3.32.6 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^3 \, dx\) [3106]

Optimal. Leaf size=460 \[ -\frac {3 (b e-a f) (d e-c f) (a d f (3+m)-b (d e+c f (2+m))) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac {3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {3 f (b e-a f) (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d (b c-a d) (4+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)^3}{(b c-a d) (4+m)} \]

[Out]

-3*(-a*f+b*e)*(-c*f+d*e)*(a*d*f*(3+m)-b*(d*e+c*f*(2+m)))*(b*x+a)^(1+m)*(d*x+c)^(-3-m)/b/d^2/(-a*d+b*c)^2/(3+m)
/(4+m)+3*(-a*f+b*e)*(a^2*d^2*f^2*(m^2+5*m+6)-2*a*b*d*f*(3+m)*(d*e+c*f*(1+m))+b^2*(2*d^2*e^2+2*c*d*e*f*(1+m)+c^
2*f^2*(m^2+3*m+2)))*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/b/d^2/(-a*d+b*c)^3/(2+m)/(3+m)/(4+m)+3*(-a*f+b*e)*(a^2*d^2*f^
2*(m^2+5*m+6)-2*a*b*d*f*(3+m)*(d*e+c*f*(1+m))+b^2*(2*d^2*e^2+2*c*d*e*f*(1+m)+c^2*f^2*(m^2+3*m+2)))*(b*x+a)^(1+
m)*(d*x+c)^(-1-m)/d^2/(-a*d+b*c)^4/(1+m)/(2+m)/(3+m)/(4+m)-3*f*(-a*f+b*e)*(b*x+a)^(1+m)*(d*x+c)^(-3-m)*(f*x+e)
/b/d/(-a*d+b*c)/(4+m)+(b*x+a)^(1+m)*(d*x+c)^(-4-m)*(f*x+e)^3/(-a*d+b*c)/(4+m)

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Rubi [A]
time = 0.32, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {96, 92, 80, 47, 37} \begin {gather*} \frac {3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{b d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac {3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac {3 (b e-a f) (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{b d^2 (m+3) (m+4) (b c-a d)^2}+\frac {(e+f x)^3 (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}-\frac {3 f (e+f x) (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d (m+4) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(b*e - a*f)*(d*e - c*f)*(b*d*e + b*c*f*(2 + m) - a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(b*d^
2*(b*c - a*d)^2*(3 + m)*(4 + m)) + (3*(b*e - a*f)*(a^2*d^2*f^2*(6 + 5*m + m^2) - 2*a*b*d*f*(3 + m)*(d*e + c*f*
(1 + m)) + b^2*(2*d^2*e^2 + 2*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m)
)/(b*d^2*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m)) + (3*(b*e - a*f)*(a^2*d^2*f^2*(6 + 5*m + m^2) - 2*a*b*d*f*(3 +
m)*(d*e + c*f*(1 + m)) + b^2*(2*d^2*e^2 + 2*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c
+ d*x)^(-1 - m))/(d^2*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (3*f*(b*e - a*f)*(a + b*x)^(1 + m)*(c +
d*x)^(-3 - m)*(e + f*x))/(b*d*(b*c - a*d)*(4 + m)) + ((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(e + f*x)^3)/((b*c
- a*d)*(4 + m))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c
, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rubi steps

\begin {align*} \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^3 \, dx &=\frac {(a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)^3}{(b c-a d) (4+m)}+\frac {(3 (b e-a f)) \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx}{(b c-a d) (4+m)}\\ &=-\frac {3 f (b e-a f) (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d (b c-a d) (4+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)^3}{(b c-a d) (4+m)}-\frac {(3 (b e-a f)) \int (a+b x)^m (c+d x)^{-4-m} \left (-b e (d e+c f (1+m))-a f (c f-d e (3+m))-(b c-a d) f^2 (2+m) x\right ) \, dx}{b d (b c-a d) (4+m)}\\ &=\frac {3 (b e-a f) (d e-c f) (b d e+b c f (2+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d)^2 (3+m) (4+m)}-\frac {3 f (b e-a f) (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d (b c-a d) (4+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)^3}{(b c-a d) (4+m)}+\frac {\left (3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right )\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{b d^2 (b c-a d)^2 (3+m) (4+m)}\\ &=\frac {3 (b e-a f) (d e-c f) (b d e+b c f (2+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac {3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}-\frac {3 f (b e-a f) (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d (b c-a d) (4+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)^3}{(b c-a d) (4+m)}+\frac {\left (3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right )\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}\\ &=\frac {3 (b e-a f) (d e-c f) (b d e+b c f (2+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac {3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {3 (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {3 f (b e-a f) (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d (b c-a d) (4+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)^3}{(b c-a d) (4+m)}\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 263, normalized size = 0.57 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left ((e+f x)^3-\frac {3 (b e-a f) (c+d x) \left ((b c-a d)^2 (-d e+c f) (1+m) (2+m) (b d e+b c f (2+m)-a d f (3+m))-\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (c+d x) (-a d (1+m)+b c (2+m)+b d x)+d (b c-a d)^3 f (1+m) (2+m) (3+m) (e+f x)\right )}{b d^2 (b c-a d)^3 (1+m) (2+m) (3+m)}\right )}{(b c-a d) (4+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*((e + f*x)^3 - (3*(b*e - a*f)*(c + d*x)*((b*c - a*d)^2*(-(d*e) + c*f)*(1
