\(\int (b+2 c x) (d+e x) (a+b x+c x^2)^2 \, dx\) [528]

Optimal result

Integrand size = 24, antiderivative size = 153 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 b d x+\frac {1}{2} a \left (2 b^2 d+2 a c d+a b e\right ) x^2+\frac {1}{3} \left (b^3 d+6 a b c d+2 a b^2 e+2 a^2 c e\right ) x^3+\frac {1}{4} \left (4 b^2 c d+4 a c^2 d+b^3 e+6 a b c e\right ) x^4+\frac {1}{5} c \left (5 b c d+4 b^2 e+4 a c e\right ) x^5+\frac {1}{6} c^2 (2 c d+5 b e) x^6+\frac {2}{7} c^3 e x^7 \] Output:

a^2*b*d*x+1/2*a*(a*b*e+2*a*c*d+2*b^2*d)*x^2+1/3*(2*a^2*c*e+2*a*b^2*e+6*a*b 
*c*d+b^3*d)*x^3+1/4*(6*a*b*c*e+4*a*c^2*d+b^3*e+4*b^2*c*d)*x^4+1/5*c*(4*a*c 
*e+4*b^2*e+5*b*c*d)*x^5+1/6*c^2*(5*b*e+2*c*d)*x^6+2/7*c^3*e*x^7
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 b d x+\frac {1}{2} a \left (2 b^2 d+2 a c d+a b e\right ) x^2+\frac {1}{3} \left (b^3 d+6 a b c d+2 a b^2 e+2 a^2 c e\right ) x^3+\frac {1}{4} \left (4 b^2 c d+4 a c^2 d+b^3 e+6 a b c e\right ) x^4+\frac {1}{5} c \left (5 b c d+4 b^2 e+4 a c e\right ) x^5+\frac {1}{6} c^2 (2 c d+5 b e) x^6+\frac {2}{7} c^3 e x^7 \] Input:

Integrate[(b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^2,x]
 

Output:

a^2*b*d*x + (a*(2*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + ((b^3*d + 6*a*b*c*d + 
2*a*b^2*e + 2*a^2*c*e)*x^3)/3 + ((4*b^2*c*d + 4*a*c^2*d + b^3*e + 6*a*b*c* 
e)*x^4)/4 + (c*(5*b*c*d + 4*b^2*e + 4*a*c*e)*x^5)/5 + (c^2*(2*c*d + 5*b*e) 
*x^6)/6 + (2*c^3*e*x^7)/7
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.

\(\displaystyle \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (x^2 \left (2 a^2 c e+2 a b^2 e+6 a b c d+b^3 d\right )+a^2 b d+c x^4 \left (4 a c e+4 b^2 e+5 b c d\right )+a x \left (a b e+2 a c d+2 b^2 d\right )+x^3 \left (6 a b c e+4 a c^2 d+b^3 e+4 b^2 c d\right )+c^2 x^5 (5 b e+2 c d)+2 c^3 e x^6\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \left (2 a^2 c e+2 a b^2 e+6 a b c d+b^3 d\right )+a^2 b d x+\frac {1}{5} c x^5 \left (4 a c e+4 b^2 e+5 b c d\right )+\frac {1}{2} a x^2 \left (a b e+2 a c d+2 b^2 d\right )+\frac {1}{4} x^4 \left (6 a b c e+4 a c^2 d+b^3 e+4 b^2 c d\right )+\frac {1}{6} c^2 x^6 (5 b e+2 c d)+\frac {2}{7} c^3 e x^7\)

Input:

Int[(b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^2,x]
 

Output:

a^2*b*d*x + (a*(2*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + ((b^3*d + 6*a*b*c*d + 
2*a*b^2*e + 2*a^2*c*e)*x^3)/3 + ((4*b^2*c*d + 4*a*c^2*d + b^3*e + 6*a*b*c* 
e)*x^4)/4 + (c*(5*b*c*d + 4*b^2*e + 4*a*c*e)*x^5)/5 + (c^2*(2*c*d + 5*b*e) 
*x^6)/6 + (2*c^3*e*x^7)/7
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {2 c^{3} e \,x^{7}}{7}+\left (\frac {5}{6} b \,c^{2} e +\frac {1}{3} d \,c^{3}\right ) x^{6}+\left (\frac {4}{5} c^{2} a e +\frac {4}{5} c e \,b^{2}+b \,c^{2} d \right ) x^{5}+\left (\frac {3}{2} a b c e +a \,c^{2} d +\frac {1}{4} e \,b^{3}+c d \,b^{2}\right ) x^{4}+\left (\frac {2}{3} a^{2} c e +\frac {2}{3} e a \,b^{2}+2 a b c d +\frac {1}{3} b^{3} d \right ) x^{3}+\left (\frac {1}{2} e \,a^{2} b +a^{2} c d +d a \,b^{2}\right ) x^{2}+a^{2} b d x\)	\(145\)
gosper	\(\frac {2}{7} c^{3} e \,x^{7}+\frac {5}{6} x^{6} b \,c^{2} e +\frac {1}{3} x^{6} d \,c^{3}+\frac {4}{5} x^{5} c^{2} a e +\frac {4}{5} x^{5} c e \,b^{2}+b \,c^{2} d \,x^{5}+\frac {3}{2} x^{4} a b c e +a \,c^{2} d \,x^{4}+\frac {1}{4} e \,b^{3} x^{4}+b^{2} c d \,x^{4}+\frac {2}{3} x^{3} a^{2} c e +\frac {2}{3} a \,b^{2} e \,x^{3}+2 a b c d \,x^{3}+\frac {1}{3} b^{3} d \,x^{3}+\frac {1}{2} e \,a^{2} b \,x^{2}+a^{2} c d \,x^{2}+a \,b^{2} d \,x^{2}+a^{2} b d x\)	\(168\)
risch	\(\frac {2}{7} c^{3} e \,x^{7}+\frac {5}{6} x^{6} b \,c^{2} e +\frac {1}{3} x^{6} d \,c^{3}+\frac {4}{5} x^{5} c^{2} a e +\frac {4}{5} x^{5} c e \,b^{2}+b \,c^{2} d \,x^{5}+\frac {3}{2} x^{4} a b c e +a \,c^{2} d \,x^{4}+\frac {1}{4} e \,b^{3} x^{4}+b^{2} c d \,x^{4}+\frac {2}{3} x^{3} a^{2} c e +\frac {2}{3} a \,b^{2} e \,x^{3}+2 a b c d \,x^{3}+\frac {1}{3} b^{3} d \,x^{3}+\frac {1}{2} e \,a^{2} b \,x^{2}+a^{2} c d \,x^{2}+a \,b^{2} d \,x^{2}+a^{2} b d x\)	\(168\)
parallelrisch	\(\frac {2}{7} c^{3} e \,x^{7}+\frac {5}{6} x^{6} b \,c^{2} e +\frac {1}{3} x^{6} d \,c^{3}+\frac {4}{5} x^{5} c^{2} a e +\frac {4}{5} x^{5} c e \,b^{2}+b \,c^{2} d \,x^{5}+\frac {3}{2} x^{4} a b c e +a \,c^{2} d \,x^{4}+\frac {1}{4} e \,b^{3} x^{4}+b^{2} c d \,x^{4}+\frac {2}{3} x^{3} a^{2} c e +\frac {2}{3} a \,b^{2} e \,x^{3}+2 a b c d \,x^{3}+\frac {1}{3} b^{3} d \,x^{3}+\frac {1}{2} e \,a^{2} b \,x^{2}+a^{2} c d \,x^{2}+a \,b^{2} d \,x^{2}+a^{2} b d x\)	\(168\)
orering	\(\frac {x \left (120 c^{3} e \,x^{6}+350 b \,c^{2} e \,x^{5}+140 c^{3} d \,x^{5}+336 a \,c^{2} e \,x^{4}+336 b^{2} c e \,x^{4}+420 b \,c^{2} d \,x^{4}+630 a b c e \,x^{3}+420 a \,c^{2} d \,x^{3}+105 b^{3} e \,x^{3}+420 b^{2} c d \,x^{3}+280 a^{2} c e \,x^{2}+280 a \,b^{2} e \,x^{2}+840 a b c d \,x^{2}+140 b^{3} d \,x^{2}+210 e \,a^{2} b x +420 a^{2} c d x +420 d a \,b^{2} x +420 a^{2} b d \right )}{420}\)	\(170\)
default	\(\frac {2 c^{3} e \,x^{7}}{7}+\frac {\left (\left (b e +2 c d \right ) c^{2}+4 b \,c^{2} e \right ) x^{6}}{6}+\frac {\left (b \,c^{2} d +2 \left (b e +2 c d \right ) b c +2 c e \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 c d \,b^{2}+\left (b e +2 c d \right ) \left (2 a c +b^{2}\right )+4 a b c e \right ) x^{4}}{4}+\frac {\left (b d \left (2 a c +b^{2}\right )+2 \left (b e +2 c d \right ) a b +2 a^{2} c e \right ) x^{3}}{3}+\frac {\left (2 d a \,b^{2}+\left (b e +2 c d \right ) a^{2}\right ) x^{2}}{2}+a^{2} b d x\)	\(176\)

