{5}+\left (a^{3} c \operatorname {c1} +6 a^{2} b^{2} \operatorname {c1} +6 a^{2} b \operatorname {b1} c +8 a \,b^{3} \operatorname {b1} \right ) x^{4}+\left (\frac {8}{3} \operatorname {c1} \,a^{3} b +\frac {4}{3} a^{3} \operatorname {b1} c +8 a^{2} b^{2} \operatorname {b1} \right ) x^{3}+\left (\frac {1}{2} \operatorname {c1} \,a^{4}+4 \operatorname {b1} \,a^{3} b \right ) x^{2}+a^{4} \operatorname {b1} x\) | \(264\) |
| gosper | \(\frac {1}{2} x^{2} \operatorname {c1} \,a^{4}+\frac {8}{3} x^{6} b^{4} \operatorname {c1} +\frac {16}{5} x^{5} b^{4} \operatorname {b1} +\frac {1}{9} x^{9} \operatorname {b1} \,c^{4}+\frac {24}{5} x^{5} a^{2} b c \operatorname {c1} +\frac {48}{5} x^{5} a \,b^{2} \operatorname {b1} c +6 a^{2} b \operatorname {b1} c \,x^{4}+8 x^{3} a^{2} b^{2} \operatorname {b1} +4 x^{2} \operatorname {b1} \,a^{3} b +a^{3} c \operatorname {c1} \,x^{4}+6 a^{2} b^{2} \operatorname {c1} \,x^{4}+8 a \,b^{3} \operatorname {b1} \,x^{4}+3 x^{8} b^{2} c^{2} \operatorname {c1} +x^{8} \operatorname {b1} b \,c^{3}+\frac {4}{7} x^{7} a \operatorname {b1} \,c^{3}+\frac {32}{7} x^{7} b^{3} c \operatorname {c1} +\frac {24}{7} x^{7} b^{2} \operatorname {b1} \,c^{2}+x^{6} a^{2} c^{2} \operatorname {c1} +\frac {16}{3} x^{6} b^{3} \operatorname {b1} c +\frac {6}{5} x^{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} x^{5} a \,b^{3} \operatorname {c1} +\frac {8}{3} x^{3} \operatorname {c1} \,a^{3} b +\frac {4}{3} x^{3} a^{3} \operatorname {b1} c +\frac {8}{9} x^{9} \operatorname {c1} b \,c^{3}+\frac {1}{2} x^{8} a \,c^{3} \operatorname {c1} +\frac {24}{7} x^{7} a b \,c^{2} \operatorname {c1} +8 x^{6} a \,b^{2} c \operatorname {c1} +4 x^{6} a b \operatorname {b1} \,c^{2}+a^{4} \operatorname {b1} x +\frac {1}{10} c^{4} \operatorname {c1} \,x^{10}\) | \(308\) |
| risch | \(\frac {1}{2} x^{2} \operatorname {c1} \,a^{4}+\frac {8}{3} x^{6} b^{4} \operatorname {c1} +\frac {16}{5} x^{5} b^{4} \operatorname {b1} +\frac {1}{9} x^{9} \operatorname {b1} \,c^{4}+\frac {24}{5} x^{5} a^{2} b c \operatorname {c1} +\frac {48}{5} x^{5} a \,b^{2} \operatorname {b1} c +6 a^{2} b \operatorname {b1} c \,x^{4}+8 x^{3} a^{2} b^{2} \operatorname {b1} +4 x^{2} \operatorname {b1} \,a^{3} b +a^{3} c \operatorname {c1} \,x^{4}+6 