3.119 \(\int (a+b x)^m (c+d x) (e+f x) (g+h x) \, dx\)

Optimal. Leaf size=167 \[ \frac{(a+b x)^{m+2} \left (3 a^2 d f h-2 a b (c f h+d e h+d f g)+b^2 (c e h+c f g+d e g)\right )}{b^4 (m+2)}+\frac{(b c-a d) (b e-a f) (b g-a h) (a+b x)^{m+1}}{b^4 (m+1)}-\frac{(a+b x)^{m+3} (3 a d f h-b (c f h+d e h+d f g))}{b^4 (m+3)}+\frac{d f h (a+b x)^{m+4}}{b^4 (m+4)} \]

[Out]

((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(1 + m))/(b^4*(1 + m)) + ((3*a^2*d*f*h + b^2*(d*e*g + c*f*g + c
*e*h) - 2*a*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)^(2 + m))/(b^4*(2 + m)) - ((3*a*d*f*h - b*(d*f*g + d*e*h + c*f
*h))*(a + b*x)^(3 + m))/(b^4*(3 + m)) + (d*f*h*(a + b*x)^(4 + m))/(b^4*(4 + m))

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Rubi [A]  time = 0.130565, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {142} \[ \frac{(a+b x)^{m+2} \left (3 a^2 d f h-2 a b (c f h+d e h+d f g)+b^2 (c e h+c f g+d e g)\right )}{b^4 (m+2)}+\frac{(b c-a d) (b e-a f) (b g-a h) (a+b x)^{m+1}}{b^4 (m+1)}-\frac{(a+b x)^{m+3} (3 a d f h-b (c f h+d e h+d f g))}{b^4 (m+3)}+\frac{d f h (a+b x)^{m+4}}{b^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)*(e + f*x)*(g + h*x),x]

[Out]

((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(1 + m))/(b^4*(1 + m)) + ((3*a^2*d*f*h + b^2*(d*e*g + c*f*g + c
*e*h) - 2*a*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)^(2 + m))/(b^4*(2 + m)) - ((3*a*d*f*h - b*(d*f*g + d*e*h + c*f
*h))*(a + b*x)^(3 + m))/(b^4*(3 + m)) + (d*f*h*(a + b*x)^(4 + m))/(b^4*(4 + m))

Rule 142

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && (IGtQ[m, 0] || IntegersQ[m, n])

Rubi steps

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

Mathematica [A]  time = 0.233434, size = 149, normalized size = 0.89 \[ \frac{(a+b x)^{m+1} \left (\frac{(a+b x) \left (3 a^2 d f h-2 a b (c f h+d e h+d f g)+b^2 (c e h+c f g+d e g)\right )}{m+2}+\frac{(a+b x)^2 (b (c f h+d e h+d f g)-3 a d f h)}{m+3}+\frac{(b c-a d) (b e-a f) (b g-a h)}{m+1}+\frac{d f h (a+b x)^3}{m+4}\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)*(e + f*x)*(g + h*x),x]

[Out]

((a + b*x)^(1 + m)*(((b*c - a*d)*(b*e - a*f)*(b*g - a*h))/(1 + m) + ((3*a^2*d*f*h + b^2*(d*e*g + c*f*g + c*e*h
) - 2*a*b*(d*f*g + d*e*h + c*f*h))*(a + b*x))/(2 + m) + ((-3*a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*(a + b*x)^2)
/(3 + m) + (d*f*h*(a + b*x)^3)/(4 + m)))/b^4

