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

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

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Rubi [A]  time = 0.13, antiderivative size = 167, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.23, size = 149, normalized size = 0.89 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^m (c+d x) (e+f x) (g+h x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.86, size = 877, normalized size = 5.25 \begin {gather*} \frac {{\left (a b^{3} c e g m^{3} + {\left (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\right )} x^{4} + {\left (8 \, b^{4} d f g + {\left (b^{4} d f g + {\left (b^{4} d e + {\left (b^{4} c + a b^{3} d\right )} f\right )} h\right )} m^{3} + {\left (7 \, b^{4} d f g + {\left (7 \, b^{4} d e + {\left (7 \, b^{4} c + 3 \, a b^{3} d\right )} f\right )} h\right )} m^{2} + 8 \, {\left (b^{4} d e + b^{4} c f\right )} h + 2 \, {\left (7 \, b^{4} d f g + {\left (7 \, b^{4} d e + {\left (7 \, b^{4} c + a b^{3} d\right )} f\right )} h\right )} m\right )} x^{3} - {\left (a^{2} b^{2} c e h + {\left (a^{2} b^{2} c f - {\left (9 \, a b^{3} c - a^{2} b^{2} d\right )} e\right )} g\right )} m^{2} + {\left (12 \, b^{4} c e h + {\left ({\left (b^{4} d e + {\left (b^{4} c + a b^{3} d\right )} f\right )} g + {\left (a b^{3} c f + {\left (b^{4} c + a b^{3} d\right )} e\right )} h\right )} m^{3} + {\left ({\left (8 \, b^{4} d e + {\left (8 \, b^{4} c + 5 \, a b^{3} d\right )} f\right )} g + {\left ({\left (8 \, b^{4} c + 5 \, a b^{3} d\right )} e + {\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} f\right )} h\right )} m^{2} + 12 \, {\left (b^{4} d e + b^{4} c f\right )} g + {\left ({\left (19 \, b^{4} d e + {\left (19 \, b^{4} c + 4 \, a b^{3} d\right )} f\right )} g + {\left ({\left (19 \, b^{4} c + 4 \, a b^{3} d\right )} e + {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} f\right )} h\right )} m\right )} x^{2} + 4 \, {\left (3 \, {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} e - {\left (3 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} f\right )} g - 2 \, {\left (2 \, {\left (3 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} e - {\left (4 \, a^{3} b c - 3 \, a^{4} d\right )} f\right )} h + {\left ({\left ({\left (26 \, a b^{3} c - 7 \, a^{2} b^{2} d\right )} e - {\left (7 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} f\right )} g + {\left (2 \, a^{3} b c f - {\left (7 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} e\right )} h\right )} m + {\left (24 \, b^{4} c e g + {\left (a b^{3} c e h + {\left (a b^{3} c f + {\left (b^{4} c + a b^{3} d\right )} e\right )} g\right )} m^{3} + {\left ({\left ({\left (9 \, b^{4} c + 7 \, a b^{3} d\right )} e + {\left (7 \, a b^{3} c - 2 \, a^{2} b^{2} d\right )} f\right )} g - {\left (2 \, a^{2} b^{2} c f - {\left (7 \, a b^{3} c - 2 \, a^{2} b^{2} d\right )} e\right )} h\right )} m^{2} + 2 \, {\left ({\left ({\left (13 \, b^{4} c + 6 \, a b^{3} d\right )} e + 2 \, {\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d\right )} f\right )} g + {\left (2 \, {\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d\right )} e - {\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} f\right )} h\right )} m\right )} x\right )} {\left (b x + a\right )}^{m}}{b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}} \end {gather*}

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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giac [B]  time = 1.03, size = 1665, normalized size = 9.97

result too large to display

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)

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maple [B]  time = 0.01, size = 726, normalized size = 4.35 \begin {gather*} -\frac {\left (-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 d f h \,x^{3} b^{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 \right ) \left (b x +a \right )^{m +1}}{\left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) b^{4}} \end {gather*}

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)^(m+1)*(-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 [B]  time = 0.54, size = 474, normalized size = 2.84 \begin {gather*} \frac {{\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} d e g}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} c f g}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} c e h}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{m + 1} c e g}{b {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} d f g}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} d e h}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} c f h}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a b^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b m x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{m} d f h}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{4}} \end {gather*}

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]

(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*d*e*g/((m^2 + 3*m + 2)*b^2) + (b^2*(m + 1)*x^2 + a*b*m*x - a^2)*
(b*x + a)^m*c*f*g/((m^2 + 3*m + 2)*b^2) + (b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*c*e*h/((m^2 + 3*m + 2)
*b^2) + (b*x + a)^(m + 1)*c*e*g/(b*(m + 1)) + ((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2*x^2 - 2*a^2*b*m*x + 2
*a^3)*(b*x + a)^m*d*f*g/((m^3 + 6*m^2 + 11*m + 6)*b^3) + ((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2*x^2 - 2*a^
2*b*m*x + 2*a^3)*(b*x + a)^m*d*e*h/((m^3 + 6*m^2 + 11*m + 6)*b^3) + ((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2
*x^2 - 2*a^2*b*m*x + 2*a^3)*(b*x + a)^m*c*f*h/((m^3 + 6*m^2 + 11*m + 6)*b^3) + ((m^3 + 6*m^2 + 11*m + 6)*b^4*x
^4 + (m^3 + 3*m^2 + 2*m)*a*b^3*x^3 - 3*(m^2 + m)*a^2*b^2*x^2 + 6*a^3*b*m*x - 6*a^4)*(b*x + a)^m*d*f*h/((m^4 +
10*m^3 + 35*m^2 + 50*m + 24)*b^4)