a^{2} b^{2} \operatorname {c1} \,x^{4}+8 a \,b^{3} \operatorname {b1} \,x^{4}+3 x^{8} b^{2} c^{2} \operatorname {c1} +x^{8} \operatorname {b1} b \,c^{3}+\frac {4}{7} x^{7} a \operatorname {b1} \,c^{3}+\frac {32}{7} x^{7} b^{3} c \operatorname {c1} +\frac {24}{7} x^{7} b^{2} \operatorname {b1} \,c^{2}+x^{6} a^{2} c^{2} \operatorname {c1} +\frac {16}{3} x^{6} b^{3} \operatorname {b1} c +\frac {6}{5} x^{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} x^{5} a \,b^{3} \operatorname {c1} +\frac {8}{3} x^{3} \operatorname {c1} \,a^{3} b +\frac {4}{3} x^{3} a^{3} \operatorname {b1} c +\frac {8}{9} x^{9} \operatorname {c1} b \,c^{3}+\frac {1}{2} x^{8} a \,c^{3} \operatorname {c1} +\frac {24}{7} x^{7} a b \,c^{2} \operatorname {c1} +8 x^{6} a \,b^{2} c \operatorname {c1} +4 x^{6} a b \operatorname {b1} \,c^{2}+a^{4} \operatorname {b1} x +\frac {1}{10} c^{4} \operatorname {c1} \,x^{10}\) | \(308\) |
| parallelrisch | \(\frac {1}{2} x^{2} \operatorname {c1} \,a^{4}+\frac {8}{3} x^{6} b^{4} \operatorname {c1} +\frac {16}{5} x^{5} b^{4} \operatorname {b1} +\frac {1}{9} x^{9} \operatorname {b1} \,c^{4}+\frac {24}{5} x^{5} a^{2} b c \operatorname {c1} +\frac {48}{5} x^{5} a \,b^{2} \operatorname {b1} c +6 a^{2} b \operatorname {b1} c \,x^{4}+8 x^{3} a^{2} b^{2} \operatorname {b1} +4 x^{2} \operatorname {b1} \,a^{3} b +a^{3} c \operatorname {c1} \,x^{4}+6 a^{2} b^{2} \operatorname {c1} \,x^{4}+8 a \,b^{3} \operatorname {b1} \,x^{4}+3 x^{8} b^{2} c^{2} \operatorname {c1} +x^{8} \operatorname {b1} b \,c^{3}+\frac {4}{7} x^{7} a \operatorname {b1} \,c^{3}+\frac {32}{7} x^{7} b^{3} c \operatorname {c1} +\frac {24}{7} x^{7} b^{2} \operatorname {b1} \,c^{2}+x^{6} a^{2} c^{2} \operatorname {c1} +\frac {16}{3} x^{6} b^{3} \operatorname {b1} c +\frac {6}{5} x^{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} x^{5} a \,b^{3} \operatorname {c1} +\frac {8}{3} x^{3} \operatorname {c1} \,a^{3} b +\frac {4}{3} x^{3} a^{3} \operatorname {b1} c +\frac {8}{9} x^{9} \operatorname {c1} b \,c^{3}+\frac {1}{2} x^{8} a \,c^{3} \operatorname {c1} +\frac {24}{7} x^{7} a b \,c^{2} \operatorname {c1} +8 x^{6} a \,b^{2} c \operatorname {c1} +4 x^{6} a b \operatorname {b1} \,c^{2}+a^{4} \operatorname {b1} x +\frac {1}{10} c^{4} \operatorname {c1} \,x^{10}\) | \(308\) |
