Integrand size = 174, antiderivative size = 35 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=x \left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n} \]
Time = 10.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=x (a+x (b+c x))^{1+m} (d+x (e+x (f+g x)))^{1+n} \]
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]
Time = 0.26 (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 |
| \(\displaystyle x \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1}\) |
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]
3.7.14.3.1 Defintions of rubi rules used
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]
Time = 43.42 (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 | \(x \left (c g \,x^{5}+b g \,x^{4}+c f \,x^{4}+a g \,x^{3}+b f \,x^{3}+c \,x^{3} e +a f \,x^{2}+e \,x^{2} b +x^{2} c d +a e x +b d x +a d \right ) \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n}\) | \(98\) |
| parallelrisch | \(\frac {x^{6} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} g^{2}+x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c \,g^{2}+x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} f g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c \,g^{2}+x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c f g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} e g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c f g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c e g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} d g +x^{2} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c e g +x^{2} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c d g +x \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c d g}{g c}\) | \(458\) |
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)
Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \]
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")
Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.71 \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx={\left (c g x^{6} + {\left (c f + b g\right )} x^{5} + {\left (c e + b f + a g\right )} x^{4} + {\left (c d + b e + a f\right )} x^{3} + a d x + {\left (b d + a e\right )} x^{2}\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )} \]
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")
(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))
Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \]
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")
Timed out. \[ \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right ) \, dx=\text {Hanged} \]
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)