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

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

[Out]

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

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Rubi [F]  time = 3.37, 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 = {} \[ \int \frac {\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (-a d+(b d m+a e n) x+(c d+b e+a f+2 c d m+b e m+b e n+2 a f n) x^2+(2 c e+2 b f+2 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(3 c f+3 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (4+2 m+3 n) x^5\right )}{x^2} \, dx \]

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

[Out]

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

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Mathematica [A]  time = 0.81, size = 34, normalized size = 0.92 \[ \frac {(a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}}{x} \]

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

[Out]

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

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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

[Out]

sage0*x

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

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

[Out]

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

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maxima [B]  time = 1.76, size = 95, normalized size = 2.57 \[ \frac {{\left (c g x^{5} + {\left (c f + b g\right )} x^{4} + {\left (c e + b f + a g\right )} x^{3} + {\left (c d + b e + a f\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )}}{x} \]

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

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mupad [B]  time = 9.68, size = 37, normalized size = 1.00 \[ \frac {{\left (c\,x^2+b\,x+a\right )}^{m+1}\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^{n+1}}{x} \]

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

[Out]

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

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

[Out]

Timed out

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