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Maple [B]  time = 0.007, size = 726, normalized size = 4.4 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( -{b}^{3}dfh{m}^{3}{x}^{3}-{b}^{3}cfh{m}^{3}{x}^{2}-{b}^{3}deh{m}^{3}{x}^{2}-{b}^{3}dfg{m}^{3}{x}^{2}-6\,{b}^{3}dfh{m}^{2}{x}^{3}+3\,a{b}^{2}dfh{m}^{2}{x}^{2}-{b}^{3}ceh{m}^{3}x-{b}^{3}cfg{m}^{3}x-7\,{b}^{3}cfh{m}^{2}{x}^{2}-{b}^{3}deg{m}^{3}x-7\,{b}^{3}deh{m}^{2}{x}^{2}-7\,{b}^{3}dfg{m}^{2}{x}^{2}-11\,{b}^{3}dfhm{x}^{3}+2\,a{b}^{2}cfh{m}^{2}x+2\,a{b}^{2}deh{m}^{2}x+2\,a{b}^{2}dfg{m}^{2}x+9\,a{b}^{2}dfhm{x}^{2}-{b}^{3}ceg{m}^{3}-8\,{b}^{3}ceh{m}^{2}x-8\,{b}^{3}cfg{m}^{2}x-14\,{b}^{3}cfhm{x}^{2}-8\,{b}^{3}deg{m}^{2}x-14\,{b}^{3}dehm{x}^{2}-14\,{b}^{3}dfgm{x}^{2}-6\,dfh{x}^{3}{b}^{3}-6\,{a}^{2}bdfhmx+a{b}^{2}ceh{m}^{2}+a{b}^{2}cfg{m}^{2}+10\,a{b}^{2}cfhmx+a{b}^{2}deg{m}^{2}+10\,a{b}^{2}dehmx+10\,a{b}^{2}dfgmx+6\,a{b}^{2}dfh{x}^{2}-9\,{b}^{3}ceg{m}^{2}-19\,{b}^{3}cehmx-19\,{b}^{3}cfgmx-8\,{b}^{3}cfh{x}^{2}-19\,{b}^{3}degmx-8\,{b}^{3}deh{x}^{2}-8\,{b}^{3}dfg{x}^{2}-2\,{a}^{2}bcfhm-2\,{a}^{2}bdehm-2\,{a}^{2}bdfgm-6\,{a}^{2}bdfhx+7\,a{b}^{2}cehm+7\,a{b}^{2}cfgm+8\,a{b}^{2}cfhx+7\,a{b}^{2}degm+8\,a{b}^{2}dehx+8\,a{b}^{2}dfgx-26\,{b}^{3}cegm-12\,{b}^{3}cehx-12\,{b}^{3}cfgx-12\,{b}^{3}degx+6\,{a}^{3}dfh-8\,{a}^{2}bcfh-8\,{a}^{2}bdeh-8\,{a}^{2}bdfg+12\,a{b}^{2}ceh+12\,a{b}^{2}cfg+12\,a{b}^{2}deg-24\,{b}^{3}ceg \right ) }{{b}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)*(f*x+e)*(h*x+g),x)

[Out]