Input:

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

Output:

2/7*c^3*e*x^7+(5/6*b*c^2*e+1/3*d*c^3)*x^6+(4/5*c^2*a*e+4/5*c*e*b^2+b*c^2*d 
)*x^5+(3/2*a*b*c*e+a*c^2*d+1/4*e*b^3+c*d*b^2)*x^4+(2/3*a^2*c*e+2/3*e*a*b^2 
+2*a*b*c*d+1/3*b^3*d)*x^3+(1/2*e*a^2*b+a^2*c*d+d*a*b^2)*x^2+a^2*b*d*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{7} \, c^{3} e x^{7} + \frac {1}{6} \, {\left (2 \, c^{3} d + 5 \, b c^{2} e\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d + 4 \, {\left (b^{2} c + a c^{2}\right )} e\right )} x^{5} + a^{2} b d x + \frac {1}{4} \, {\left (4 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 2 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{2} b e + 2 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} \] Input:

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

Output:

2/7*c^3*e*x^7 + 1/6*(2*c^3*d + 5*b*c^2*e)*x^6 + 1/5*(5*b*c^2*d + 4*(b^2*c 
+ a*c^2)*e)*x^5 + a^2*b*d*x + 1/4*(4*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e 
)*x^4 + 1/3*((b^3 + 6*a*b*c)*d + 2*(a*b^2 + a^2*c)*e)*x^3 + 1/2*(a^2*b*e + 
 2*(a*b^2 + a^2*c)*d)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.10 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^{2} b d x + \frac {2 c^{3} e x^{7}}{7} + x^{6} \cdot \left (\frac {5 b c^{2} e}{6} + \frac {c^{3} d}{3}\right ) + x^{5} \cdot \left (\frac {4 a c^{2} e}{5} + \frac {4 b^{2} c e}{5} + b c^{2} d\right ) + x^{4} \cdot \left (\frac {3 a b c e}{2} + a c^{2} d + \frac {b^{3} e}{4} + b^{2} c d\right ) + x^{3} \cdot \left (\frac {2 a^{2} c e}{3} + \frac {2 a b^{2} e}{3} + 2 a b c d + \frac {b^{3} d}{3}\right ) + x^{2} \left (\frac {a^{2} b e}{2} + a^{2} c d + a b^{2} d\right ) \] Input:

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

Output:

a**2*b*d*x + 2*c**3*e*x**7/7 + x**6*(5*b*c**2*e/6 + c**3*d/3) + x**5*(4*a* 
c**2*e/5 + 4*b**2*c*e/5 + b*c**2*d) + x**4*(3*a*b*c*e/2 + a*c**2*d + b**3* 
e/4 + b**2*c*d) + x**3*(2*a**2*c*e/3 + 2*a*b**2*e/3 + 2*a*b*c*d + b**3*d/3 
) + x**2*(a**2*b*e/2 + a**2*c*d + a*b**2*d)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{7} \, c^{3} e x^{7} + \frac {1}{6} \, {\left (2 \, c^{3} d + 5 \, b c^{2} e\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d + 4 \, {\left (b^{2} c + a c^{2}\right )} e\right )} x^{5} + a^{2} b d x + \frac {1}{4} \, {\left (4 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 2 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{2} b e + 2 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} \] Input:

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

Output:

2/7*c^3*e*x^7 + 1/6*(2*c^3*d + 5*b*c^2*e)*x^6 + 1/5*(5*b*c^2*d + 4*(b^2*c 
+ a*c^2)*e)*x^5 + a^2*b*d*x + 1/4*(4*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e 
)*x^4 + 1/3*((b^3 + 6*a*b*c)*d + 2*(a*b^2 + a^2*c)*e)*x^3 + 1/2*(a^2*b*e + 
 2*(a*b^2 + a^2*c)*d)*x^2
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.09 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{7} \, c^{3} e x^{7} + \frac {1}{3} \, c^{3} d x^{6} + \frac {5}{6} \, b c^{2} e x^{6} + b c^{2} d x^{5} + \frac {4}{5} \, b^{2} c e x^{5} + \frac {4}{5} \, a c^{2} e x^{5} + b^{2} c d x^{4} + a c^{2} d x^{4} + \frac {1}{4} \, b^{3} e x^{4} + \frac {3}{2} \, a b c e x^{4} + \frac {1}{3} \, b^{3} d x^{3} + 2 \, a b c d x^{3} + \frac {2}{3} \, a b^{2} e x^{3} + \frac {2}{3} \, a^{2} c e x^{3} + a b^{2} d x^{2} + a^{2} c d x^{2} + \frac {1}{2} \, a^{2} b e x^{2} + a^{2} b d x \] Input:

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

Output:

2/7*c^3*e*x^7 + 1/3*c^3*d*x^6 + 5/6*b*c^2*e*x^6 + b*c^2*d*x^5 + 4/5*b^2*c* 
e*x^5 + 4/5*a*c^2*e*x^5 + b^2*c*d*x^4 + a*c^2*d*x^4 + 1/4*b^3*e*x^4 + 3/2* 
a*b*c*e*x^4 + 1/3*b^3*d*x^3 + 2*a*b*c*d*x^3 + 2/3*a*b^2*e*x^3 + 2/3*a^2*c* 
e*x^3 + a*b^2*d*x^2 + a^2*c*d*x^2 + 1/2*a^2*b*e*x^2 + a^2*b*d*x
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=x^6\,\left (\frac {d\,c^3}{3}+\frac {5\,b\,e\,c^2}{6}\right )+x^2\,\left (\frac {e\,a^2\,b}{2}+c\,d\,a^2+d\,a\,b^2\right )+x^5\,\left (\frac {4\,e\,b^2\,c}{5}+d\,b\,c^2+\frac {4\,a\,e\,c^2}{5}\right )+x^3\,\left (\frac {2\,c\,e\,a^2}{3}+\frac {2\,e\,a\,b^2}{3}+2\,c\,d\,a\,b+\frac {d\,b^3}{3}\right )+x^4\,\left (\frac {e\,b^3}{4}+d\,b^2\,c+\frac {3\,a\,e\,b\,c}{2}+a\,d\,c^2\right )+\frac {2\,c^3\,e\,x^7}{7}+a^2\,b\,d\,x \] Input:

int((b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^2,x)
 

Output:

x^6*((c^3*d)/3 + (5*b*c^2*e)/6) + x^2*(a*b^2*d + (a^2*b*e)/2 + a^2*c*d) + 
x^5*((4*a*c^2*e)/5 + b*c^2*d + (4*b^2*c*e)/5) + x^3*((b^3*d)/3 + (2*a*b^2* 
e)/3 + (2*a^2*c*e)/3 + 2*a*b*c*d) + x^4*((b^3*e)/4 + a*c^2*d + b^2*c*d + ( 
3*a*b*c*e)/2) + (2*c^3*e*x^7)/7 + a^2*b*d*x
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {x \left (120 c^{3} e \,x^{6}+350 b \,c^{2} e \,x^{5}+140 c^{3} d \,x^{5}+336 a \,c^{2} e \,x^{4}+336 b^{2} c e \,x^{4}+420 b \,c^{2} d \,x^{4}+630 a b c e \,x^{3}+420 a \,c^{2} d \,x^{3}+105 b^{3} e \,x^{3}+420 b^{2} c d \,x^{3}+280 a^{2} c e \,x^{2}+280 a \,b^{2} e \,x^{2}+840 a b c d \,x^{2}+140 b^{3} d \,x^{2}+210 a^{2} b e x +420 a^{2} c d x +420 a \,b^{2} d x +420 a^{2} b d \right )}{420} \] Input:

int((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^2,x)
 

Output:

(x*(420*a**2*b*d + 210*a**2*b*e*x + 420*a**2*c*d*x + 280*a**2*c*e*x**2 + 4 
20*a*b**2*d*x + 280*a*b**2*e*x**2 + 840*a*b*c*d*x**2 + 630*a*b*c*e*x**3 + 
420*a*c**2*d*x**3 + 336*a*c**2*e*x**4 + 140*b**3*d*x**2 + 105*b**3*e*x**3 
+ 420*b**2*c*d*x**3 + 336*b**2*c*e*x**4 + 420*b*c**2*d*x**4 + 350*b*c**2*e 
*x**5 + 140*c**3*d*x**5 + 120*c**3*e*x**6))/420