| orering | \(\frac {x \left (63 \operatorname {c1} \,c^{4} x^{9}+560 b \,c^{3} \operatorname {c1} \,x^{8}+70 \operatorname {b1} \,c^{4} x^{8}+315 a \,c^{3} \operatorname {c1} \,x^{7}+1890 b^{2} c^{2} \operatorname {c1} \,x^{7}+630 b \operatorname {b1} \,c^{3} x^{7}+2160 a b \,c^{2} \operatorname {c1} \,x^{6}+360 a \operatorname {b1} \,c^{3} x^{6}+2880 b^{3} c \operatorname {c1} \,x^{6}+2160 b^{2} \operatorname {b1} \,c^{2} x^{6}+630 a^{2} c^{2} \operatorname {c1} \,x^{5}+5040 a \,b^{2} c \operatorname {c1} \,x^{5}+2520 a b \operatorname {b1} \,c^{2} x^{5}+1680 b^{4} \operatorname {c1} \,x^{5}+3360 b^{3} \operatorname {b1} c \,x^{5}+3024 a^{2} b c \operatorname {c1} \,x^{4}+756 a^{2} \operatorname {b1} \,c^{2} x^{4}+4032 a \,b^{3} \operatorname {c1} \,x^{4}+6048 a \,b^{2} \operatorname {b1} c \,x^{4}+2016 b^{4} \operatorname {b1} \,x^{4}+630 a^{3} c \operatorname {c1} \,x^{3}+3780 a^{2} b^{2} \operatorname {c1} \,x^{3}+3780 a^{2} b \operatorname {b1} c \,x^{3}+5040 a \,b^{3} \operatorname {b1} \,x^{3}+1680 a^{3} b \operatorname {c1} \,x^{2}+840 a^{3} \operatorname {b1} c \,x^{2}+5040 a^{2} b^{2} \operatorname {b1} \,x^{2}+315 a^{4} \operatorname {c1} x +2520 a^{3} b \operatorname {b1} x +630 \operatorname {b1} \,a^{4}\right )}{630}\) | \(310\) |
| default | \(\frac {c^{4} \operatorname {c1} \,x^{10}}{10}+\frac {\left (8 \operatorname {c1} b \,c^{3}+\operatorname {b1} \,c^{4}\right ) x^{9}}{9}+\frac {\left (8 \operatorname {b1} b \,c^{3}+\operatorname {c1} \left (2 \left (2 a c +4 b^{2}\right ) c^{2}+16 b^{2} c^{2}\right )\right ) x^{8}}{8}+\frac {\left (\operatorname {b1} \left (2 \left (2 a c +4 b^{2}\right ) c^{2}+16 b^{2} c^{2}\right )+\operatorname {c1} \left (8 a b \,c^{2}+8 \left (2 a c +4 b^{2}\right ) b c \right )\right ) x^{7}}{7}+\frac {\left (\operatorname {b1} \left (8 a b \,c^{2}+8 \left (2 a c +4 b^{2}\right ) b c \right )+\operatorname {c1} \left (2 a^{2} c^{2}+32 a \,b^{2} c +\left (2 a c +4 b^{2}\right )^{2}\right )\right ) x^{6}}{6}+\frac {\left (\operatorname {b1} \left (2 a^{2} c^{2}+32 a \,b^{2} c +\left (2 a c +4 b^{2}\right )^{2}\right )+\operatorname {c1} \left (8 a^{2} b c +8 a b \left (2 a c +4 