3.12.27 \(\int (A+B x) (d+e x)^m (b x+c x^2)^3 \, dx\) [1127]

Optimal. Leaf size=484 \[ -\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) (d+e x)^{2+m}}{e^8 (2+m)}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}+\frac {\left (3 A c e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B \left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{5+m}}{e^8 (5+m)}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {B c^3 (d+e x)^{8+m}}{e^8 (8+m)} \]

[Out]

-d^3*(-A*e+B*d)*(-b*e+c*d)^3*(e*x+d)^(1+m)/e^8/(1+m)+d^2*(-b*e+c*d)^2*(B*d*(-4*b*e+7*c*d)-3*A*e*(-b*e+2*c*d))*
(e*x+d)^(2+m)/e^8/(2+m)+3*d*(-b*e+c*d)*(A*e*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)-B*d*(2*b^2*e^2-8*b*c*d*e+7*c^2*d^2))
*(e*x+d)^(3+m)/e^8/(3+m)+(B*d*(-4*b^3*e^3+30*b^2*c*d*e^2-60*b*c^2*d^2*e+35*c^3*d^3)-A*e*(-b^3*e^3+12*b^2*c*d*e
^2-30*b*c^2*d^2*e+20*c^3*d^3))*(e*x+d)^(4+m)/e^8/(4+m)+(3*A*c*e*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)-B*(-b^3*e^3+15*b
^2*c*d*e^2-45*b*c^2*d^2*e+35*c^3*d^3))*(e*x+d)^(5+m)/e^8/(5+m)-3*c*(A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-6*b*c*d*e+7*
c^2*d^2))*(e*x+d)^(6+m)/e^8/(6+m)-c^2*(-A*c*e-3*B*b*e+7*B*c*d)*(e*x+d)^(7+m)/e^8/(7+m)+B*c^3*(e*x+d)^(8+m)/e^8
/(8+m)

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Rubi [A]
time = 0.27, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \begin {gather*} \frac {3 d (c d-b e) (d+e x)^{m+3} \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+3)}-\frac {3 c (d+e x)^{m+6} \left (A c e (2 c d-b e)-B \left (b^2 e^2-6 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+6)}+\frac {(d+e x)^{m+4} \left (B d \left (-4 b^3 e^3+30 b^2 c d e^2-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8 (m+4)}+\frac {(d+e x)^{m+5} \left (3 A c e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B \left (-b^3 e^3+15 b^2 c d e^2-45 b c^2 d^2 e+35 c^3 d^3\right )\right )}{e^8 (m+5)}-\frac {c^2 (d+e x)^{m+7} (-A c e-3 b B e+7 B c d)}{e^8 (m+7)}-\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^{m+1}}{e^8 (m+1)}+\frac {d^2 (c d-b e)^2 (d+e x)^{m+2} (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (m+2)}+\frac {B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

-((d^3*(B*d - A*e)*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) -
3*A*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^8*(2 + m)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^
2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*(d + e*x)^(3 + m))/(e^8*(3 + m)) + ((B*d*(35*c^3*d^3 - 60*b*c^2*