-(b*x+a)^(1+m)*(-b^3*d*f*h*m^3*x^3-b^3*c*f*h*m^3*x^2-b^3*d*e*h*m^3*x^2-b^3*d*f*g*m^3*x^2-6*b^3*d*f*h*m^2*x^3+3
*a*b^2*d*f*h*m^2*x^2-b^3*c*e*h*m^3*x-b^3*c*f*g*m^3*x-7*b^3*c*f*h*m^2*x^2-b^3*d*e*g*m^3*x-7*b^3*d*e*h*m^2*x^2-7
*b^3*d*f*g*m^2*x^2-11*b^3*d*f*h*m*x^3+2*a*b^2*c*f*h*m^2*x+2*a*b^2*d*e*h*m^2*x+2*a*b^2*d*f*g*m^2*x+9*a*b^2*d*f*
h*m*x^2-b^3*c*e*g*m^3-8*b^3*c*e*h*m^2*x-8*b^3*c*f*g*m^2*x-14*b^3*c*f*h*m*x^2-8*b^3*d*e*g*m^2*x-14*b^3*d*e*h*m*
x^2-14*b^3*d*f*g*m*x^2-6*b^3*d*f*h*x^3-6*a^2*b*d*f*h*m*x+a*b^2*c*e*h*m^2+a*b^2*c*f*g*m^2+10*a*b^2*c*f*h*m*x+a*
b^2*d*e*g*m^2+10*a*b^2*d*e*h*m*x+10*a*b^2*d*f*g*m*x+6*a*b^2*d*f*h*x^2-9*b^3*c*e*g*m^2-19*b^3*c*e*h*m*x-19*b^3*
c*f*g*m*x-8*b^3*c*f*h*x^2-19*b^3*d*e*g*m*x-8*b^3*d*e*h*x^2-8*b^3*d*f*g*x^2-2*a^2*b*c*f*h*m-2*a^2*b*d*e*h*m-2*a
^2*b*d*f*g*m-6*a^2*b*d*f*h*x+7*a*b^2*c*e*h*m+7*a*b^2*c*f*g*m+8*a*b^2*c*f*h*x+7*a*b^2*d*e*g*m+8*a*b^2*d*e*h*x+8
*a*b^2*d*f*g*x-26*b^3*c*e*g*m-12*b^3*c*e*h*x-12*b^3*c*f*g*x-12*b^3*d*e*g*x+6*a^3*d*f*h-8*a^2*b*c*f*h-8*a^2*b*d
*e*h-8*a^2*b*d*f*g+12*a*b^2*c*e*h+12*a*b^2*c*f*g+12*a*b^2*d*e*g-24*b^3*c*e*g)/b^4/(m^4+10*m^3+35*m^2+50*m+24)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)*(f*x+e)*(h*x+g),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60691, size = 1847, normalized size = 11.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)*(f*x+e)*(h*x+g),x, algorithm="fricas")

[Out]

(a*b^3*c*e*g*m^3 + (b^4*d*f*h*m^3 + 6*b^4*d*f*h*m^2 + 11*b^4*d*f*h*m + 6*b^4*d*f*h)*x^4 + (8*b^4*d*f*g + (b^4*
d*f*g + (b^4*d*e + (b^4*c + a*b^3*d)*f)*h)*m^3 + (7*b^4*d*f*g + (7*b^4*d*e + (7*b^4*c + 3*a*b^3*d)*f)*h)*m^2 +
8*(b^4*d*e + b^4*c*f)*h + 2*(7*b^4*d*f*g + (7*b^4*d*e + (7*b^4*c + a*b^3*d)*f)*h)*m)*x^3 - (a^2*b^2*c*e*h + (
a^2*b^2*c*f - (9*a*b^3*c - a^2*b^2*d)*e)*g)*m^2 + (12*b^4*c*e*h + ((b^4*d*e + (b^4*c + a*b^3*d)*f)*g + (a*b^3*
c*f + (b^4*c + a*b^3*d)*e)*h)*m^3 + ((8*b^4*d*e + (8*b^4*c + 5*a*b^3*d)*f)*g + ((8*b^4*c + 5*a*b^3*d)*e + (5*a
*b^3*c - 3*a^2*b^2*d)*f)*h)*m^2 + 12*(b^4*d*e + b^4*c*f)*g + ((19*b^4*d*e + (19*b^4*c + 4*a*b^3*d)*f)*g + ((19
*b^4*c + 4*a*b^3*d)*e + (4*a*b^3*c - 3*a^2*b^2*d)*f)*h)*m)*x^2 + 4*(3*(2*a*b^3*c - a^2*b^2*d)*e - (3*a^2*b^2*c
- 2*a^3*b*d)*f)*g - 2*(2*(3*a^2*b^2*c - 2*a^3*b*d)*e - (4*a^3*b*c - 3*a^4*d)*f)*h + (((26*a*b^3*c - 7*a^2*b^2
*d)*e - (7*a^2*b^2*c - 2*a^3*b*d)*f)*g + (2*a^3*b*c*f - (7*a^2*b^2*c - 2*a^3*b*d)*e)*h)*m + (24*b^4*c*e*g + (a
*b^3*c*e*h + (a*b^3*c*f + (b^4*c + a*b^3*d)*e)*g)*m^3 + (((9*b^4*c + 7*a*b^3*d)*e + (7*a*b^3*c - 2*a^2*b^2*d)*
f)*g - (2*a^2*b^2*c*f - (7*a*b^3*c - 2*a^2*b^2*d)*e)*h)*m^2 + 2*(((13*b^4*c + 6*a*b^3*d)*e + 2*(3*a*b^3*c - 2*
a^2*b^2*d)*f)*g + (2*(3*a*b^3*c - 2*a^2*b^2*d)*e - (4*a^2*b^2*c - 3*a^3*b*d)*f)*h)*m)*x)*(b*x + a)^m/(b^4*m^4
+ 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)