+ m)*(2 + m)*(b*d*e + b*c*f*(2 + m) - a*d*f*(3 + m)) - (a^2*d^2*f^2*(6 + 5*m + m^2) - 2*a*b*d*f*(3 + m)*(d*e
+ c*f*(1 + m)) + b^2*(2*d^2*e^2 + 2*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(c + d*x)*(-(a*d*(1 + m)) + b*
c*(2 + m) + b*d*x) + d*(b*c - a*d)^3*f*(1 + m)*(2 + m)*(3 + m)*(e + f*x)))/(b*d^2*(b*c - a*d)^3*(1 + m)*(2 + m
)*(3 + m))))/((b*c - a*d)*(4 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2480\) vs. \(2(460)=920\).
time = 0.12, size = 2481, normalized size = 5.39

method	result	size
gosper	\(\text {Expression too large to display}\)	\(2481\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^3,x,method=_RETURNVERBOSE)

[Out]

-(d*x+c)^(-4-m)*(b*x+a)^(1+m)*(a^3*d^3*f^3*m^3*x^3-3*a^2*b*c*d^2*f^3*m^3*x^3+3*a*b^2*c^2*d*f^3*m^3*x^3-b^3*c^3
*f^3*m^3*x^3+3*a^3*d^3*e*f^2*m^3*x^2+9*a^3*d^3*f^3*m^2*x^3-9*a^2*b*c*d^2*e*f^2*m^3*x^2-24*a^2*b*c*d^2*f^3*m^2*
x^3-3*a^2*b*d^3*e*f^2*m^2*x^3+9*a*b^2*c^2*d*e*f^2*m^3*x^2+21*a*b^2*c^2*d*f^3*m^2*x^3+6*a*b^2*c*d^2*e*f^2*m^2*x
^3-3*b^3*c^3*e*f^2*m^3*x^2-6*b^3*c^3*f^3*m^2*x^3-3*b^3*c^2*d*e*f^2*m^2*x^3+3*a^3*c*d^2*f^3*m^2*x^2+3*a^3*d^3*e
^2*f*m^3*x+24*a^3*d^3*e*f^2*m^2*x^2+26*a^3*d^3*f^3*m*x^3-6*a^2*b*c^2*d*f^3*m^2*x^2-9*a^2*b*c*d^2*e^2*f*m^3*x-6
9*a^2*b*c*d^2*e*f^2*m^2*x^2-57*a^2*b*c*d^2*f^3*m*x^3-6*a^2*b*d^3*e^2*f*m^2*x^2-21*a^2*b*d^3*e*f^2*m*x^3+3*a*b^
2*c^3*f^3*m^2*x^2+9*a*b^2*c^2*d*e^2*f*m^3*x+66*a*b^2*c^2*d*e*f^2*m^2*x^2+42*a*b^2*c^2*d*f^3*m*x^3+12*a*b^2*c*d
^2*e^2*f*m^2*x^2+30*a*b^2*c*d^2*e*f^2*m*x^3+6*a*b^2*d^3*e^2*f*m*x^3-3*b^3*c^3*e^2*f*m^3*x-21*b^3*c^3*e*f^2*m^2
*x^2-11*b^3*c^3*f^3*m*x^3-6*b^3*c^2*d*e^2*f*m^2*x^2-9*b^3*c^2*d*e*f^2*m*x^3-6*b^3*c*d^2*e^2*f*m*x^3+6*a^3*c*d^
2*e*f^2*m^2*x+21*a^3*c*d^2*f^3*m*x^2+a^3*d^3*e^3*m^3+21*a^3*d^3*e^2*f*m^2*x+57*a^3*d^3*e*f^2*m*x^2+24*a^3*d^3*
f^3*x^3-12*a^2*b*c^2*d*e*f^2*m^2*x-30*a^2*b*c^2*d*f^3*m*x^2-3*a^2*b*c*d^2*e^3*m^3-66*a^2*b*c*d^2*e^2*f*m^2*x-1
74*a^2*b*c*d^2*e*f^2*m*x^2-36*a^2*b*c*d^2*f^3*x^3-3*a^2*b*d^3*e^3*m^2*x-30*a^2*b*d^3*e^2*f*m*x^2-36*a^2*b*d^3*
e*f^2*x^3+6*a*b^2*c^3*e*f^2*m^2*x+9*a*b^2*c^3*f^3*m*x^2+3*a*b^2*c^2*d*e^3*m^3+69*a*b^2*c^2*d*e^2*f*m^2*x+159*a
*b^2*c^2*d*e*f^2*m*x^2+24*a*b^2*c^2*d*f^3*x^3+6*a*b^2*c*d^2*e^3*m^2*x+60*a*b^2*c*d^2*e^2*f*m*x^2+24*a*b^2*c*d^
2*e*f^2*x^3+6*a*b^2*d^3*e^3*m*x^2+24*a*b^2*d^3*e^2*f*x^3-b^3*c^3*e^3*m^3-24*b^3*c^3*e^2*f*m^2*x-42*b^3*c^3*e*f
^2*m*x^2-6*b^3*c^3*f^3*x^3-3*b^3*c^2*d*e^3*m^2*x-30*b^3*c^2*d*e^2*f*m*x^2-6*b^3*c^2*d*e*f^2*x^3-6*b^3*c*d^2*e^
3*m*x^2-6*b^3*c*d^2*e^2*f*x^3-6*b^3*d^3*e^3*x^3+6*a^3*c^2*d*f^3*m*x+3*a^3*c*d^2*e^2*f*m^2+30*a^3*c*d^2*e*f^2*m
*x+36*a^3*c*d^2*f^3*x^2+6*a^3*d^3*e^3*m^2+42*a^3*d^3*e^2*f*m*x+36*a^3*d^3*e*f^2*x^2-6*a^2*b*c^3*f^3*m*x-6*a^2*
