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

Optimal. Leaf size=35 \[ x \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]

[Out]

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

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Rubi [A]  time = 0.0367923, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 174, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.006 \[ x \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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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*(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)

[Out]

Timed out

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Mathematica [A]  time = 0.578451, size = 32, normalized size = 0.91 \[ x (a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.034, size = 36, normalized size = 1. \[ x \left ( c{x}^{2}+bx+a \right ) ^{1+m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) ^{1+n} \]

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*(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)

[Out]

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

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Maxima [A]  time = 0.95988, size = 128, normalized size = 3.66 \[{\left (c g x^{6} +{\left (c f + b g\right )} x^{5} +{\left (c e + b f + a g\right )} x^{4} +{\left (c d + b e + a f\right )} x^{3} + a d x +{\left (b d + a e\right )} x^{2}\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

[Out]

Timed out

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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*(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)

[Out]

Timed out

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

[Out]

Timed out