b^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (\operatorname {b1} \left (8 a^{2} b c +8 a b \left (2 a c +4 b^{2}\right )\right )+\operatorname {c1} \left (2 a^{2} \left (2 a c +4 b^{2}\right )+16 b^{2} a^{2}\right )\right ) x^{4}}{4}+\frac {\left (\operatorname {b1} \left (2 a^{2} \left (2 a c +4 b^{2}\right )+16 b^{2} a^{2}\right )+8 \operatorname {c1} \,a^{3} b \right ) x^{3}}{3}+\frac {\left (\operatorname {c1} \,a^{4}+8 \operatorname {b1} \,a^{3} b \right ) x^{2}}{2}+a^{4} \operatorname {b1} x\) | \(363\) |
Input:
int((c1*x+b1)*(c*x^2+2*b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/10*c^4*c1*x^10+(8/9*c1*b*c^3+1/9*b1*c^4)*x^9+(1/2*a*c^3*c1+3*b^2*c^2*c1+ b1*b*c^3)*x^8+(24/7*a*b*c^2*c1+4/7*a*b1*c^3+32/7*b^3*c*c1+24/7*b^2*b1*c^2) *x^7+(a^2*c^2*c1+8*a*b^2*c*c1+4*a*b*b1*c^2+8/3*b^4*c1+16/3*b^3*b1*c)*x^6+( 24/5*a^2*b*c*c1+6/5*a^2*b1*c^2+32/5*a*b^3*c1+48/5*a*b^2*b1*c+16/5*b^4*b1)* x^5+(a^3*c*c1+6*a^2*b^2*c1+6*a^2*b*b1*c+8*a*b^3*b1)*x^4+(8/3*c1*a^3*b+4/3* a^3*b1*c+8*a^2*b^2*b1)*x^3+(1/2*c1*a^4+4*b1*a^3*b)*x^2+a^4*b1*x
Time = 0.06 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} c_{1} x^{10} + \frac {1}{9} \, {\left (b_{1} c^{4} + 8 \, b c^{3} c_{1}\right )} x^{9} + \frac {1}{2} \, {\left (2 \, b b_{1} c^{3} + {\left (6 \, b^{2} c^{2} + a c^{3}\right )} c_{1}\right )} x^{8} + \frac {4}{7} \, {\left (6 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} + 2 \, {\left (4 \, b^{3} c + 3 \, a b c^{2}\right )} c_{1}\right )} x^{7} + \frac {1}{3} \, {\left (16 \, b^{3} b_{1} c + 12 \, a b b_{1} c^{2} + {\left (8 \, b^{4} + 24 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} c_{1}\right )} x^{6} + a^{4} b_{1} x + \frac {2}{5} \, {\left (8 \, b^{4} b_{1} + 24 \, a b^{2} b_{1} c + 3 \, a^{2} b_{1} c^{2} + 4 \, {\left (4 \, a b^{3} + 3 \, a^{2} b c\right )} c_{1}\right )} x^{5} + {\left (8 \, a b^{3} b_{1} + 6 \, a^{2} b b_{1} c + {\left (6 \, a^{2} b^{2} + a^{3} c\right )} c_{1}\right )} x^{4} + \frac {4}{3} \, {\left (6 \, a^{2} b^{2} b_{1} + a^{3} b_{1} c + 2 \, a^{3} b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (8 \, a^{3} b b_{1} + a^{4} c_{1}\right )} x^{2} \] Input:
integrate((c1*x+b1)*(c*x^2+2*b*x+a)^4,x, algorithm="fricas")
Output:
1/10*c^4*c1*x^10 + 1/9*(b1*c^4 + 8*b*c^3*c1)*x^9 + 1/2*(2*b*b1*c^3 + (6*b^ 2*c^2 + a*c^3)*c1)*x^8 + 4/7*(6*b^2*b1*c^2 + a*b1*c^3 + 2*(4*b^3*c + 3*a*b *c^2)*c1)*x^7 + 1/3*(16*b^3*b1*c + 12*a*b*b1*c^2 + (8*b^4 + 24*a*b^2*c + 3 *a^2*c^2)*c1)*x^6 + a^4*b1*x + 2/5*(8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^ 2 + 4*(4*a*b^3 + 3*a^2*b*c)*c1)*x^5 + (8*a*b^3*b1 + 6*a^2*b*b1*c + (6*a^2* b^2 + a^3*c)*c1)*x^4 + 4/3*(6*a^2*b^2*b1 + a^3*b1*c + 2*a^3*b*c1)*x^3 + 1/ 2*(8*a^3*b*b1 + a^4*c1)*x^2
Time = 0.04 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.19 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^{4} b_{1} x + \frac {c^{4} c_{1} x^{10}}{10} + x^{9} \cdot \left (\frac {8 b c^{3} c_{1}}{9} + \frac {b_{1} c^{4}}{9}\right ) + x^{8} \left (\frac {a c^{3} c_{1}}{2} + 3 b^{2} c^{2} c_{1} + b b_{1} c^{3}\right ) + x^{7} \cdot \left (\frac {24 a b c^{2} c_{1}}{7} + \frac {4 a b_{1} c^{3}}{7} + \frac {32 b^{3} c c_{1}}{7} + \frac {24 b^{2} b_{1} c^{2}}{7}\right ) + x^{6} \left (a^{2} c^{2} c_{1} + 8 a b^{2} c c_{1} + 4 a b b_{1} c^{2} + \frac {8 b^{4} c_{1}}{3} + \frac {16 b^{3} b_{1} c}{3}\right ) + x^{5} \cdot \left (\frac {24 a^{2} b c c_{1}}{5} + \frac {6 a^{2} b_{1} c^{2}}{5} + \frac {32 a b^{3} c_{1}}{5} + \frac {48 a b^{2} b_{1} c}{5} + \frac {16 b^{4} b_{1}}{5}\right ) + x^{4} \left (a^{3} c c_{1} + 6 a^{2} b^{2} c_{1} + 6 a^{2} b b_{1} c + 8 a b^{3} b_{1}\right ) + x^{3} \cdot \left (\frac {8 a^{3} b c_{1}}{3} + \frac {4 a^{3} b_{1} c}{3} + 8 a^{2} b^{2} b_{1}\right ) + x^{2} \left (\frac {a^{4} c_{1}}{2} + 4 a^{3} b b_{1}\right ) \] Input:
integrate((c1*x+b1)*(c*x**2+2*b*x+a)**4,x)
Output:
a**4*b1*x + c**4*c1*x**10/10 + x**9*(8*b*c**3*c1/9 + b1*c**4/9) + x**8*(a* c**3*c1/2 + 3*b**2*c**2*c1 + b*b1*c**3) + x**7*(24*a*b*c**2*c1/7 + 4*a*b1* c**3/7 + 32*b**3*c*c1/7 + 24*b**2*b1*c**2/7) + x**6*(a**2*c**2*c1 + 8*a*b* *2*c*c1 + 4*a*b*b1*c**2 + 8*b**4*c1/3 + 16*b**3*b1*c/3) + x**5*(24*a**2*b* c*c1/5 + 6*a**2*b1*c**2/5 + 32*a*b**3*c1/5 + 48*a*b**2*b1*c/5 + 16*b**4*b1 /5) + x**4*(a**3*c*c1 + 6*a**2*b**2*c1 + 6*a**2*b*b1*c + 8*a*b**3*b1) + x* *3*(8*a**3*b*c1/3 + 4*a**3*b1*c/3 + 8*a**2*b**2*b1) + x**2*(a**4*c1/2 + 4* a**3*b*b1)