b*c^2*d*e^2*f*m^2-60*a^2*b*c^2*d*e*f^2*m*x-24*a^2*b*c^2*d*f^3*x^2-21*a^2*b*c*d^2*e^3*m^2-159*a^2*b*c*d^2*e^2*f
*m*x-168*a^2*b*c*d^2*e*f^2*x^2-9*a^2*b*d^3*e^3*m*x-24*a^2*b*d^3*e^2*f*x^2+3*a*b^2*c^3*e^2*f*m^2+30*a*b^2*c^3*e
*f^2*m*x+6*a*b^2*c^3*f^3*x^2+24*a*b^2*c^2*d*e^3*m^2+174*a*b^2*c^2*d*e^2*f*m*x+102*a*b^2*c^2*d*e*f^2*x^2+30*a*b
^2*c*d^2*e^3*m*x+102*a*b^2*c*d^2*e^2*f*x^2+6*a*b^2*d^3*e^3*x^2-9*b^3*c^3*e^3*m^2-57*b^3*c^3*e^2*f*m*x-24*b^3*c
^3*e*f^2*x^2-21*b^3*c^2*d*e^3*m*x-24*b^3*c^2*d*e^2*f*x^2-24*b^3*c*d^2*e^3*x^2+6*a^3*c^2*d*e*f^2*m+24*a^3*c^2*d
*f^3*x+9*a^3*c*d^2*e^2*f*m+24*a^3*c*d^2*e*f^2*x+11*a^3*d^3*e^3*m+24*a^3*d^3*e^2*f*x-6*a^2*b*c^3*e*f^2*m-6*a^2*
b*c^3*f^3*x-30*a^2*b*c^2*d*e^2*f*m-102*a^2*b*c^2*d*e*f^2*x-42*a^2*b*c*d^2*e^3*m-102*a^2*b*c*d^2*e^2*f*x-6*a^2*
b*d^3*e^3*x+21*a*b^2*c^3*e^2*f*m+24*a*b^2*c^3*e*f^2*x+57*a*b^2*c^2*d*e^3*m+168*a*b^2*c^2*d*e^2*f*x+24*a*b^2*c*
d^2*e^3*x-26*b^3*c^3*e^3*m-36*b^3*c^3*e^2*f*x-36*b^3*c^2*d*e^3*x+6*a^3*c^3*f^3+6*a^3*c^2*d*e*f^2+6*a^3*c*d^2*e
^2*f+6*a^3*d^3*e^3-24*a^2*b*c^3*e*f^2-24*a^2*b*c^2*d*e^2*f-24*a^2*b*c*d^2*e^3+36*a*b^2*c^3*e^2*f+36*a*b^2*c^2*
d*e^3-24*b^3*c^3*e^3)/(a^4*d^4*m^4-4*a^3*b*c*d^3*m^4+6*a^2*b^2*c^2*d^2*m^4-4*a*b^3*c^3*d*m^4+b^4*c^4*m^4+10*a^
4*d^4*m^3-40*a^3*b*c*d^3*m^3+60*a^2*b^2*c^2*d^2*m^3-40*a*b^3*c^3*d*m^3+10*b^4*c^4*m^3+35*a^4*d^4*m^2-140*a^3*b
*c*d^3*m^2+210*a^2*b^2*c^2*d^2*m^2-140*a*b^3*c^3*d*m^2+35*b^4*c^4*m^2+50*a^4*d^4*m-200*a^3*b*c*d^3*m+300*a^2*b
^2*c^2*d^2*m-200*a*b^3*c^3*d*m+50*b^4*c^4*m+24*a^4*d^4-96*a^3*b*c*d^3+144*a^2*b^2*c^2*d^2-96*a*b^3*c^3*d+24*b^
4*c^4)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^3,x, algorithm="maxima")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3383 vs. \(2 (470) = 940\).
time = 1.00, size = 3383, normalized size = 7.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^3,x, algorithm="fricas")

[Out]

-(6*a^4*c^4*f^3 - ((b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*f^3*m^3 + 3*(2*b^4*c^3*d - 7*a*
b^3*c^2*d^2 + 8*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*f^3*m^2 + (11*b^4*c^3*d - 42*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 - 2
6*a^3*b*d^4)*f^3*m + 6*(b^4*c^3*d - 4*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*f^3)*x^5 - ((b^4*c^4 - 2*
a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f^3*m^3 + 3*(2*b^4*c^4 - 6*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 + 4*a^3*b*c*
d^3 - 3*a^4*d^4)*f^3*m^2 + (11*b^4*c^4 - 40*a*b^3*c^3*d + 45*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 - 26*a^4*d^4)*f^