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Sympy [A]  time = 7.09604, size = 8218, normalized size = 49.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)*(f*x+e)*(h*x+g),x)

[Out]

Piecewise((a**m*(c*e*g*x + c*e*h*x**2/2 + c*f*g*x**2/2 + c*f*h*x**3/3 + d*e*g*x**2/2 + d*e*h*x**3/3 + d*f*g*x*
*3/3 + d*f*h*x**4/4), Eq(b, 0)), (6*a**3*d*f*h*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6
*b**7*x**3) + 11*a**3*d*f*h/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*a**2*b*c*f*h/(6*
a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*a**2*b*d*e*h/(6*a**3*b**4 + 18*a**2*b**5*x + 18
*a*b**6*x**2 + 6*b**7*x**3) - 2*a**2*b*d*f*g/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 1
8*a**2*b*d*f*h*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*d*f*h*
x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - a*b**2*c*e*h/(6*a**3*b**4 + 18*a**2*b**5*x +
18*a*b**6*x**2 + 6*b**7*x**3) - a*b**2*c*f*g/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) -
6*a*b**2*c*f*h*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - a*b**2*d*e*g/(6*a**3*b**4 + 1
8*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*a*b**2*d*e*h*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**
2 + 6*b**7*x**3) - 6*a*b**2*d*f*g*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*
d*f*h*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*d*f*h*x**2/(
6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*c*e*g/(6*a**3*b**4 + 18*a**2*b**5*x + 18
*a*b**6*x**2 + 6*b**7*x**3) - 3*b**3*c*e*h*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3
*b**3*c*f*g*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*b**3*c*f*h*x**2/(6*a**3*b**4 +
18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*b**3*d*e*g*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**
2 + 6*b**7*x**3) - 6*b**3*d*e*h*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*b**3*d*
f*g*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*d*f*h*x**3*log(a/b + x)/(6*a**
3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(m, -4)), (-6*a**4*d*f*h*log(a/b + x)/(2*a**3*b**4
+ 4*a**2*b**5*x + 2*a*b**6*x**2) - 3*a**4*d*f*h/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 2*a**3*b*c*f*h
*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + a**3*b*c*f*h/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*
b**6*x**2) + 2*a**3*b*d*e*h*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + a**3*b*d*e*h/(2*a**3*
b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 2*a**3*b*d*f*g*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x*
*2) + a**3*b*d*f*g/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 12*a**3*b*d*f*h*x*log(a/b + x)/(2*a**3*b**4
+ 4*a**2*b**5*x + 2*a*b**6*x**2) + 4*a**2*b**2*c*f*h*x*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x
**2) + 4*a**2*b**2*d*e*h*x*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 4*a**2*b**2*d*f*g*x*lo
g(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 6*a**2*b**2*d*f*h*x**2*log(a/b + x)/(2*a**3*b**4 +
4*a**2*b**5*x + 2*a*b**6*x**2) + 6*a**2*b**2*d*f*h*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - a*b**3
*c*e*g/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 2*a*b**3*c*f*h*x**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*
b**5*x + 2*a*b**6*x**2) - 2*a*b**3*c*f*h*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 2*a*b**3*d*e*h*x
**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 2*a*b**3*d*e*h*x**2/(2*a**3*b**4 + 4*a**2*b**
5*x + 2*a*b**6*x**2) + 2*a*b**3*d*f*g*x**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 2*a*b*
*3*d*f*g*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 2*a*b**3*d*f*h*x**3/(2*a**3*b**4 + 4*a**2*b**5*x
+ 2*a*b**6*x**2) + b**4*c*e*h*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + b**4*c*f*g*x**2/(2*a**3*b*
*4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + b**4*d*e*g*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2), Eq(m, -3)
), (6*a**3*d*f*h*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*d*f*h/(2*a*b**4 + 2*b**5*x) - 4*a**2*b*c*f*h*log(
a/b + x)/(2*a*b**4 + 2*b**5*x) - 4*a**2*b*c*f*h/(2*a*b**4 + 2*b**5*x) - 4*a**2*b*d*e*h*log(a/b + x)/(2*a*b**4
+ 2*b**5*x) - 4*a**2*b*d*e*h/(2*a*b**4 + 2*b**5*x) - 4*a**2*b*d*f*g*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 4*a**
2*b*d*f*g/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d*f*h*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*c*e*h*log(a/b
