3.615 \(\int (a+b x+c x^2)^m (d+e x+f x^2+g x^3)^n (b d+a e+b d m+a e n+(2 c d+2 b e+2 a f+2 c d m+b e m+b e n+2 a f n) x+(3 c e+3 b f+3 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^2+(4 c f+4 b g+2 c f m+b g m+2 c f n+3 b g n) x^3+c g (5+2 m+3 n) x^4) \, dx\)
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)
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Rubi [A] time = 0.11651, 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, Rules used
= {1590} \[ \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]
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)
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 \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 &=\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 = 0.320869, 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)
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Maple [A] time = 0.02, size = 35, normalized size = 1. \begin{align*} \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((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)
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Maxima [B] time = 1.5879, size = 124, normalized size = 3.65 \begin{align*}{\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 )} \end{align*}
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, 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))
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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((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, 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((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
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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((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, algorithm="giac")
[Out]
Timed out