d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3) - A*e*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3))*(d + e*x)^
(4 + m))/(e^8*(4 + m)) + ((3*A*c*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*(35*c^3*d^3 - 45*b*c^2*d^2*e + 15*b^2
*c*d*e^2 - b^3*e^3))*(d + e*x)^(5 + m))/(e^8*(5 + m)) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 - 6*b*c*d*e +
b^2*e^2))*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)
) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + m))

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx &=\int \left (-\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^m}{e^7}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) (d+e x)^{1+m}}{e^7}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) (d+e x)^{2+m}}{e^7}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{3+m}}{e^7}+\frac {\left (3 A c e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B \left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{4+m}}{e^7}+\frac {3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) (d+e x)^{5+m}}{e^7}+\frac {c^2 (-7 B c d+3 b B e+A c e) (d+e x)^{6+m}}{e^7}+\frac {B c^3 (d+e x)^{7+m}}{e^7}\right ) \, dx\\ &=-\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) (d+e x)^{2+m}}{e^8 (2+m)}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}+\frac {\left (3 A c e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B \left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{5+m}}{e^8 (5+m)}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {B c^3 (d+e x)^{8+m}}{e^8 (8+m)}\\ \end {align*}

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Mathematica [A]
time = 0.95, size = 525, normalized size = 1.08 \begin {gather*} \frac {(d+e x)^{1+m} \left (A e \left (\frac {d^3 (c d-b e)^3}{1+m}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)}{2+m}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^3}{4+m}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^4}{5+m}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{6+m}+\frac {c^3 (d+e x)^6}{7+m}\right )+B \left (-\frac {d^4 (c d-b e)^3}{1+m}+\frac {d^3 (7 c d-4 b e) (c d-b e)^2 (d+e x)}{2+m}-\frac {3 d^2 (c d-b e) \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right ) (d+e x)^2}{3+m}+\frac {d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right ) (d+e x)^3}{4+m}-\frac {\left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right ) (d+e x)^4}{5+m}+\frac {3 c \left (7 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^5}{6+m}-\frac {c^2 (7 c d-3 b e) (d+e x)^6}{7+m}+\frac {c^3 (d+e x)^7}{8+m}\right )\right )}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*(A*e*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x))/(2 + m) +