+ x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*c*e*h/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*c*f*g*log(a/b + x)/(2*a*b**4 + 2
*b**5*x) + 2*a*b**2*c*f*g/(2*a*b**4 + 2*b**5*x) - 4*a*b**2*c*f*h*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b*
*2*d*e*g*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*d*e*g/(2*a*b**4 + 2*b**5*x) - 4*a*b**2*d*e*h*x*log(a/b
+ x)/(2*a*b**4 + 2*b**5*x) - 4*a*b**2*d*f*g*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d*f*h*x**2/(2*a*b*
*4 + 2*b**5*x) - 2*b**3*c*e*g/(2*a*b**4 + 2*b**5*x) + 2*b**3*c*e*h*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*b*
*3*c*f*g*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*b**3*c*f*h*x**2/(2*a*b**4 + 2*b**5*x) + 2*b**3*d*e*g*x*log(a
/b + x)/(2*a*b**4 + 2*b**5*x) + 2*b**3*d*e*h*x**2/(2*a*b**4 + 2*b**5*x) + 2*b**3*d*f*g*x**2/(2*a*b**4 + 2*b**5
*x) + b**3*d*f*h*x**3/(2*a*b**4 + 2*b**5*x), Eq(m, -2)), (-a**3*d*f*h*log(a/b + x)/b**4 + a**2*c*f*h*log(a/b +
x)/b**3 + a**2*d*e*h*log(a/b + x)/b**3 + a**2*d*f*g*log(a/b + x)/b**3 + a**2*d*f*h*x/b**3 - a*c*e*h*log(a/b +
x)/b**2 - a*c*f*g*log(a/b + x)/b**2 - a*c*f*h*x/b**2 - a*d*e*g*log(a/b + x)/b**2 - a*d*e*h*x/b**2 - a*d*f*g*x
/b**2 - a*d*f*h*x**2/(2*b**2) + c*e*g*log(a/b + x)/b + c*e*h*x/b + c*f*g*x/b + c*f*h*x**2/(2*b) + d*e*g*x/b +
d*e*h*x**2/(2*b) + d*f*g*x**2/(2*b) + d*f*h*x**3/(3*b), Eq(m, -1)), (-6*a**4*d*f*h*(a + b*x)**m/(b**4*m**4 + 1
0*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 2*a**3*b*c*f*h*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 +
35*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*a**3*b*c*f*h*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 5
0*b**4*m + 24*b**4) + 2*a**3*b*d*e*h*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*
b**4) + 8*a**3*b*d*e*h*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 2*a**3*b
*d*f*g*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*a**3*b*d*f*g*(a + b*
x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 6*a**3*b*d*f*h*m*x*(a + b*x)**m/(b**4*
m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - a**2*b**2*c*e*h*m**2*(a + b*x)**m/(b**4*m**4 + 10*
b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 7*a**2*b**2*c*e*h*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 +
35*b**4*m**2 + 50*b**4*m + 24*b**4) - 12*a**2*b**2*c*e*h*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**
2 + 50*b**4*m + 24*b**4) - a**2*b**2*c*f*g*m**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**
4*m + 24*b**4) - 7*a**2*b**2*c*f*g*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b*
*4) - 12*a**2*b**2*c*f*g*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 2*a**2
*b**2*c*f*h*m**2*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 8*a**2*b**2*
c*f*h*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - a**2*b**2*d*e*g*m**2*
(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 7*a**2*b**2*d*e*g*m*(a + b*x)**
m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 12*a**2*b**2*d*e*g*(a + b*x)**m/(b**4*m**4
+ 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 2*a**2*b**2*d*e*h*m**2*x*(a + b*x)**m/(b**4*m**4 + 10*
b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 8*a**2*b**2*d*e*h*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3
+ 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 2*a**2*b**2*d*f*g*m**2*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*
b**4*m**2 + 50*b**4*m + 24*b**4) - 8*a**2*b**2*d*f*g*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2
+ 50*b**4*m + 24*b**4) - 3*a**2*b**2*d*f*h*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 +
50*b**4*m + 24*b**4) - 3*a**2*b**2*d*f*h*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**
4*m + 24*b**4) + a*b**3*c*e*g*m**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4
) + 9*a*b**3*c*e*g*m**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 26*a*b*
*3*c*e*g*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 24*a*b**3*c*e*g*(a +
b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + a*b**3*c*e*h*m**3*x*(a + b*x)**m/(b
**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 7*a*b**3*c*e*h*m**2*x*(a + b*x)**m/(b**4*m**4
+ 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 12*a*b**3*c*e*h*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m
**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + a*b**3*c*f*g*m**3*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b*
*4*m**2 + 50*b**4*m + 24*b**4) + 7*a*b**3*c*f*g*m**2*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 +
