\(\int (a+b x+c x^2)^m (d+e x+f x^2+g x^3)^n (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5) \, dx\) [97]

Optimal result

Integrand size = 174, antiderivative size = 35 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=x \left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n} \] Output:

x*(c*x^2+b*x+a)^(1+m)*(g*x^3+f*x^2+e*x+d)^(1+n)
 

Mathematica [A] (verified)

Time = 10.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=x (a+x (b+c x))^{1+m} (d+x (e+x (f+g x)))^{1+n} \] Input:

Integrate[(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(a*d + (2*b*d + 
2*a*e + b*d*m + a*e*n)*x + (3*c*d + 3*b*e + 3*a*f + 2*c*d*m + b*e*m + b*e* 
n + 2*a*f*n)*x^2 + (4*c*e + 4*b*f + 4*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b* 
f*n + 3*a*g*n)*x^3 + (5*c*f + 5*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n) 
*x^4 + c*g*(6 + 2*m + 3*n)*x^5),x]
 

Output:

x*(a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {2023}

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.

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

\(\Big \downarrow \) 2023

\(\displaystyle x \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1}\)

Input:

Int[(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(a*d + (2*b*d + 2*a*e 
+ b*d*m + a*e*n)*x + (3*c*d + 3*b*e + 3*a*f + 2*c*d*m + b*e*m + b*e*n + 2* 
a*f*n)*x^2 + (4*c*e + 4*b*f + 4*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 
3*a*g*n)*x^3 + (5*c*f + 5*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + 
 c*g*(6 + 2*m + 3*n)*x^5),x]
 

Output:

x*(a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n)
 

Defintions of rubi rules used

rule 2023

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = 
 Expon[Qq, x], r = Expon[Rr, x]}, Simp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq 
^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r])) 
, x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q 
]*Coeff[Rr, x, r]*Pp, Coeff[Pp, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr 
+ (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n}, x] && P 
olyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]
 

Maple [A] (verified)

Time = 25.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03

method	result	size
gosper	\(x \left (c \,x^{2}+b x +a \right )^{1+m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{1+n}\)	\(36\)
risch	\(x \left (c g \,x^{5}+b g \,x^{4}+c f \,x^{4}+a g \,x^{3}+b f \,x^{3}+c e \,x^{3}+a f \,x^{2}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n}\)	\(98\)
orering	\(\frac {x \left (g \,x^{3}+f \,x^{2}+e x +d \right ) \left (c \,x^{2}+b x +a \right ) \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} \left (a d +\left (a e n +b d m +2 a e +2 b d \right ) x +\left (2 a f n +b e m +b e n +2 c d m +3 a f +3 b e +3 c d \right ) x^{2}+\left (3 a g n +b f m +2 b f n +2 c e m +c e n +4 a g +4 b f +4 c e \right ) x^{3}+\left (b g m +3 b g n +2 c f m +2 c f n +5 b g +5 c f \right ) x^{4}+c g \left (6+2 m +3 n \right ) x^{5}\right )}{2 c g m \,x^{5}+3 c g n \,x^{5}+b g m \,x^{4}+3 b g n \,x^{4}+2 c f m \,x^{4}+2 c f n \,x^{4}+6 c g \,x^{5}+3 a g n \,x^{3}+b f m \,x^{3}+2 b f n \,x^{3}+5 b g \,x^{4}+2 c e m \,x^{3}+c e n \,x^{3}+5 c f \,x^{4}+2 a f n \,x^{2}+4 a g \,x^{3}+b e m \,x^{2}+b e n \,x^{2}+4 b f \,x^{3}+2 c d m \,x^{2}+4 c e \,x^{3}+a e n x +3 a f \,x^{2}+b d m x +3 b e \,x^{2}+3 c d \,x^{2}+2 a e x +2 b d x +a d}\)	\(405\)
parallelrisch	\(\frac {x^{6} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} g^{2}+x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c \,g^{2}+x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} f g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c \,g^{2}+x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c f g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} e g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c f g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c e g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} d g +x^{2} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c e g +x^{2} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c d g +x \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c d g}{c g}\)	\(458\)

Input:

int((c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(a*d+(a*e*n+b*d*m+2*a*e+2*b*d)*x 
+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+3*a*f+3*b*e+3*c*d)*x^2+(3*a*g*n+b*f*m+2*b*f* 
n+2*c*e*m+c*e*n+4*a*g+4*b*f+4*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f*n+5*b* 
g+5*c*f)*x^4+c*g*(6+2*m+3*n)*x^5),x,method=_RETURNVERBOSE)
 