(3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c
*d*e + b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4)/(5 + m) - (3*c^2*(2
*c*d - b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + e*x)^6)/(7 + m)) + B*(-((d^4*(c*d - b*e)^3)/(1 + m)) + (d^3*(7*c*
d - 4*b*e)*(c*d - b*e)^2*(d + e*x))/(2 + m) - (3*d^2*(c*d - b*e)*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2)*(d + e*x)
^2)/(3 + m) + (d*(35*c^3*d^3 - 60*b*c^2*d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3)*(d + e*x)^3)/(4 + m) - ((35*c^3*d^
3 - 45*b*c^2*d^2*e + 15*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^4)/(5 + m) + (3*c*(7*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(
d + e*x)^5)/(6 + m) - (c^2*(7*c*d - 3*b*e)*(d + e*x)^6)/(7 + m) + (c^3*(d + e*x)^7)/(8 + m))))/e^8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2223\) vs. \(2(484)=968\).
time = 0.69, size = 2224, normalized size = 4.60

method	result	size
norman	\(\text {Expression too large to display}\)	\(2224\)
gosper	\(\text {Expression too large to display}\)	\(4138\)
risch	\(\text {Expression too large to display}\)	\(4678\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

B*c^3/(8+m)*x^8*exp(m*ln(e*x+d))+(3*A*b^2*c*e^3*m^3+3*A*b*c^2*d*e^2*m^3+B*b^3*e^3*m^3+3*B*b^2*c*d*e^2*m^3+63*A
*b^2*c*e^3*m^2+45*A*b*c^2*d*e^2*m^2-6*A*c^3*d^2*e*m^2+21*B*b^3*e^3*m^2+45*B*b^2*c*d*e^2*m^2-18*B*b*c^2*d^2*e*m
^2+438*A*b^2*c*e^3*m+168*A*b*c^2*d*e^2*m-48*A*c^3*d^2*e*m+146*B*b^3*e^3*m+168*B*b^2*c*d*e^2*m-144*B*b*c^2*d^2*
e*m+42*B*c^3*d^3*m+1008*A*b^2*c*e^3+336*B*b^3*e^3)/e^3/(m^4+26*m^3+251*m^2+1066*m+1680)*x^5*exp(m*ln(e*x+d))+(
A*b^3*e^4*m^4+3*A*b^2*c*d*e^3*m^4+B*b^3*d*e^3*m^4+26*A*b^3*e^4*m^3+63*A*b^2*c*d*e^3*m^3-15*A*b*c^2*d^2*e^2*m^3
+21*B*b^3*d*e^3*m^3-15*B*b^2*c*d^2*e^2*m^3+251*A*b^3*e^4*m^2+438*A*b^2*c*d*e^3*m^2-225*A*b*c^2*d^2*e^2*m^2+30*
A*c^3*d^3*e*m^2+146*B*b^3*d*e^3*m^2-225*B*b^2*c*d^2*e^2*m^2+90*B*b*c^2*d^3*e*m^2+1066*A*b^3*e^4*m+1008*A*b^2*c
*d*e^3*m-840*A*b*c^2*d^2*e^2*m+240*A*c^3*d^3*e*m+336*B*b^3*d*e^3*m-840*B*b^2*c*d^2*e^2*m+720*B*b*c^2*d^3*e*m-2
10*B*c^3*d^4*m+1680*A*b^3*e^4)/e^4/(m^5+30*m^4+355*m^3+2070*m^2+5944*m+6720)*x^4*exp(m*ln(e*x+d))+(A*c*e*m+3*B
*b*e*m+B*c*d*m+8*A*c*e+24*B*b*e)*c^2/e/(m^2+15*m+56)*x^7*exp(m*ln(e*x+d))+(3*A*b*c*e^2*m^2+A*c^2*d*e*m^2+3*B*b