50*b**4*m + 24*b**4) + 12*a*b**3*c*f*g*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m
+ 24*b**4) + a*b**3*c*f*h*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**
4) + 5*a*b**3*c*f*h*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 4
*a*b**3*c*f*h*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + a*b**3*d*e
*g*m**3*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 7*a*b**3*d*e*g*m**2*x
*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 12*a*b**3*d*e*g*m*x*(a + b*x)*
*m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + a*b**3*d*e*h*m**3*x**2*(a + b*x)**m/(b**4
*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 5*a*b**3*d*e*h*m**2*x**2*(a + b*x)**m/(b**4*m**4
+ 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 4*a*b**3*d*e*h*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4
*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + a*b**3*d*f*g*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 +
35*b**4*m**2 + 50*b**4*m + 24*b**4) + 5*a*b**3*d*f*g*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**
4*m**2 + 50*b**4*m + 24*b**4) + 4*a*b**3*d*f*g*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 +
50*b**4*m + 24*b**4) + a*b**3*d*f*h*m**3*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*
m + 24*b**4) + 3*a*b**3*d*f*h*m**2*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24
*b**4) + 2*a*b**3*d*f*h*m*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) +
b**4*c*e*g*m**3*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 9*b**4*c*e*g*
m**2*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 26*b**4*c*e*g*m*x*(a + b
*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 24*b**4*c*e*g*x*(a + b*x)**m/(b**4*m*
*4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*c*e*h*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b*
*4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*b**4*c*e*h*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3
+ 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 19*b**4*c*e*h*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*
m**2 + 50*b**4*m + 24*b**4) + 12*b**4*c*e*h*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b*
*4*m + 24*b**4) + b**4*c*f*g*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*
b**4) + 8*b**4*c*f*g*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) +
19*b**4*c*f*g*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 12*b**4*c*
f*g*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*c*f*h*m**3*x**3*(
a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 7*b**4*c*f*h*m**2*x**3*(a + b*x)
**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 14*b**4*c*f*h*m*x**3*(a + b*x)**m/(b**4*
m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*b**4*c*f*h*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**
4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*d*e*g*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 3
5*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*b**4*d*e*g*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m
**2 + 50*b**4*m + 24*b**4) + 19*b**4*d*e*g*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b
**4*m + 24*b**4) + 12*b**4*d*e*g*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b
**4) + b**4*d*e*h*m**3*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 7*b
**4*d*e*h*m**2*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 14*b**4*d*e
*h*m*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*b**4*d*e*h*x**3*(a
+ b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*d*f*g*m**3*x**3*(a + b*x)**m/
(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 7*b**4*d*f*g*m**2*x**3*(a + b*x)**m/(b**4*m*
*4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 14*b**4*d*f*g*m*x**3*(a + b*x)**m/(b**4*m**4 + 10*b*
*4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 8*b**4*d*f*g*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*
b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*d*f*h*m**3*x**4*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2
+ 50*b**4*m + 24*b**4) + 6*b**4*d*f*h*m**2*x**4*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**
4*m + 24*b**4) + 11*b**4*d*f*h*m*x**4*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b
**4) + 6*b**4*d*f*h*x**4*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4), True))

