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

Optimal. Leaf size=34 \[ \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0411668, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 164, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.006 \[ \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.536428, size = 31, normalized size = 0.91 \[ (a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.031, size = 35, normalized size = 1. \[ \left ( c{x}^{2}+bx+a \right ) ^{1+m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) ^{1+n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.919864, size = 124, normalized size = 3.65 \[{\left (c g x^{5} +{\left (c f + b g\right )} x^{4} +{\left (c e + b f + a g\right )} x^{3} +{\left (c d + b e + a f\right )} x^{2} + a d +{\left (b d + a e\right )} x\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out