3*m + 6*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - 4*a^4*d^4)*f^3)*x^4 + (60*a^4*c*d^3*f^3
- (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*f^3*m^3 - 3*(a*b^3*c^4 - 6*a^2*b^2*c^3*d + 9*a^
3*b*c^2*d^2 - 4*a^4*c*d^3)*f^3*m^2 - (2*a*b^3*c^4 - 15*a^2*b^2*c^3*d + 60*a^3*b*c^2*d^2 - 47*a^4*c*d^3)*f^3*m)
*x^3 + 3*(20*a^4*c^2*d^2*f^3 + (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f^3*m^2 + (a^2*b^2*c^4 - 10*a^3*b*c
^3*d + 9*a^4*c^2*d^2)*f^3*m)*x^2 + 6*(5*a^4*c^3*d*f^3 - (a^3*b*c^4 - a^4*c^3*d)*f^3*m)*x - (6*b^4*d^4*x^5 + 24
*a*b^3*c^4 - 36*a^2*b^2*c^3*d + 24*a^3*b*c^2*d^2 - 6*a^4*c*d^3 + 6*(5*b^4*c*d^3 + (b^4*c*d^3 - a*b^3*d^4)*m)*x
^4 + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*m^3 + 3*(20*b^4*c^2*d^2 + (b^4*c^2*d^2 - 2*a*
b^3*c*d^3 + a^2*b^2*d^4)*m^2 + (9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*m)*x^3 + 3*(3*a*b^3*c^4 - 8*a^2*
b^2*c^3*d + 7*a^3*b*c^2*d^2 - 2*a^4*c*d^3)*m^2 + (60*b^4*c^3*d + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^
3 - a^3*b*d^4)*m^3 + 3*(4*b^4*c^3*d - 9*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 - a^3*b*d^4)*m^2 + (47*b^4*c^3*d - 60*
a*b^3*c^2*d^2 + 15*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*m)*x^2 + (26*a*b^3*c^4 - 57*a^2*b^2*c^3*d + 42*a^3*b*c^2*d^2 -
11*a^4*c*d^3)*m + (24*b^4*c^4 + 24*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 24*a^3*b*c*d^3 - 6*a^4*d^4 + (b^4*c^4 -
2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*m^3 + 3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 6*a^3*b*c*d
^3 - 2*a^4*d^4)*m^2 + (26*b^4*c^4 - 10*a*b^3*c^3*d - 45*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 - 11*a^4*d^4)*m)*x)*e
^3 - 3*(2*((b^4*c*d^3 - a*b^3*d^4)*f*m + (b^4*c*d^3 - 4*a*b^3*d^4)*f)*x^5 + 2*((b^4*c^2*d^2 - 2*a*b^3*c*d^3 +
a^2*b^2*d^4)*f*m^2 + 2*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*f*m + 5*(b^4*c^2*d^2 - 4*a*b^3*c*d^3)*f
)*x^4 - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f*m^2 + ((b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 -
a^3*b*d^4)*f*m^3 + 5*(2*b^4*c^3*d - 5*a*b^3*c^2*d^2 + 4*a^2*b^2*c*d^3 - a^3*b*d^4)*f*m^2 + (29*b^4*c^3*d - 66*
a*b^3*c^2*d^2 + 41*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*f*m + 20*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*f)*x^3 - (7*a^2*b^2*c^4
- 10*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f*m + ((b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f*m^3 + (8*b^4*c
^4 - 14*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 - 7*a^4*d^4)*f*m^2 + (19*b^4*c^4 - 36*a*b^3*c^3*d - 1