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Giac [B]  time = 2.65803, size = 2248, normalized size = 13.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)*(f*x+e)*(h*x+g),x, algorithm="giac")

[Out]

((b*x + a)^m*b^4*d*f*h*m^3*x^4 + (b*x + a)^m*b^4*d*f*g*m^3*x^3 + (b*x + a)^m*b^4*c*f*h*m^3*x^3 + (b*x + a)^m*a
*b^3*d*f*h*m^3*x^3 + 6*(b*x + a)^m*b^4*d*f*h*m^2*x^4 + (b*x + a)^m*b^4*d*h*m^3*x^3*e + (b*x + a)^m*b^4*c*f*g*m
^3*x^2 + (b*x + a)^m*a*b^3*d*f*g*m^3*x^2 + (b*x + a)^m*a*b^3*c*f*h*m^3*x^2 + 7*(b*x + a)^m*b^4*d*f*g*m^2*x^3 +
7*(b*x + a)^m*b^4*c*f*h*m^2*x^3 + 3*(b*x + a)^m*a*b^3*d*f*h*m^2*x^3 + 11*(b*x + a)^m*b^4*d*f*h*m*x^4 + (b*x +
a)^m*b^4*d*g*m^3*x^2*e + (b*x + a)^m*b^4*c*h*m^3*x^2*e + (b*x + a)^m*a*b^3*d*h*m^3*x^2*e + 7*(b*x + a)^m*b^4*
d*h*m^2*x^3*e + (b*x + a)^m*a*b^3*c*f*g*m^3*x + 8*(b*x + a)^m*b^4*c*f*g*m^2*x^2 + 5*(b*x + a)^m*a*b^3*d*f*g*m^
2*x^2 + 5*(b*x + a)^m*a*b^3*c*f*h*m^2*x^2 - 3*(b*x + a)^m*a^2*b^2*d*f*h*m^2*x^2 + 14*(b*x + a)^m*b^4*d*f*g*m*x
^3 + 14*(b*x + a)^m*b^4*c*f*h*m*x^3 + 2*(b*x + a)^m*a*b^3*d*f*h*m*x^3 + 6*(b*x + a)^m*b^4*d*f*h*x^4 + (b*x + a
)^m*b^4*c*g*m^3*x*e + (b*x + a)^m*a*b^3*d*g*m^3*x*e + (b*x + a)^m*a*b^3*c*h*m^3*x*e + 8*(b*x + a)^m*b^4*d*g*m^
2*x^2*e + 8*(b*x + a)^m*b^4*c*h*m^2*x^2*e + 5*(b*x + a)^m*a*b^3*d*h*m^2*x^2*e + 14*(b*x + a)^m*b^4*d*h*m*x^3*e
+ 7*(b*x + a)^m*a*b^3*c*f*g*m^2*x - 2*(b*x + a)^m*a^2*b^2*d*f*g*m^2*x - 2*(b*x + a)^m*a^2*b^2*c*f*h*m^2*x + 1