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mupad [B]  time = 2.95, size = 819, normalized size = 4.90 \begin {gather*} \frac {x\,{\left (a+b\,x\right )}^m\,\left (24\,b^4\,c\,e\,g+9\,b^4\,c\,e\,g\,m^2+b^4\,c\,e\,g\,m^3+26\,b^4\,c\,e\,g\,m+12\,a\,b^3\,c\,e\,h\,m+12\,a\,b^3\,c\,f\,g\,m+12\,a\,b^3\,d\,e\,g\,m+6\,a^3\,b\,d\,f\,h\,m+7\,a\,b^3\,c\,e\,h\,m^2+7\,a\,b^3\,c\,f\,g\,m^2+7\,a\,b^3\,d\,e\,g\,m^2+a\,b^3\,c\,e\,h\,m^3+a\,b^3\,c\,f\,g\,m^3+a\,b^3\,d\,e\,g\,m^3-8\,a^2\,b^2\,c\,f\,h\,m-8\,a^2\,b^2\,d\,e\,h\,m-8\,a^2\,b^2\,d\,f\,g\,m-2\,a^2\,b^2\,c\,f\,h\,m^2-2\,a^2\,b^2\,d\,e\,h\,m^2-2\,a^2\,b^2\,d\,f\,g\,m^2\right )}{b^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {{\left (a+b\,x\right )}^m\,\left (6\,a^4\,d\,f\,h+12\,a^2\,b^2\,c\,e\,h+12\,a^2\,b^2\,c\,f\,g+12\,a^2\,b^2\,d\,e\,g-24\,a\,b^3\,c\,e\,g-8\,a^3\,b\,c\,f\,h-8\,a^3\,b\,d\,e\,h-8\,a^3\,b\,d\,f\,g-26\,a\,b^3\,c\,e\,g\,m-2\,a^3\,b\,c\,f\,h\,m-2\,a^3\,b\,d\,e\,h\,m-2\,a^3\,b\,d\,f\,g\,m-9\,a\,b^3\,c\,e\,g\,m^2-a\,b^3\,c\,e\,g\,m^3+7\,a^2\,b^2\,c\,e\,h\,m+7\,a^2\,b^2\,c\,f\,g\,m+7\,a^2\,b^2\,d\,e\,g\,m+a^2\,b^2\,c\,e\,h\,m^2+a^2\,b^2\,c\,f\,g\,m^2+a^2\,b^2\,d\,e\,g\,m^2\right )}{b^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^3\,{\left (a+b\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,b\,c\,f\,h+4\,b\,d\,e\,h+4\,b\,d\,f\,g+a\,d\,f\,h\,m+b\,c\,f\,h\,m+b\,d\,e\,h\,m+b\,d\,f\,g\,m\right )}{b\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (a+b\,x\right )}^m\,\left (12\,b^2\,c\,e\,h+12\,b^2\,c\,f\,g+12\,b^2\,d\,e\,g+b^2\,c\,e\,h\,m^2+b^2\,c\,f\,g\,m^2+b^2\,d\,e\,g\,m^2+7\,b^2\,c\,e\,h\,m+7\,b^2\,c\,f\,g\,m+7\,b^2\,d\,e\,g\,m-3\,a^2\,d\,f\,h\,m+a\,b\,c\,f\,h\,m^2+a\,b\,d\,e\,h\,m^2+a\,b\,d\,f\,g\,m^2+4\,a\,b\,c\,f\,h\,m+4\,a\,b\,d\,e\,h\,m+4\,a\,b\,d\,f\,g\,m\right )}{b^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {d\,f\,h\,x^4\,{\left (a+b\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 8.12, size = 8221, normalized size = 49.23

result too large to display

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**3*d*f*h*log(a/b + x)/(2*a**2*b**4
+ 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d*f*h/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*a**2*b*c*f*h*log(a/b +
 x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 3*a**2*b*c*f*h/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*a**
2*b*d*e*h*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 3*a**2*b*d*e*h/(2*a**2*b**4 + 4*a*b**5*x + 2
*b**6*x**2) + 2*a**2*b*d*f*g*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 3*a**2*b*d*f*g/(2*a**2*b*
*4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d*f*h*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*
a**2*b*d*f*h*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - a*b**2*c*e*h/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2
) - a*b**2*c*f*g/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 4*a*b**2*c*f*h*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b
**5*x + 2*b**6*x**2) + 4*a*b**2*c*f*h*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - a*b**2*d*e*g/(2*a**2*b**4 +
 4*a*b**5*x + 2*b**6*x**2) + 4*a*b**2*d*e*h*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 4*a*b**2
*d*e*h*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 4*a*b**2*d*f*g*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x +
2*b**6*x**2) + 4*a*b**2*d*f*g*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*d*f*h*x**2*log(a/b + x)/(2
*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - b**3*c*e*g/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*b**3*c*e*h*x/
(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*b**3*c*f*g*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*c*
f*h*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*b**3*d*e*g*x/(2*a**2*b**4 + 4*a*b**5*x + 2*
b**6*x**2) + 2*b**3*d*e*h*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d*f*g*x**2*log(a
/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d*f*h*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*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 + 10*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 + 50*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 + 3
5*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 + 2
4*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 + 2
4*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 + 35*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) + 1
4*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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