5*a^2*b^2*c^2*d^2 + 46*a^3*b*c*d^3 - 14*a^4*d^4)*f*m + 4*(3*b^4*c^4 - 12*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 8*
a^3*b*c*d^3 - 2*a^4*d^4)*f)*x^2 - 2*(6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*f + ((a*b^3*c^4 - 3*a^2*b^2*
c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*f*m^3 + (7*a*b^3*c^4 - 22*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 8*a^4*c*d^3)
*f*m^2 + (12*a*b^3*c^4 - 55*a^2*b^2*c^3*d + 60*a^3*b*c^2*d^2 - 17*a^4*c*d^3)*f*m - 10*(6*a^2*b^2*c^3*d - 4*a^3
*b*c^2*d^2 + a^4*c*d^3)*f)*x)*e^2 - 3*(((b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f^2*m^2 + (3*b^4*c^2*d^2 -
10*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*f^2*m + 2*(b^4*c^2*d^2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*f^2)*x^5 + ((b^4*c^3*
d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*f^2*m^3 + (8*b^4*c^3*d - 23*a*b^3*c^2*d^2 + 22*a^2*b^2*c*d^
3 - 7*a^3*b*d^4)*f^2*m^2 + (17*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 55*a^2*b^2*c*d^3 - 12*a^3*b*d^4)*f^2*m + 10*(b^4
*c^3*d - 4*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3)*f^2)*x^4 + 2*(a^3*b*c^4 - a^4*c^3*d)*f^2*m + ((b^4*c^4 - 2*a*b^3*c
^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f^2*m^3 + (7*b^4*c^4 - 16*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 8
*a^4*d^4)*f^2*m^2 + (14*b^4*c^4 - 46*a*b^3*c^3*d + 15*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 19*a^4*d^4)*f^2*m + 4
*(2*b^4*c^4 - 8*a*b^3*c^3*d + 12*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 3*a^4*d^4)*f^2)*x^3 + 2*(4*a^3*b*c^4 - a^4
*c^3*d)*f^2 + ((a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*f^2*m^3 + 5*(a*b^3*c^4 - 4*a^2*b^2*
c^3*d + 5*a^3*b*c^2*d^2 - 2*a^4*c*d^3)*f^2*m^2 + (4*a*b^3*c^4 - 41*a^2*b^2*c^3*d + 66*a^3*b*c^2*d^2 - 29*a^4*c
*d^3)*f^2*m + 20*(4*a^3*b*c^2*d^2 - a^4*c*d^3)*f^2)*x^2 - 2*((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f^2*m
^2 + 2*(2*a^2*b^2*c^4 - 5*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f^2*m - 5*(4*a^3*b*c^3*d - a^4*c^2*d^2)*f^2)*x)*e)*(b*x
+ a)^m*(d*x + c)^(-m - 5)/(24*b^4*c^4 - 96*a*b^3*c^3*d + 144*a^2*b^2*c^2*d^2 - 96*a^3*b*c*d^3 + 24*a^4*d^4 +
(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*m^4 + 10*(b^4*c^4 - 4*a*b^3*c^3*d + 6*
a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*m^3 + 35*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d
^3 + a^4*d^4)*m^2 + 50*(b^4*c^4 - 4*a*b^3*c^3*d...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**(-5-m)*(f*x+e)**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Mupad [B]
time = 6.85, size = 2500, normalized size = 5.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^3*(a + b*x)^m)/(c + d*x)^(m + 5),x)

