\(\int (d+e x)^2 (3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4) \, dx\) [125]

Optimal result

Integrand size = 36, antiderivative size = 151 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {7 d^2 x^2}{2}+\frac {7}{3} d (3 d+2 e) x^3-\frac {1}{4} \left (4 d^2-42 d e-7 e^2\right ) x^4+\frac {1}{5} \left (17 d^2-8 d e+21 e^2\right ) x^5-\frac {1}{6} \left (17 d^2-34 d e+4 e^2\right ) x^6+\frac {1}{7} \left (20 d^2-34 d e+17 e^2\right ) x^7+\frac {1}{8} (40 d-17 e) e x^8+\frac {20 e^2 x^9}{9}+\frac {2 (d+e x)^3}{e} \] Output:

7/2*d^2*x^2+7/3*d*(3*d+2*e)*x^3-1/4*(4*d^2-42*d*e-7*e^2)*x^4+1/5*(17*d^2-8 
*d*e+21*e^2)*x^5-1/6*(17*d^2-34*d*e+4*e^2)*x^6+1/7*(20*d^2-34*d*e+17*e^2)* 
x^7+1/8*(40*d-17*e)*e*x^8+20/9*e^2*x^9+2*(e*x+d)^3/e
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {e^2 x^3 \left (5040+4410 x+10584 x^2-1680 x^3+6120 x^4-5355 x^5+5600 x^6\right )}{2520}+d^2 \left (6 x+\frac {7 x^2}{2}+7 x^3-x^4+\frac {17 x^5}{5}-\frac {17 x^6}{6}+\frac {20 x^7}{7}\right )+d e \left (6 x^2+\frac {14 x^3}{3}+\frac {21 x^4}{2}-\frac {8 x^5}{5}+\frac {17 x^6}{3}-\frac {34 x^7}{7}+5 x^8\right ) \] Input:

Integrate[(d + e*x)^2*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]
 

Output:

(e^2*x^3*(5040 + 4410*x + 10584*x^2 - 1680*x^3 + 6120*x^4 - 5355*x^5 + 560 
0*x^6))/2520 + d^2*(6*x + (7*x^2)/2 + 7*x^3 - x^4 + (17*x^5)/5 - (17*x^6)/ 
6 + (20*x^7)/7) + d*e*(6*x^2 + (14*x^3)/3 + (21*x^4)/2 - (8*x^5)/5 + (17*x 
^6)/3 - (34*x^7)/7 + 5*x^8)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2159, 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.

\(\displaystyle \int \left (5 x^2+2 x+3\right ) \left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)^2 \, dx\)

\(\Big \downarrow \) 2159

\(\displaystyle \int \left (x^6 \left (20 d^2-34 d e+17 e^2\right )-x^5 \left (17 d^2-34 d e+4 e^2\right )+x^4 \left (17 d^2-8 d e+21 e^2\right )-x^3 \left (4 d^2-42 d e-7 e^2\right )+x^2 \left (21 d^2+14 d e+6 e^2\right )+6 d^2+e x^7 (40 d-17 e)+d x (7 d+12 e)+20 e^2 x^8\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} x^7 \left (20 d^2-34 d e+17 e^2\right )-\frac {1}{6} x^6 \left (17 d^2-34 d e+4 e^2\right )+\frac {1}{5} x^5 \left (17 d^2-8 d e+21 e^2\right )-\frac {1}{4} x^4 \left (4 d^2-42 d e-7 e^2\right )+\frac {1}{3} x^3 \left (21 d^2+14 d e+6 e^2\right )+6 d^2 x+\frac {1}{8} e x^8 (40 d-17 e)+\frac {1}{2} d x^2 (7 d+12 e)+\frac {20 e^2 x^9}{9}\)

Input:

Int[(d + e*x)^2*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]
 

Output:

6*d^2*x + (d*(7*d + 12*e)*x^2)/2 + ((21*d^2 + 14*d*e + 6*e^2)*x^3)/3 - ((4 
*d^2 - 42*d*e - 7*e^2)*x^4)/4 + ((17*d^2 - 8*d*e + 21*e^2)*x^5)/5 - ((17*d 
^2 - 34*d*e + 4*e^2)*x^6)/6 + ((20*d^2 - 34*d*e + 17*e^2)*x^7)/7 + ((40*d 
- 17*e)*e*x^8)/8 + (20*e^2*x^9)/9
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2159

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.92

method	result	size
norman	\(\frac {20 e^{2} x^{9}}{9}+\left (5 d e -\frac {17}{8} e^{2}\right ) x^{8}+\left (\frac {20}{7} d^{2}-\frac {34}{7} d e +\frac {17}{7} e^{2}\right ) x^{7}+\left (-\frac {17}{6} d^{2}+\frac {17}{3} d e -\frac {2}{3} e^{2}\right ) x^{6}+\left (\frac {17}{5} d^{2}-\frac {8}{5} d e +\frac {21}{5} e^{2}\right ) x^{5}+\left (-d^{2}+\frac {21}{2} d e +\frac {7}{4} e^{2}\right ) x^{4}+\left (7 d^{2}+\frac {14}{3} d e +2 e^{2}\right ) x^{3}+\left (\frac {7}{2} d^{2}+6 d e \right ) x^{2}+6 d^{2} x\)	\(139\)
default	\(\frac {20 e^{2} x^{9}}{9}+\frac {\left (40 d e -17 e^{2}\right ) x^{8}}{8}+\frac {\left (20 d^{2}-34 d e +17 e^{2}\right ) x^{7}}{7}+\frac {\left (-17 d^{2}+34 d e -4 e^{2}\right ) x^{6}}{6}+\frac {\left (17 d^{2}-8 d e +21 e^{2}\right ) x^{5}}{5}+\frac {\left (-4 d^{2}+42 d e +7 e^{2}\right ) x^{4}}{4}+\frac {\left (21 d^{2}+14 d e +6 e^{2}\right ) x^{3}}{3}+\frac {\left (7 d^{2}+12 d e \right ) x^{2}}{2}+6 d^{2} x\)	\(146\)
orering	\(\frac {x \left (5600 x^{8} e^{2}+12600 x^{7} d e -5355 x^{7} e^{2}+7200 x^{6} d^{2}-12240 x^{6} d e +6120 x^{6} e^{2}-7140 x^{5} d^{2}+14280 x^{5} d e -1680 e^{2} x^{5}+8568 x^{4} d^{2}-4032 x^{4} d e +10584 x^{4} e^{2}-2520 x^{3} d^{2}+26460 x^{3} d e +4410 e^{2} x^{3}+17640 d^{2} x^{2}+11760 x^{2} d e +5040 e^{2} x^{2}+8820 d^{2} x +15120 d e x +15120 d^{2}\right )}{2520}\)	\(159\)
gosper	\(\frac {20}{9} e^{2} x^{9}+5 x^{8} d e -\frac {17}{8} x^{8} e^{2}+\frac {20}{7} x^{7} d^{2}-\frac {34}{7} x^{7} d e +\frac {17}{7} x^{7} e^{2}-\frac {17}{6} x^{6} d^{2}+\frac {17}{3} x^{6} d e -\frac {2}{3} x^{6} e^{2}+\frac {17}{5} x^{5} d^{2}-\frac {8}{5} x^{5} d e +\frac {21}{5} e^{2} x^{5}-x^{4} d^{2}+\frac {21}{2} x^{4} d e +\frac {7}{4} x^{4} e^{2}+7 x^{3} d^{2}+\frac {14}{3} x^{3} d e +2 e^{2} x^{3}+\frac {7}{2} d^{2} x^{2}+6 x^{2} d e +6 d^{2} x\)	\(161\)
risch	\(\frac {20}{9} e^{2} x^{9}+5 x^{8} d e -\frac {17}{8} x^{8} e^{2}+\frac {20}{7} x^{7} d^{2}-\frac {34}{7} x^{7} d e +\frac {17}{7} x^{7} e^{2}-\frac {17}{6} x^{6} d^{2}+\frac {17}{3} x^{6} d e -\frac {2}{3} x^{6} e^{2}+\frac {17}{5} x^{5} d^{2}-\frac {8}{5} x^{5} d e +\frac {21}{5} e^{2} x^{5}-x^{4} d^{2}+\frac {21}{2} x^{4} d e +\frac {7}{4} x^{4} e^{2}+7 x^{3} d^{2}+\frac {14}{3} x^{3} d e +2 e^{2} x^{3}+\frac {7}{2} d^{2} x^{2}+6 x^{2} d e +6 d^{2} x\)	\(161\)
parallelrisch	\(\frac {20}{9} e^{2} x^{9}+5 x^{8} d e -\frac {17}{8} x^{8} e^{2}+\frac {20}{7} x^{7} d^{2}-\frac {34}{7} x^{7} d e +\frac {17}{7} x^{7} e^{2}-\frac {17}{6} x^{6} d^{2}+\frac {17}{3} x^{6} d e -\frac {2}{3} x^{6} e^{2}+\frac {17}{5} x^{5} d^{2}-\frac {8}{5} x^{5} d e +\frac {21}{5} e^{2} x^{5}-x^{4} d^{2}+\frac {21}{2} x^{4} d e +\frac {7}{4} x^{4} e^{2}+7 x^{3} d^{2}+\frac {14}{3} x^{3} d e +2 e^{2} x^{3}+\frac {7}{2} d^{2} x^{2}+6 x^{2} d e +6 d^{2} x\)	\(161\)

