3.617 \(\int \frac{(a+b x+c x^2)^m (d+e x+f x^2+g x^3)^n (-2 a d+(-b d-a e+b d m+a e n) x+(2 c d m+b e m+b e n+2 a f n) x^2+(c e+b f+a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(2 c f+2 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (3+2 m+3 n) x^5)}{x^3} \, dx\)

Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1}}{x^2} \]

[Out]

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

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Rubi [F]  time = 2.99845, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (-2 a d+(-b d-a e+b d m+a e n) x+(2 c d m+b e m+b e n+2 a f n) x^2+(c e+b f+a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(2 c f+2 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (3+2 m+3 n) x^5\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c*e*(1 + 2*m + n) + b*f*(1 + m + 2*n) + a*g*(1 + 3*n))*Defer[Int][(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^
3)^n, x] - 2*a*d*Defer[Int][((a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n)/x^3, x] - (b*d*(1 - m) + a*e*(1
- n))*Defer[Int][((a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n)/x^2, x] + (2*c*d*m + 2*a*f*n + b*e*(m + n))
*Defer[Int][((a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n)/x, x] + (2*c*f*(1 + m + n) + b*g*(2 + m + 3*n))*
Defer[Int][x*(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n, x] + c*g*(3 + 2*m + 3*n)*Defer[Int][x^2*(a + b*x
+ c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n, x]

Rubi steps

\begin{align*} \int \frac{\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+(-b d-a e+b d m+a e n) x+(2 c d m+b e m+b e n+2 a f n) x^2+(c e+b f+a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(2 c f+2 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (3+2 m+3 n) x^5\right )}{x^3} \, dx &=\int \left (c e \left (1+\frac{c e (2 m+n)+b f (1+m+2 n)+a (g+3 g n)}{c e}\right ) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n-\frac{2 a d \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^3}+\frac{(-b d (1-m)-a e (1-n)) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^2}+\frac{(2 c d m+2 a f n+b e (m+n)) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x}+(2 c f (1+m+n)+b g (2+m+3 n)) x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n+c g (3+2 m+3 n) x^2 \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n\right ) \, dx\\ &=-\left ((2 a d) \int \frac{\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^3} \, dx\right )+(-b d (1-m)-a e (1-n)) \int \frac{\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^2} \, dx+(c g (3+2 m+3 n)) \int x^2 \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx+(2 c d m+2 a f n+b e (m+n)) \int \frac{\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x} \, dx+(c e (1+2 m+n)+b f (1+m+2 n)+a g (1+3 n)) \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx+(2 c f (1+m+n)+b g (2+m+3 n)) \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx\\ \end{align*}

Mathematica [A]  time = 1.39784, size = 34, normalized size = 0.92 \[ \frac{(a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}}{x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 38, normalized size = 1. \begin{align*}{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) ^{1+n}}{{x}^{2}}} \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*(-2*a*d+(a*e*n+b*d*m-a*e-b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m)*x^2+(3
*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+a*g+b*f+c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f*n+2*b*g+2*c*f)*x^4+c*g*(3+2*m
+3*n)*x^5)/x^3,x)

[Out]

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

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Maxima [B]  time = 1.61362, size = 128, normalized size = 3.46 \begin{align*} \frac{{\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 )}}{x^{2}} \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*(-2*a*d+(a*e*n+b*d*m-a*e-b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m)*
x^2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+a*g+b*f+c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f*n+2*b*g+2*c*f)*x^4+c*g*
(3+2*m+3*n)*x^5)/x^3,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))/x^2

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c g{\left (2 \, m + 3 \, n + 3\right )} x^{5} +{\left (2 \, c f m + b g m + 2 \, c f n + 3 \, b g n + 2 \, c f + 2 \, b g\right )} x^{4} +{\left (2 \, c e m + b f m + c e n + 2 \, b f n + 3 \, a g n + c e + b f + a g\right )} x^{3} +{\left (2 \, c d m + b e m + b e n + 2 \, a f n\right )} x^{2} - 2 \, a d +{\left (b d m + a e n - b d - a e\right )} x\right )}{\left (g x^{3} + f x^{2} + e x + d\right )}^{n}{\left (c x^{2} + b x + a\right )}^{m}}{x^{3}}\,{d x} \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*(-2*a*d+(a*e*n+b*d*m-a*e-b*d)*x+(2*a*f*n+b*e*m+b*e*n+2*c*d*m)*
x^2+(3*a*g*n+b*f*m+2*b*f*n+2*c*e*m+c*e*n+a*g+b*f+c*e)*x^3+(b*g*m+3*b*g*n+2*c*f*m+2*c*f*n+2*b*g+2*c*f)*x^4+c*g*
(3+2*m+3*n)*x^5)/x^3,x, algorithm="giac")

[Out]

integrate((c*g*(2*m + 3*n + 3)*x^5 + (2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n + 2*c*f + 2*b*g)*x^4 + (2*c*e*m + b*
f*m + c*e*n + 2*b*f*n + 3*a*g*n + c*e + b*f + a*g)*x^3 + (2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 - 2*a*d + (b*
d*m + a*e*n - b*d - a*e)*x)*(g*x^3 + f*x^2 + e*x + d)^n*(c*x^2 + b*x + a)^m/x^3, x)