^2*e^2*m^2+3*B*b*c*d*e*m^2+45*A*b*c*e^2*m+8*A*c^2*d*e*m+45*B*b^2*e^2*m+24*B*b*c*d*e*m-7*B*c^2*d^2*m+168*A*b*c*
e^2+168*B*b^2*e^2)*c/e^2/(m^3+21*m^2+146*m+336)*x^6*exp(m*ln(e*x+d))+m*d*(A*b^3*e^4*m^4+26*A*b^3*e^4*m^3-12*A*
b^2*c*d*e^3*m^3-4*B*b^3*d*e^3*m^3+251*A*b^3*e^4*m^2-252*A*b^2*c*d*e^3*m^2+60*A*b*c^2*d^2*e^2*m^2-84*B*b^3*d*e^
3*m^2+60*B*b^2*c*d^2*e^2*m^2+1066*A*b^3*e^4*m-1752*A*b^2*c*d*e^3*m+900*A*b*c^2*d^2*e^2*m-120*A*c^3*d^3*e*m-584
*B*b^3*d*e^3*m+900*B*b^2*c*d^2*e^2*m-360*B*b*c^2*d^3*e*m+1680*A*b^3*e^4-4032*A*b^2*c*d*e^3+3360*A*b*c^2*d^2*e^
2-960*A*c^3*d^3*e-1344*B*b^3*d*e^3+3360*B*b^2*c*d^2*e^2-2880*B*b*c^2*d^3*e+840*B*c^3*d^4)/e^5/(m^6+33*m^5+445*
m^4+3135*m^3+12154*m^2+24552*m+20160)*x^3*exp(m*ln(e*x+d))-6*d^4*(A*b^3*e^4*m^4+26*A*b^3*e^4*m^3-12*A*b^2*c*d*
e^3*m^3-4*B*b^3*d*e^3*m^3+251*A*b^3*e^4*m^2-252*A*b^2*c*d*e^3*m^2+60*A*b*c^2*d^2*e^2*m^2-84*B*b^3*d*e^3*m^2+60
*B*b^2*c*d^2*e^2*m^2+1066*A*b^3*e^4*m-1752*A*b^2*c*d*e^3*m+900*A*b*c^2*d^2*e^2*m-120*A*c^3*d^3*e*m-584*B*b^3*d
*e^3*m+900*B*b^2*c*d^2*e^2*m-360*B*b*c^2*d^3*e*m+1680*A*b^3*e^4-4032*A*b^2*c*d*e^3+3360*A*b*c^2*d^2*e^2-960*A*
c^3*d^3*e-1344*B*b^3*d*e^3+3360*B*b^2*c*d^2*e^2-2880*B*b*c^2*d^3*e+840*B*c^3*d^4)/e^8/(m^8+36*m^7+546*m^6+4536
*m^5+22449*m^4+67284*m^3+118124*m^2+109584*m+40320)*exp(m*ln(e*x+d))+6/e^7*m*d^3*(A*b^3*e^4*m^4+26*A*b^3*e^4*m
^3-12*A*b^2*c*d*e^3*m^3-4*B*b^3*d*e^3*m^3+251*A*b^3*e^4*m^2-252*A*b^2*c*d*e^3*m^2+60*A*b*c^2*d^2*e^2*m^2-84*B*
b^3*d*e^3*m^2+60*B*b^2*c*d^2*e^2*m^2+1066*A*b^3*e^4*m-1752*A*b^2*c*d*e^3*m+900*A*b*c^2*d^2*e^2*m-120*A*c^3*d^3
*e*m-584*B*b^3*d*e^3*m+900*B*b^2*c*d^2*e^2*m-360*B*b*c^2*d^3*e*m+1680*A*b^3*e^4-4032*A*b^2*c*d*e^3+3360*A*b*c^
2*d^2*e^2-960*A*c^3*d^3*e-1344*B*b^3*d*e^3+3360*B*b^2*c*d^2*e^2-2880*B*b*c^2*d^3*e+840*B*c^3*d^4)/(m^8+36*m^7+
546*m^6+4536*m^5+22449*m^4+67284*m^3+118124*m^2+109584*m+40320)*x*exp(m*ln(e*x+d))-3*(A*b^3*e^4*m^4+26*A*b^3*e
^4*m^3-12*A*b^2*c*d*e^3*m^3-4*B*b^3*d*e^3*m^3+251*A*b^3*e^4*m^2-252*A*b^2*c*d*e^3*m^2+60*A*b*c^2*d^2*e^2*m^2-8
4*B*b^3*d*e^3*m^2+60*B*b^2*c*d^2*e^2*m^2+1066*A*b^3*e^4*m-1752*A*b^2*c*d*e^3*m+900*A*b*c^2*d^2*e^2*m-120*A*c^3
*d^3*e*m-584*B*b^3*d*e^3*m+900*B*b^2*c*d^2*e^2*m-360*B*b*c^2*d^3*e*m+1680*A*b^3*e^4-4032*A*b^2*c*d*e^3+3360*A*
b*c^2*d^2*e^2-960*A*c^3*d^3*e-1344*B*b^3*d*e^3+3360*B*b^2*c*d^2*e^2-2880*B*b*c^2*d^3*e+840*B*c^3*d^4)*d^2/e^6*
m/(m^7+35*m^6+511*m^5+4025*m^4+18424*m^3+48860*m^2+69264*m+40320)*x^2*exp(m*ln(e*x+d))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1511 vs. \(2 (495) = 990\).