Output:

x*(c*x^2+b*x+a)^(1+m)*(g*x^3+f*x^2+e*x+d)^(1+n)
 

Fricas [F(-1)]

Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(a*d+(a*e*n+b*d*m+2*a*e+2* 
b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+3*a*f+3*b*e+3*c*d)*x^2+(3*a*g*n+b*f*m+ 
2*b*f*n+2*c*e*m+c*e*n+4*a*g+4*b*f+4*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f* 
n+5*b*g+5*c*f)*x^4+c*g*(6+2*m+3*n)*x^5),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**m*(g*x**3+f*x**2+e*x+d)**n*(a*d+(a*e*n+b*d*m+2*a 
*e+2*b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+3*a*f+3*b*e+3*c*d)*x**2+(3*a*g*n+ 
b*f*m+2*b*f*n+2*c*e*m+c*e*n+4*a*g+4*b*f+4*c*e)*x**3+(b*g*m+3*b*g*n+2*c*f*m 
+2*c*f*n+5*b*g+5*c*f)*x**4+c*g*(6+2*m+3*n)*x**5),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (35) = 70\).

Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.71 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx={\left (c g x^{6} + {\left (c f + b g\right )} x^{5} + {\left (c e + b f + a g\right )} x^{4} + {\left (c d + b e + a f\right )} x^{3} + a d x + {\left (b d + a e\right )} x^{2}\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )} \] Input:

integrate((c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(a*d+(a*e*n+b*d*m+2*a*e+2* 
b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+3*a*f+3*b*e+3*c*d)*x^2+(3*a*g*n+b*f*m+ 
2*b*f*n+2*c*e*m+c*e*n+4*a*g+4*b*f+4*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f* 
n+5*b*g+5*c*f)*x^4+c*g*(6+2*m+3*n)*x^5),x, algorithm="maxima")
 

Output:

(c*g*x^6 + (c*f + b*g)*x^5 + (c*e + b*f + a*g)*x^4 + (c*d + b*e + a*f)*x^3 
 + a*d*x + (b*d + a*e)*x^2)*e^(n*log(g*x^3 + f*x^2 + e*x + d) + m*log(c*x^ 
2 + b*x + a))
 

Giac [F(-1)]

Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(a*d+(a*e*n+b*d*m+2*a*e+2* 
b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+3*a*f+3*b*e+3*c*d)*x^2+(3*a*g*n+b*f*m+ 
2*b*f*n+2*c*e*m+c*e*n+4*a*g+4*b*f+4*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f* 
n+5*b*g+5*c*f)*x^4+c*g*(6+2*m+3*n)*x^5),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Hanged} \] Input:

int((a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(a*d + x^4*(5*b*g + 5* 
c*f + b*g*m + 2*c*f*m + 3*b*g*n + 2*c*f*n) + x^2*(3*a*f + 3*b*e + 3*c*d + 
b*e*m + 2*c*d*m + 2*a*f*n + b*e*n) + x*(2*a*e + 2*b*d + b*d*m + a*e*n) + x 
^3*(4*a*g + 4*b*f + 4*c*e + b*f*m + 2*c*e*m + 3*a*g*n + 2*b*f*n + c*e*n) + 
 c*g*x^5*(2*m + 3*n + 6)),x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.77 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} \left (c \,x^{2}+b x +a \right )^{m} x \left (c g \,x^{5}+b g \,x^{4}+c f \,x^{4}+a g \,x^{3}+b f \,x^{3}+c e \,x^{3}+a f \,x^{2}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) \] Input:

int((c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(a*d+(a*e*n+b*d*m+2*a*e+2*b*d)*x 
+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+3*a*f+3*b*e+3*c*d)*x^2+(3*a*g*n+b*f*m+2*b*f* 
n+2*c*e*m+c*e*n+4*a*g+4*b*f+4*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f*n+5*b* 
g+5*c*f)*x^4+c*g*(6+2*m+3*n)*x^5),x)
 

Output:

(d + e*x + f*x**2 + g*x**3)**n*(a + b*x + c*x**2)**m*x*(a*d + a*e*x + a*f* 
x**2 + a*g*x**3 + b*d*x + b*e*x**2 + b*f*x**3 + b*g*x**4 + c*d*x**2 + c*e* 
x**3 + c*f*x**4 + c*g*x**5)