[Out]

(x^3*(a + b*x)^m*(24*b^4*c^4*e*f^2 - 36*a^4*d^4*e*f^2 - 60*a^4*c*d^3*f^3 + 60*b^4*c^2*d^2*e^3 + 3*a^2*b^2*d^4*
e^3*m^2 + 3*b^4*c^2*d^2*e^3*m^2 + 60*b^4*c^3*d*e^2*f + 2*a*b^3*c^4*f^3*m - 47*a^4*c*d^3*f^3*m - 57*a^4*d^4*e*f
^2*m + 42*b^4*c^4*e*f^2*m + 3*a^2*b^2*d^4*e^3*m + 3*a*b^3*c^4*f^3*m^2 + a*b^3*c^4*f^3*m^3 + 27*b^4*c^2*d^2*e^3
*m - 12*a^4*c*d^3*f^3*m^2 - a^4*c*d^3*f^3*m^3 - 24*a^4*d^4*e*f^2*m^2 + 21*b^4*c^4*e*f^2*m^2 - 3*a^4*d^4*e*f^2*
m^3 + 3*b^4*c^4*e*f^2*m^3 - 240*a*b^3*c^2*d^2*e^2*f - 6*a*b^3*c*d^3*e^3*m^2 - 15*a^2*b^2*c^3*d*f^3*m + 60*a^3*
b*c^2*d^2*f^3*m - 15*a^3*b*d^4*e^2*f*m^2 - 3*a^3*b*d^4*e^2*f*m^3 + 30*b^4*c^3*d*e^2*f*m^2 + 3*b^4*c^3*d*e^2*f*
m^3 + 144*a^2*b^2*c^2*d^2*e*f^2 - 18*a^2*b^2*c^3*d*f^3*m^2 + 27*a^3*b*c^2*d^2*f^3*m^2 - 3*a^2*b^2*c^3*d*f^3*m^
3 + 3*a^3*b*c^2*d^2*f^3*m^3 - 96*a*b^3*c^3*d*e*f^2 + 144*a^3*b*c*d^3*e*f^2 - 30*a*b^3*c*d^3*e^3*m - 12*a^3*b*d
^4*e^2*f*m + 87*b^4*c^3*d*e^2*f*m - 138*a*b^3*c^3*d*e*f^2*m + 108*a^3*b*c*d^3*e*f^2*m + 9*a^2*b^2*c^2*d^2*e*f^
2*m^2 - 198*a*b^3*c^2*d^2*e^2*f*m + 123*a^2*b^2*c*d^3*e^2*f*m - 48*a*b^3*c^3*d*e*f^2*m^2 + 42*a^3*b*c*d^3*e*f^
2*m^2 - 6*a*b^3*c^3*d*e*f^2*m^3 + 6*a^3*b*c*d^3*e*f^2*m^3 + 45*a^2*b^2*c^2*d^2*e*f^2*m - 75*a*b^3*c^2*d^2*e^2*
f*m^2 + 60*a^2*b^2*c*d^3*e^2*f*m^2 - 9*a*b^3*c^2*d^2*e^2*f*m^3 + 9*a^2*b^2*c*d^3*e^2*f*m^3))/((a*d - b*c)^4*(c
+ d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - ((a + b*x)^m*(6*a^4*c^4*f^3 - 24*a*b^3*c^4*e^3 + 6*a^4*
c*d^3*e^3 - 24*a^3*b*c^4*e*f^2 + 6*a^4*c^3*d*e*f^2 - 26*a*b^3*c^4*e^3*m + 11*a^4*c*d^3*e^3*m + 36*a^2*b^2*c^3*
d*e^3 - 24*a^3*b*c^2*d^2*e^3 + 36*a^2*b^2*c^4*e^2*f + 6*a^4*c^2*d^2*e^2*f - 9*a*b^3*c^4*e^3*m^2 - a*b^3*c^4*e^
3*m^3 + 6*a^4*c*d^3*e^3*m^2 + a^4*c*d^3*e^3*m^3 + 57*a^2*b^2*c^3*d*e^3*m - 42*a^3*b*c^2*d^2*e^3*m + 21*a^2*b^2
*c^4*e^2*f*m + 9*a^4*c^2*d^2*e^2*f*m + 24*a^2*b^2*c^3*d*e^3*m^2 - 21*a^3*b*c^2*d^2*e^3*m^2 + 3*a^2*b^2*c^3*d*e
^3*m^3 - 3*a^3*b*c^2*d^2*e^3*m^3 + 3*a^2*b^2*c^4*e^2*f*m^2 + 3*a^4*c^2*d^2*e^2*f*m^2 - 24*a^3*b*c^3*d*e^2*f -
6*a^3*b*c^4*e*f^2*m + 6*a^4*c^3*d*e*f^2*m - 30*a^3*b*c^3*d*e^2*f*m - 6*a^3*b*c^3*d*e^2*f*m^2))/((a*d - b*c)^4*
(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^4*(a + b*x)^m*(6*b^4*c^4*f^3 - 24*a^4*d^4*f^3 + 30
*b^4*c*d^3*e^3 - 26*a^4*d^4*f^3*m + 11*b^4*c^4*f^3*m - 9*a^4*d^4*f^3*m^2 + 6*b^4*c^4*f^3*m^2 - a^4*d^4*f^3*m^3
+ b^4*c^4*f^3*m^3 + 36*a^2*b^2*c^2*d^2*f^3 - 24*a*b^3*c^3*d*f^3 - 24*a^3*b*c*d^3*f^3 + 30*b^4*c^3*d*e*f^2 - 6
*a*b^3*d^4*e^3*m + 6*b^4*c*d^3*e^3*m + 30*b^4*c^2*d^2*e^2*f - 120*a*b^3*c^2*d^2*e*f^2 + 180*a^2*b^2*c*d^3*e*f^