time = 0.32, size = 1511, normalized size = 3.12 \begin {gather*} \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} A b^{3} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} B b^{3} e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} A b^{2} c e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} x^{6} e^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d x^{5} e^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} x^{4} e^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} x^{3} e^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} x^{2} e^{2} + 120 \, d^{5} m x e - 120 \, d^{6}\right )} B b^{2} c e^{\left (m \log \left (x e + d\right ) - 6\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} x^{6} e^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d x^{5} e^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} x^{4} e^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} x^{3} e^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} x^{2} e^{2} + 120 \, d^{5} m x e - 120 \, d^{6}\right )} A b c^{2} e^{\left (m \log \left (x e + d\right ) - 6\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} + \frac {3 \, {\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} x^{7} e^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d x^{6} e^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} x^{5} e^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} x^{4} e^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} x^{3} e^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} x^{2} e^{2} - 720 \, d^{6} m x e + 720 \, d^{7}\right )} B b c^{2} e^{\left (m \log \left (x e + d\right ) - 7\right )}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} x^{7} e^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d x^{6} e^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} x^{5} e^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} x^{4} e^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} x^{3} e^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} x^{2} e^{2} - 720 \, d^{6} m x e + 720 \, d^{7}\right )} A c^{3} e^{\left (m \log \left (x e + d\right ) - 7\right )}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} + \frac {{\left ({\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} x^{8} e^{8} + {\left (m^{7} + 21 \, m^{6} + 175 \, m^{5} + 735 \, m^{4} + 1624 \, m^{3} + 1764 \, m^{2} + 720 \, m\right )} d x^{7} e^{7} - 7 \, {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d^{2} x^{6} e^{6} + 42 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{3} x^{5} e^{5} - 210 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{4} x^{4} e^{4} + 840 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{5} x^{3} e^{3} - 2520 \, {\left (m^{2} + m\right )} d^{6} x^{2} e^{2} + 5040 \, d^{7} m x e - 5040 \, d^{8}\right )} B c^{3} e^{\left (m \log \left (x e + d\right ) - 8\right )}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*
d^4)*A*b^3*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x
^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2
- 24*d^4*m*x*e + 24*d^5)*B*b^3*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 3*((m
^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2
*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*A*b^2*c*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85
*m^3 + 225*m^2 + 274*m + 120) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*x^6*e^6 + (m^5 + 10*m^4 + 3
5*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3
*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*x*e - 120*d^6)*B*b^2*c*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*
m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*x^6*e^6 + (m^5 +
10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2
*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*x*e - 120*d^6)*A*b*c^2*e^(m*log(x*e + d) - 6)/(m^6 + 21
*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + 3*((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m
+ 720)*x^7*e^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*x^6*e^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 5
0*m^2 + 24*m)*d^2*x^5*e^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*x^4*e^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*x^3*e^3
+ 360*(m^2 + m)*d^5*x^2*e^2 - 720*d^6*m*x*e + 720*d^7)*B*b*c^2*e^(m*log(x*e + d) - 7)/(m^7 + 28*m^6 + 322*m^5
+ 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m +
720)*x^7*e^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*x^6*e^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*
m^2 + 24*m)*d^2*x^5*e^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*x^4*e^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*x^3*e^3 +
360*(m^2 + m)*d^5*x^2*e^2 - 720*d^6*m*x*e + 720*d^7)*A*c^3*e^(m*log(x*e + d) - 7)/(m^7 + 28*m^6 + 322*m^5 + 19
60*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) + ((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 +
13068*m + 5040)*x^8*e^8 + (m^7 + 21*m^6 + 175*m^5 + 735*m^4 + 1624*m^3 + 1764*m^2 + 720*m)*d*x^7*e^7 - 7*(m^6
+ 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d^2*x^6*e^6 + 42*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^3*x
^5*e^5 - 210*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^4*x^4*e^4 + 840*(m^3 + 3*m^2 + 2*m)*d^5*x^3*e^3 - 2520*(m^2 + m)*d
^6*x^2*e^2 + 5040*d^7*m*x*e - 5040*d^8)*B*c^3*e^(m*log(x*e + d) - 8)/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 2244
9*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2986 vs. \(2 (495) = 990\).