9*(b*x + a)^m*b^4*c*f*g*m*x^2 + 4*(b*x + a)^m*a*b^3*d*f*g*m*x^2 + 4*(b*x + a)^m*a*b^3*c*f*h*m*x^2 - 3*(b*x + a
)^m*a^2*b^2*d*f*h*m*x^2 + 8*(b*x + a)^m*b^4*d*f*g*x^3 + 8*(b*x + a)^m*b^4*c*f*h*x^3 + (b*x + a)^m*a*b^3*c*g*m^
3*e + 9*(b*x + a)^m*b^4*c*g*m^2*x*e + 7*(b*x + a)^m*a*b^3*d*g*m^2*x*e + 7*(b*x + a)^m*a*b^3*c*h*m^2*x*e - 2*(b
*x + a)^m*a^2*b^2*d*h*m^2*x*e + 19*(b*x + a)^m*b^4*d*g*m*x^2*e + 19*(b*x + a)^m*b^4*c*h*m*x^2*e + 4*(b*x + a)^
m*a*b^3*d*h*m*x^2*e + 8*(b*x + a)^m*b^4*d*h*x^3*e - (b*x + a)^m*a^2*b^2*c*f*g*m^2 + 12*(b*x + a)^m*a*b^3*c*f*g
*m*x - 8*(b*x + a)^m*a^2*b^2*d*f*g*m*x - 8*(b*x + a)^m*a^2*b^2*c*f*h*m*x + 6*(b*x + a)^m*a^3*b*d*f*h*m*x + 12*
(b*x + a)^m*b^4*c*f*g*x^2 + 9*(b*x + a)^m*a*b^3*c*g*m^2*e - (b*x + a)^m*a^2*b^2*d*g*m^2*e - (b*x + a)^m*a^2*b^
2*c*h*m^2*e + 26*(b*x + a)^m*b^4*c*g*m*x*e + 12*(b*x + a)^m*a*b^3*d*g*m*x*e + 12*(b*x + a)^m*a*b^3*c*h*m*x*e -
8*(b*x + a)^m*a^2*b^2*d*h*m*x*e + 12*(b*x + a)^m*b^4*d*g*x^2*e + 12*(b*x + a)^m*b^4*c*h*x^2*e - 7*(b*x + a)^m
*a^2*b^2*c*f*g*m + 2*(b*x + a)^m*a^3*b*d*f*g*m + 2*(b*x + a)^m*a^3*b*c*f*h*m + 26*(b*x + a)^m*a*b^3*c*g*m*e -
7*(b*x + a)^m*a^2*b^2*d*g*m*e - 7*(b*x + a)^m*a^2*b^2*c*h*m*e + 2*(b*x + a)^m*a^3*b*d*h*m*e + 24*(b*x + a)^m*b
^4*c*g*x*e - 12*(b*x + a)^m*a^2*b^2*c*f*g + 8*(b*x + a)^m*a^3*b*d*f*g + 8*(b*x + a)^m*a^3*b*c*f*h - 6*(b*x + a
)^m*a^4*d*f*h + 24*(b*x + a)^m*a*b^3*c*g*e - 12*(b*x + a)^m*a^2*b^2*d*g*e - 12*(b*x + a)^m*a^2*b^2*c*h*e + 8*(
b*x + a)^m*a^3*b*d*h*e)/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)