3.613 \(\int x (a+b x+c x^2)^m (d+e x+f x^2+g x^3)^n (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5) \, dx\)
Optimal. Leaf size=37 \[ x^2 \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]
[Out]
x^2*(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 = 0.0923031, antiderivative size = 37, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 176, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.006, Rules used
= {1590} \[ x^2 \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(2*a*d + (3*b*d + 3*a*e + b*d*m + a*e*n)*x + (4*c*d + 4*
b*e + 4*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (5*c*e + 5*b*f + 5*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*
f*n + 3*a*g*n)*x^3 + (6*c*f + 6*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(7 + 2*m + 3*n)*x^5),x]
[Out]
x^2*(a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n)
Rule 1590
Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(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] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]
Rubi steps
\begin{align*} \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx &=x^2 \left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n}\\ \end{align*}
Mathematica [A] time = 0.363181, size = 34, normalized size = 0.92 \[ x^2 (a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(2*a*d + (3*b*d + 3*a*e + b*d*m + a*e*n)*x + (4*c*
d + 4*b*e + 4*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (5*c*e + 5*b*f + 5*a*g + 2*c*e*m + b*f*m + c*e*n
+ 2*b*f*n + 3*a*g*n)*x^3 + (6*c*f + 6*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(7 + 2*m + 3*n)*x^5
),x]
[Out]
x^2*(a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n)
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Maple [A] time = 0.023, size = 38, normalized size = 1. \begin{align*}{x}^{2} \left ( c{x}^{2}+bx+a \right ) ^{1+m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) ^{1+n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(2*a*d+(a*e*n+b*d*m+3*a*e+3*b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m+4*
a*f+4*b*e+4*c*d)*x^2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+5*a*g+5*b*f+5*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f*
n+6*b*g+6*c*f)*x^4+c*g*(7+2*m+3*n)*x^5),x)
[Out]
x^2*(c*x^2+b*x+a)^(1+m)*(g*x^3+f*x^2+e*x+d)^(1+n)
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Maxima [B] time = 1.56011, size = 131, normalized size = 3.54 \begin{align*}{\left (c g x^{7} +{\left (c f + b g\right )} x^{6} +{\left (c e + b f + a g\right )} x^{5} +{\left (c d + b e + a f\right )} x^{4} + a d x^{2} +{\left (b d + a e\right )} x^{3}\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(2*a*d+(a*e*n+b*d*m+3*a*e+3*b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*
d*m+4*a*f+4*b*e+4*c*d)*x^2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+5*a*g+5*b*f+5*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+
2*c*f*n+6*b*g+6*c*f)*x^4+c*g*(7+2*m+3*n)*x^5),x, algorithm="maxima")
[Out]
(c*g*x^7 + (c*f + b*g)*x^6 + (c*e + b*f + a*g)*x^5 + (c*d + b*e + a*f)*x^4 + a*d*x^2 + (b*d + a*e)*x^3)*e^(n*l
og(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. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(2*a*d+(a*e*n+b*d*m+3*a*e+3*b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*
d*m+4*a*f+4*b*e+4*c*d)*x^2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+5*a*g+5*b*f+5*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+
2*c*f*n+6*b*g+6*c*f)*x^4+c*g*(7+2*m+3*n)*x^5),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(c*x**2+b*x+a)**m*(g*x**3+f*x**2+e*x+d)**n*(2*a*d+(a*e*n+b*d*m+3*a*e+3*b*d)*x+(2*a*f*n+b*e*m+b*e*n
+2*c*d*m+4*a*f+4*b*e+4*c*d)*x**2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+5*a*g+5*b*f+5*c*e)*x**3+(b*g*m+3*b*g*n+2
*c*f*m+2*c*f*n+6*b*g+6*c*f)*x**4+c*g*(7+2*m+3*n)*x**5),x)
[Out]
Timed out
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(c*x^2+b*x+a)^m*(g*x^3+f*x^2+e*x+d)^n*(2*a*d+(a*e*n+b*d*m+3*a*e+3*b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*
d*m+4*a*f+4*b*e+4*c*d)*x^2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+5*a*g+5*b*f+5*c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+
2*c*f*n+6*b*g+6*c*f)*x^4+c*g*(7+2*m+3*n)*x^5),x, algorithm="giac")
[Out]
Timed out