2 - 18*a*b^3*c^3*d*f^3*m^2 + 12*a^3*b*c*d^3*f^3*m^2 - 2*a*b^3*c^3*d*f^3*m^3 + 2*a^3*b*c*d^3*f^3*m^3 + 24*a^2*b
^2*d^4*e^2*f*m - 21*a^3*b*d^4*e*f^2*m^2 - 3*a^3*b*d^4*e*f^2*m^3 + 36*b^4*c^2*d^2*e^2*f*m + 24*b^4*c^3*d*e*f^2*
m^2 + 3*b^4*c^3*d*e*f^2*m^3 + 45*a^2*b^2*c^2*d^2*f^3*m + 6*a^2*b^2*d^4*e^2*f*m^2 + 6*b^4*c^2*d^2*e^2*f*m^2 - 1
20*a*b^3*c*d^3*e^2*f - 40*a*b^3*c^3*d*f^3*m + 10*a^3*b*c*d^3*f^3*m - 36*a^3*b*d^4*e*f^2*m + 51*b^4*c^3*d*e*f^2
*m + 9*a^2*b^2*c^2*d^2*f^3*m^2 - 60*a*b^3*c*d^3*e^2*f*m - 180*a*b^3*c^2*d^2*e*f^2*m + 165*a^2*b^2*c*d^3*e*f^2*
m - 12*a*b^3*c*d^3*e^2*f*m^2 - 69*a*b^3*c^2*d^2*e*f^2*m^2 + 66*a^2*b^2*c*d^3*e*f^2*m^2 - 9*a*b^3*c^2*d^2*e*f^2
*m^3 + 9*a^2*b^2*c*d^3*e*f^2*m^3))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - (x*
(a + b*x)^m*(6*a^4*d^4*e^3 - 24*b^4*c^4*e^3 + 30*a^4*c^3*d*f^3 + 11*a^4*d^4*e^3*m - 26*b^4*c^4*e^3*m + 6*a^4*d
^4*e^3*m^2 - 9*b^4*c^4*e^3*m^2 + a^4*d^4*e^3*m^3 - b^4*c^4*e^3*m^3 + 36*a^2*b^2*c^2*d^2*e^3 - 24*a*b^3*c^3*d*e
^3 - 24*a^3*b*c*d^3*e^3 + 30*a^4*c*d^3*e^2*f - 6*a^3*b*c^4*f^3*m + 6*a^4*c^3*d*f^3*m + 30*a^4*c^2*d^2*e*f^2 +
180*a^2*b^2*c^3*d*e^2*f - 120*a^3*b*c^2*d^2*e^2*f + 12*a*b^3*c^3*d*e^3*m^2 - 18*a^3*b*c*d^3*e^3*m^2 + 2*a*b^3*
c^3*d*e^3*m^3 - 2*a^3*b*c*d^3*e^3*m^3 + 24*a^2*b^2*c^4*e*f^2*m - 21*a*b^3*c^4*e^2*f*m^2 - 3*a*b^3*c^4*e^2*f*m^
3 + 36*a^4*c^2*d^2*e*f^2*m + 24*a^4*c*d^3*e^2*f*m^2 + 3*a^4*c*d^3*e^2*f*m^3 + 45*a^2*b^2*c^2*d^2*e^3*m + 6*a^2
*b^2*c^4*e*f^2*m^2 + 6*a^4*c^2*d^2*e*f^2*m^2 - 120*a^3*b*c^3*d*e*f^2 + 10*a*b^3*c^3*d*e^3*m - 40*a^3*b*c*d^3*e
^3*m - 36*a*b^3*c^4*e^2*f*m + 51*a^4*c*d^3*e^2*f*m + 9*a^2*b^2*c^2*d^2*e^3*m^2 - 60*a^3*b*c^3*d*e*f^2*m + 165*
a^2*b^2*c^3*d*e^2*f*m - 180*a^3*b*c^2*d^2*e^2*f*m - 12*a^3*b*c^3*d*e*f^2*m^2 + 66*a^2*b^2*c^3*d*e^2*f*m^2 - 69
*a^3*b*c^2*d^2*e^2*f*m^2 + 9*a^2*b^2*c^3*d*e^2*f*m^3 - 9*a^3*b*c^2*d^2*e^2*f*m^3))/((a*d - b*c)^4*(c + d*x)^(m
+ 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^5*(a + b*x)^m*(6*b^4*d^4*e^3 - 24*a^3*b*d^4*f^3 + 6*b^4*c^3*d*
f^3 - 24*a*b^3*d^4*e^2*f + 6*b^4*c*d^3*e^2*f - 26*a^3*b*d^4*f^3*m + 11*b^4*c^3*d*f^3*m - 24*a*b^3*c^2*d^2*f^3
+ 36*a^2*b^2*c*d^3*f^3 + 36*a^2*b^2*d^4*e*f^2 + 6*b^4*c^2*d^2*e*f^2 - 9*a^3*b*d^4*f^3*m^2 - a^3*b*d^4*f^3*m^3
+ 6*b^4*c^3*d*f^3*m^2 + b^4*c^3*d*f^3*m^3 - 42*a*b^3*c^2*d^2*f^3*m + 57*a^2*b^2*c*d^3*f^3*m + 21*a^2*b^2*d^4*e
*f^2*m + 9*b^4*c^2*d^2*e*f^2*m - 21*a*b^3*c^2*d^2*f^3*m^2 + 24*a^2*b^2*c*d^3*f^3*m^2 - 3*a*b^3*c^2*d^2*f^3*m^3
+ 3*a^2*b^2*c*d^3*f^3*m^3 + 3*a^2*b^2*d^4*e*f^...

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