Input:

int((e*x+d)^2*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x,method=_RETURNVERBOS 
E)
 

Output:

20/9*e^2*x^9+(5*d*e-17/8*e^2)*x^8+(20/7*d^2-34/7*d*e+17/7*e^2)*x^7+(-17/6* 
d^2+17/3*d*e-2/3*e^2)*x^6+(17/5*d^2-8/5*d*e+21/5*e^2)*x^5+(-d^2+21/2*d*e+7 
/4*e^2)*x^4+(7*d^2+14/3*d*e+2*e^2)*x^3+(7/2*d^2+6*d*e)*x^2+6*d^2*x
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {20}{9} \, e^{2} x^{9} + \frac {1}{8} \, {\left (40 \, d e - 17 \, e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (20 \, d^{2} - 34 \, d e + 17 \, e^{2}\right )} x^{7} - \frac {1}{6} \, {\left (17 \, d^{2} - 34 \, d e + 4 \, e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (17 \, d^{2} - 8 \, d e + 21 \, e^{2}\right )} x^{5} - \frac {1}{4} \, {\left (4 \, d^{2} - 42 \, d e - 7 \, e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, d^{2} + 14 \, d e + 6 \, e^{2}\right )} x^{3} + 6 \, d^{2} x + \frac {1}{2} \, {\left (7 \, d^{2} + 12 \, d e\right )} x^{2} \] Input:

integrate((e*x+d)^2*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fr 
icas")
 

Output:

20/9*e^2*x^9 + 1/8*(40*d*e - 17*e^2)*x^8 + 1/7*(20*d^2 - 34*d*e + 17*e^2)* 
x^7 - 1/6*(17*d^2 - 34*d*e + 4*e^2)*x^6 + 1/5*(17*d^2 - 8*d*e + 21*e^2)*x^ 
5 - 1/4*(4*d^2 - 42*d*e - 7*e^2)*x^4 + 1/3*(21*d^2 + 14*d*e + 6*e^2)*x^3 + 
 6*d^2*x + 1/2*(7*d^2 + 12*d*e)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=6 d^{2} x + \frac {20 e^{2} x^{9}}{9} + x^{8} \cdot \left (5 d e - \frac {17 e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {20 d^{2}}{7} - \frac {34 d e}{7} + \frac {17 e^{2}}{7}\right ) + x^{6} \left (- \frac {17 d^{2}}{6} + \frac {17 d e}{3} - \frac {2 e^{2}}{3}\right ) + x^{5} \cdot \left (\frac {17 d^{2}}{5} - \frac {8 d e}{5} + \frac {21 e^{2}}{5}\right ) + x^{4} \left (- d^{2} + \frac {21 d e}{2} + \frac {7 e^{2}}{4}\right ) + x^{3} \cdot \left (7 d^{2} + \frac {14 d e}{3} + 2 e^{2}\right ) + x^{2} \cdot \left (\frac {7 d^{2}}{2} + 6 d e\right ) \] Input:

integrate((e*x+d)**2*(5*x**2+2*x+3)*(4*x**4-5*x**3+3*x**2+x+2),x)
 

Output:

6*d**2*x + 20*e**2*x**9/9 + x**8*(5*d*e - 17*e**2/8) + x**7*(20*d**2/7 - 3 
4*d*e/7 + 17*e**2/7) + x**6*(-17*d**2/6 + 17*d*e/3 - 2*e**2/3) + x**5*(17* 
d**2/5 - 8*d*e/5 + 21*e**2/5) + x**4*(-d**2 + 21*d*e/2 + 7*e**2/4) + x**3* 
(7*d**2 + 14*d*e/3 + 2*e**2) + x**2*(7*d**2/2 + 6*d*e)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {20}{9} \, e^{2} x^{9} + \frac {1}{8} \, {\left (40 \, d e - 17 \, e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (20 \, d^{2} - 34 \, d e + 17 \, e^{2}\right )} x^{7} - \frac {1}{6} \, {\left (17 \, d^{2} - 34 \, d e + 4 \, e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (17 \, d^{2} - 8 \, d e + 21 \, e^{2}\right )} x^{5} - \frac {1}{4} \, {\left (4 \, d^{2} - 42 \, d e - 7 \, e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, d^{2} + 14 \, d e + 6 \, e^{2}\right )} x^{3} + 6 \, d^{2} x + \frac {1}{2} \, {\left (7 \, d^{2} + 12 \, d e\right )} x^{2} \] Input:

integrate((e*x+d)^2*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="ma 
xima")
 

Output:

20/9*e^2*x^9 + 1/8*(40*d*e - 17*e^2)*x^8 + 1/7*(20*d^2 - 34*d*e + 17*e^2)* 
x^7 - 1/6*(17*d^2 - 34*d*e + 4*e^2)*x^6 + 1/5*(17*d^2 - 8*d*e + 21*e^2)*x^ 
5 - 1/4*(4*d^2 - 42*d*e - 7*e^2)*x^4 + 1/3*(21*d^2 + 14*d*e + 6*e^2)*x^3 + 
 6*d^2*x + 1/2*(7*d^2 + 12*d*e)*x^2
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.06 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {20}{9} \, e^{2} x^{9} + 5 \, d e x^{8} - \frac {17}{8} \, e^{2} x^{8} + \frac {20}{7} \, d^{2} x^{7} - \frac {34}{7} \, d e x^{7} + \frac {17}{7} \, e^{2} x^{7} - \frac {17}{6} \, d^{2} x^{6} + \frac {17}{3} \, d e x^{6} - \frac {2}{3} \, e^{2} x^{6} + \frac {17}{5} \, d^{2} x^{5} - \frac {8}{5} \, d e x^{5} + \frac {21}{5} \, e^{2} x^{5} - d^{2} x^{4} + \frac {21}{2} \, d e x^{4} + \frac {7}{4} \, e^{2} x^{4} + 7 \, d^{2} x^{3} + \frac {14}{3} \, d e x^{3} + 2 \, e^{2} x^{3} + \frac {7}{2} \, d^{2} x^{2} + 6 \, d e x^{2} + 6 \, d^{2} x \] Input:

integrate((e*x+d)^2*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="gi 
ac")
                                                                                    
