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]

(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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Rubi [F]  time = 3.00, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 1.15, 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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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*(-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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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*(-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]

Timed out

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maple [A]  time = 0.03, size = 38, normalized size = 1.03 \[ \frac {\left (c \,x^{2}+b x +a \right )^{m +1} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n +1}}{x^{2}} \]

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

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maxima [B]  time = 1.81, size = 95, normalized size = 2.57 \[ \frac {{\left (c g x^{5} + {\left (c f + b g\right )} x^{4} + {\left (c e + b f + a g\right )} x^{3} + {\left (c d + b e + a f\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )}}{x^{2}} \]

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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mupad [B]  time = 9.22, size = 146, normalized size = 3.95 \[ {\left (c\,x^2+b\,x+a\right )}^m\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^n\,\left (a\,f+b\,e+c\,d+c\,g\,x^3+a\,g\,x+b\,f\,x+c\,e\,x+b\,g\,x^2+c\,f\,x^2\right )+\frac {\left (a\,e+b\,d\right )\,{\left (c\,x^2+b\,x+a\right )}^m\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^n}{x}+\frac {a\,d\,{\left (c\,x^2+b\,x+a\right )}^m\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^n}{x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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*(-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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