Time = 0.03 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} c_{1} x^{10} + \frac {1}{9} \, {\left (b_{1} c^{4} + 8 \, b c^{3} c_{1}\right )} x^{9} + \frac {1}{2} \, {\left (2 \, b b_{1} c^{3} + {\left (6 \, b^{2} c^{2} + a c^{3}\right )} c_{1}\right )} x^{8} + \frac {4}{7} \, {\left (6 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} + 2 \, {\left (4 \, b^{3} c + 3 \, a b c^{2}\right )} c_{1}\right )} x^{7} + \frac {1}{3} \, {\left (16 \, b^{3} b_{1} c + 12 \, a b b_{1} c^{2} + {\left (8 \, b^{4} + 24 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} c_{1}\right )} x^{6} + a^{4} b_{1} x + \frac {2}{5} \, {\left (8 \, b^{4} b_{1} + 24 \, a b^{2} b_{1} c + 3 \, a^{2} b_{1} c^{2} + 4 \, {\left (4 \, a b^{3} + 3 \, a^{2} b c\right )} c_{1}\right )} x^{5} + {\left (8 \, a b^{3} b_{1} + 6 \, a^{2} b b_{1} c + {\left (6 \, a^{2} b^{2} + a^{3} c\right )} c_{1}\right )} x^{4} + \frac {4}{3} \, {\left (6 \, a^{2} b^{2} b_{1} + a^{3} b_{1} c + 2 \, a^{3} b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (8 \, a^{3} b b_{1} + a^{4} c_{1}\right )} x^{2} \] Input:
integrate((c1*x+b1)*(c*x^2+2*b*x+a)^4,x, algorithm="maxima")
Output:
1/10*c^4*c1*x^10 + 1/9*(b1*c^4 + 8*b*c^3*c1)*x^9 + 1/2*(2*b*b1*c^3 + (6*b^ 2*c^2 + a*c^3)*c1)*x^8 + 4/7*(6*b^2*b1*c^2 + a*b1*c^3 + 2*(4*b^3*c + 3*a*b *c^2)*c1)*x^7 + 1/3*(16*b^3*b1*c + 12*a*b*b1*c^2 + (8*b^4 + 24*a*b^2*c + 3 *a^2*c^2)*c1)*x^6 + a^4*b1*x + 2/5*(8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^ 2 + 4*(4*a*b^3 + 3*a^2*b*c)*c1)*x^5 + (8*a*b^3*b1 + 6*a^2*b*b1*c + (6*a^2* b^2 + a^3*c)*c1)*x^4 + 4/3*(6*a^2*b^2*b1 + a^3*b1*c + 2*a^3*b*c1)*x^3 + 1/ 2*(8*a^3*b*b1 + a^4*c1)*x^2
Time = 0.12 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.17 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} c_{1} x^{10} + \frac {1}{9} \, b_{1} c^{4} x^{9} + \frac {8}{9} \, b c^{3} c_{1} x^{9} + b b_{1} c^{3} x^{8} + 3 \, b^{2} c^{2} c_{1} x^{8} + \frac {1}{2} \, a c^{3} c_{1} x^{8} + \frac {24}{7} \, b^{2} b_{1} c^{2} x^{7} + \frac {4}{7} \, a b_{1} c^{3} x^{7} + \frac {32}{7} \, b^{3} c c_{1} x^{7} + \frac {24}{7} \, a b c^{2} c_{1} x^{7} + \frac {16}{3} \, b^{3} b_{1} c x^{6} + 4 \, a b b_{1} c^{2} x^{6} + \frac {8}{3} \, b^{4} c_{1} x^{6} + 8 \, a b^{2} c c_{1} x^{6} + a^{2} c^{2} c_{1} x^{6} + \frac {16}{5} \, b^{4} b_{1} x^{5} + \frac {48}{5} \, a b^{2} b_{1} c x^{5} + \frac {6}{5} \, a^{2} b_{1} c^{2} x^{5} + \frac {32}{5} \, a b^{3} c_{1} x^{5} + \frac {24}{5} \, a^{2} b c c_{1} x^{5} + 8 \, a b^{3} b_{1} x^{4} + 6 \, a^{2} b b_{1} c x^{4} + 6 \, a^{2} b^{2} c_{1} x^{4} + a^{3} c c_{1} x^{4} + 8 \, a^{2} b^{2} b_{1} x^{3} + \frac {4}{3} \, a^{3} b_{1} c x^{3} + \frac {8}{3} \, a^{3} b c_{1} x^{3} + 4 \, a^{3} b b_{1} x^{2} + \frac {1}{2} \, a^{4} c_{1} x^{2} + a^{4} b_{1} x \] Input:
integrate((c1*x+b1)*(c*x^2+2*b*x+a)^4,x, algorithm="giac")
Output:
1/10*c^4*c1*x^10 + 1/9*b1*c^4*x^9 + 8/9*b*c^3*c1*x^9 + b*b1*c^3*x^8 + 3*b^ 2*c^2*c1*x^8 + 1/2*a*c^3*c1*x^8 + 24/7*b^2*b1*c^2*x^7 + 4/7*a*b1*c^3*x^7 + 32/7*b^3*c*c1*x^7 + 24/7*a*b*c^2*c1*x^7 + 16/3*b^3*b1*c*x^6 + 4*a*b*b1*c^ 2*x^6 + 8/3*b^4*c1*x^6 + 8*a*b^2*c*c1*x^6 + a^2*c^2*c1*x^6 + 16/5*b^4*b1*x ^5 + 48/5*a*b^2*b1*c*x^5 + 6/5*a^2*b1*c^2*x^5 + 32/5*a*b^3*c1*x^5 + 24/5*a ^2*b*c*c1*x^5 + 8*a*b^3*b1*x^4 + 6*a^2*b*b1*c*x^4 + 6*a^2*b^2*c1*x^4 + a^3 *c*c1*x^4 + 8*a^2*b^2*b1*x^3 + 4/3*a^3*b1*c*x^3 + 8/3*a^3*b*c1*x^3 + 4*a^3 *b*b1*x^2 + 1/2*a^4*c1*x^2 + a^4*b1*x
Time = 0.15 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=x^9\,\left (\frac {b_{1}\,c^4}{9}+\frac {8\,b\,c_{1}\,c^3}{9}\right )+x^3\,\left (\frac {8\,c_{1}\,a^3\,b}{3}+\frac {4\,b_{1}\,c\,a^3}{3}+8\,b_{1}\,a^2\,b^2\right )+x^8\,\left (3\,c_{1}\,b^2\,c^2+b_{1}\,b\,c^3+\frac {a\,c_{1}\,c^3}{2}\right )+x^5\,\left (\frac {24\,c_{1}\,a^2\,b\,c}{5}+\frac {6\,b_{1}\,a^2\,c^2}{5}+\frac {32\,c_{1}\,a\,b^3}{5}+\frac {48\,b_{1}\,a\,b^2\,c}{5}+\frac {16\,b_{1}\,b^4}{5}\right )+x^6\,\left (c_{1}\,a^2\,c^2+8\,c_{1}\,a\,b^2\,c+4\,b_{1}\,a\,b\,c^2+\frac {8\,c_{1}\,b^4}{3}+\frac {16\,b_{1}\,b^3\,c}{3}\right )+x^4\,\left (c\,c_{1}\,a^3+6\,c_{1}\,a^2\,b^2+6\,b_{1}\,c\,a^2\,b+8\,b_{1}\,a\,b^3\right )+x^7\,\left (\frac {32\,c_{1}\,b^3\,c}{7}+\frac {24\,b_{1}\,b^2\,c^2}{7}+\frac {24\,a\,c_{1}\,b\,c^2}{7}+\frac {4\,a\,b_{1}\,c^3}{7}\right )+x^2\,\left (\frac {c_{1}\,a^4}{2}+4\,b\,b_{1}\,a^3\right )+\frac {c^4\,c_{1}\,x^{10}}{10}+a^4\,b_{1}\,x \] Input:
int((b1 + c1*x)*(a + 2*b*x + c*x^2)^4,x)
Output:
x^9*((b1*c^4)/9 + (8*b*c^3*c1)/9) + x^3*(8*a^2*b^2*b1 + (8*a^3*b*c1)/3 + ( 4*a^3*b1*c)/3) + x^8*(3*b^2*c^2*c1 + (a*c^3*c1)/2 + b*b1*c^3) + x^5*((16*b ^4*b1)/5 + (6*a^2*b1*c^2)/5 + (32*a*b^3*c1)/5 + (48*a*b^2*b1*c)/5 + (24*a^ 2*b*c*c1)/5) + x^6*((8*b^4*c1)/3 + a^2*c^2*c1 + (16*b^3*b1*c)/3 + 4*a*b*b1 *c^2 + 8*a*b^2*c*c1) + x^4*(6*a^2*b^2*c1 + 8*a*b^3*b1 + a^3*c*c1 + 6*a^2*b *b1*c) + x^7*((24*b^2*b1*c^2)/7 + (4*a*b1*c^3)/7 + (32*b^3*c*c1)/7 + (24*a *b*c^2*c1)/7) + x^2*((a^4*c1)/2 + 4*a^3*b*b1) + (c^4*c1*x^10)/10 + a^4*b1* x
Time = 0.16 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.17 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {x \left (63 c^{4} \mathit {c1} \,x^{9}+560 b \,c^{3} \mathit {c1} \,x^{8}+70 \mathit {b1} \,c^{4} x^{8}+315 a \,c^{3} \mathit {c1} \,x^{7}+1890 b^{2} c^{2} \mathit {c1} \,x^{7}+630 b \mathit {b1} \,c^{3} x^{7}+2160 a b \,c^{2} \mathit {c1} \,x^{6}+360 a \mathit {b1} \,c^{3} x^{6}+2880 b^{3} c \mathit {c1} \,x^{6}+2160 b^{2} \mathit {b1} \,c^{2} x^{6}+630 a^{2} c^{2} \mathit {c1} \,x^{5}+5040 a \,b^{2} c \mathit {c1} \,x^{5}+2520 a b \mathit {b1} \,c^{2} x^{5}+1680 b^{4} \mathit {c1} \,x^{5}+3360 b^{3} \mathit {b1} c \,x^{5}+3024 a^{2} b c \mathit {c1} \,x^{4}+756 a^{2} \mathit {b1} \,c^{2} x^{4}+4032 a \,b^{3} \mathit {c1} \,x^{4}+6048 a \,b^{2} \mathit {b1} c \,x^{4}+2016 b^{4} \mathit {b1} \,x^{4}+630 a^{3} c \mathit {c1} \,x^{3}+3780 a^{2} b^{2} \mathit {c1} \,x^{3}+3780 a^{2} b \mathit {b1} c \,x^{3}+5040 a \,b^{3} \mathit {b1} \,x^{3}+1680 a^{3} b \mathit {c1} \,x^{2}+840 a^{3} \mathit {b1} c \,x^{2}+5040 a^{2} b^{2} \mathit {b1} \,x^{2}+315 a^{4} \mathit {c1} x +2520 a^{3} b \mathit {b1} x +630 a^{4} \mathit {b1} \right )}{630} \] Input:
int((c1*x+b1)*(c*x^2+2*b*x+a)^4,x)
Output:
(x*(630*a**4*b1 + 315*a**4*c1*x + 2520*a**3*b*b1*x + 1680*a**3*b*c1*x**2 + 840*a**3*b1*c*x**2 + 630*a**3*c*c1*x**3 + 5040*a**2*b**2*b1*x**2 + 3780*a **2*b**2*c1*x**3 + 3780*a**2*b*b1*c*x**3 + 3024*a**2*b*c*c1*x**4 + 756*a** 2*b1*c**2*x**4 + 630*a**2*c**2*c1*x**5 + 5040*a*b**3*b1*x**3 + 4032*a*b**3 *c1*x**4 + 6048*a*b**2*b1*c*x**4 + 5040*a*b**2*c*c1*x**5 + 2520*a*b*b1*c** 2*x**5 + 2160*a*b*c**2*c1*x**6 + 360*a*b1*c**3*x**6 + 315*a*c**3*c1*x**7 + 2016*b**4*b1*x**4 + 1680*b**4*c1*x**5 + 3360*b**3*b1*c*x**5 + 2880*b**3*c *c1*x**6 + 2160*b**2*b1*c**2*x**6 + 1890*b**2*c**2*c1*x**7 + 630*b*b1*c**3 *x**7 + 560*b*c**3*c1*x**8 + 70*b1*c**4*x**8 + 63*c**4*c1*x**9))/630