                                                                                    
 

Output:

20/9*e^2*x^9 + 5*d*e*x^8 - 17/8*e^2*x^8 + 20/7*d^2*x^7 - 34/7*d*e*x^7 + 17 
/7*e^2*x^7 - 17/6*d^2*x^6 + 17/3*d*e*x^6 - 2/3*e^2*x^6 + 17/5*d^2*x^5 - 8/ 
5*d*e*x^5 + 21/5*e^2*x^5 - d^2*x^4 + 21/2*d*e*x^4 + 7/4*e^2*x^4 + 7*d^2*x^ 
3 + 14/3*d*e*x^3 + 2*e^2*x^3 + 7/2*d^2*x^2 + 6*d*e*x^2 + 6*d^2*x
 

Mupad [B] (verification not implemented)

Time = 18.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=x^3\,\left (7\,d^2+\frac {14\,d\,e}{3}+2\,e^2\right )+x^4\,\left (-d^2+\frac {21\,d\,e}{2}+\frac {7\,e^2}{4}\right )-x^6\,\left (\frac {17\,d^2}{6}-\frac {17\,d\,e}{3}+\frac {2\,e^2}{3}\right )+x^5\,\left (\frac {17\,d^2}{5}-\frac {8\,d\,e}{5}+\frac {21\,e^2}{5}\right )+x^7\,\left (\frac {20\,d^2}{7}-\frac {34\,d\,e}{7}+\frac {17\,e^2}{7}\right )+6\,d^2\,x+\frac {20\,e^2\,x^9}{9}+\frac {d\,x^2\,\left (7\,d+12\,e\right )}{2}+\frac {e\,x^8\,\left (40\,d-17\,e\right )}{8} \] Input:

int((d + e*x)^2*(2*x + 5*x^2 + 3)*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2),x)
 

Output:

x^3*((14*d*e)/3 + 7*d^2 + 2*e^2) + x^4*((21*d*e)/2 - d^2 + (7*e^2)/4) - x^ 
6*((17*d^2)/6 - (17*d*e)/3 + (2*e^2)/3) + x^5*((17*d^2)/5 - (8*d*e)/5 + (2 
1*e^2)/5) + x^7*((20*d^2)/7 - (34*d*e)/7 + (17*e^2)/7) + 6*d^2*x + (20*e^2 
*x^9)/9 + (d*x^2*(7*d + 12*e))/2 + (e*x^8*(40*d - 17*e))/8
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {x \left (5600 e^{2} x^{8}+12600 d e \,x^{7}-5355 e^{2} x^{7}+7200 d^{2} x^{6}-12240 d e \,x^{6}+6120 e^{2} x^{6}-7140 d^{2} x^{5}+14280 d e \,x^{5}-1680 e^{2} x^{5}+8568 d^{2} x^{4}-4032 d e \,x^{4}+10584 e^{2} x^{4}-2520 d^{2} x^{3}+26460 d e \,x^{3}+4410 e^{2} x^{3}+17640 d^{2} x^{2}+11760 d e \,x^{2}+5040 e^{2} x^{2}+8820 d^{2} x +15120 d e x +15120 d^{2}\right )}{2520} \] Input:

int((e*x+d)^2*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x)
 

Output:

(x*(7200*d**2*x**6 - 7140*d**2*x**5 + 8568*d**2*x**4 - 2520*d**2*x**3 + 17 
640*d**2*x**2 + 8820*d**2*x + 15120*d**2 + 12600*d*e*x**7 - 12240*d*e*x**6 
 + 14280*d*e*x**5 - 4032*d*e*x**4 + 26460*d*e*x**3 + 11760*d*e*x**2 + 1512 
0*d*e*x + 5600*e**2*x**8 - 5355*e**2*x**7 + 6120*e**2*x**6 - 1680*e**2*x** 
5 + 10584*e**2*x**4 + 4410*e**2*x**3 + 5040*e**2*x**2))/2520