time = 4.10, size = 2986, normalized size = 6.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-(5040*B*c^3*d^8 - ((B*c^3*m^7 + 28*B*c^3*m^6 + 322*B*c^3*m^5 + 1960*B*c^3*m^4 + 6769*B*c^3*m^3 + 13132*B*c^3*
m^2 + 13068*B*c^3*m + 5040*B*c^3)*x^8 + ((3*B*b*c^2 + A*c^3)*m^7 + 29*(3*B*b*c^2 + A*c^3)*m^6 + 343*(3*B*b*c^2
+ A*c^3)*m^5 + 2135*(3*B*b*c^2 + A*c^3)*m^4 + 17280*B*b*c^2 + 5760*A*c^3 + 7504*(3*B*b*c^2 + A*c^3)*m^3 + 147
56*(3*B*b*c^2 + A*c^3)*m^2 + 14832*(3*B*b*c^2 + A*c^3)*m)*x^7 + 3*((B*b^2*c + A*b*c^2)*m^7 + 30*(B*b^2*c + A*b
*c^2)*m^6 + 366*(B*b^2*c + A*b*c^2)*m^5 + 2340*(B*b^2*c + A*b*c^2)*m^4 + 6720*B*b^2*c + 6720*A*b*c^2 + 8409*(B
*b^2*c + A*b*c^2)*m^3 + 16830*(B*b^2*c + A*b*c^2)*m^2 + 17144*(B*b^2*c + A*b*c^2)*m)*x^6 + ((B*b^3 + 3*A*b^2*c
)*m^7 + 31*(B*b^3 + 3*A*b^2*c)*m^6 + 391*(B*b^3 + 3*A*b^2*c)*m^5 + 2581*(B*b^3 + 3*A*b^2*c)*m^4 + 8064*B*b^3 +
24192*A*b^2*c + 9544*(B*b^3 + 3*A*b^2*c)*m^3 + 19564*(B*b^3 + 3*A*b^2*c)*m^2 + 20304*(B*b^3 + 3*A*b^2*c)*m)*x
^5 + (A*b^3*m^7 + 32*A*b^3*m^6 + 418*A*b^3*m^5 + 2864*A*b^3*m^4 + 10993*A*b^3*m^3 + 23312*A*b^3*m^2 + 24876*A*
b^3*m + 10080*A*b^3)*x^4)*e^8 - ((B*c^3*d*m^7 + 21*B*c^3*d*m^6 + 175*B*c^3*d*m^5 + 735*B*c^3*d*m^4 + 1624*B*c^
3*d*m^3 + 1764*B*c^3*d*m^2 + 720*B*c^3*d*m)*x^7 + ((3*B*b*c^2 + A*c^3)*d*m^7 + 23*(3*B*b*c^2 + A*c^3)*d*m^6 +
205*(3*B*b*c^2 + A*c^3)*d*m^5 + 905*(3*B*b*c^2 + A*c^3)*d*m^4 + 2074*(3*B*b*c^2 + A*c^3)*d*m^3 + 2312*(3*B*b*c
^2 + A*c^3)*d*m^2 + 960*(3*B*b*c^2 + A*c^3)*d*m)*x^6 + 3*((B*b^2*c + A*b*c^2)*d*m^7 + 25*(B*b^2*c + A*b*c^2)*d
*m^6 + 241*(B*b^2*c + A*b*c^2)*d*m^5 + 1135*(B*b^2*c + A*b*c^2)*d*m^4 + 2734*(B*b^2*c + A*b*c^2)*d*m^3 + 3160*
(B*b^2*c + A*b*c^2)*d*m^2 + 1344*(B*b^2*c + A*b*c^2)*d*m)*x^5 + ((B*b^3 + 3*A*b^2*c)*d*m^7 + 27*(B*b^3 + 3*A*b
^2*c)*d*m^6 + 283*(B*b^3 + 3*A*b^2*c)*d*m^5 + 1449*(B*b^3 + 3*A*b^2*c)*d*m^4 + 3748*(B*b^3 + 3*A*b^2*c)*d*m^3
+ 4572*(B*b^3 + 3*A*b^2*c)*d*m^2 + 2016*(B*b^3 + 3*A*b^2*c)*d*m)*x^4 + (A*b^3*d*m^7 + 29*A*b^3*d*m^6 + 331*A*b
^3*d*m^5 + 1871*A*b^3*d*m^4 + 5380*A*b^3*d*m^3 + 7172*A*b^3*d*m^2 + 3360*A*b^3*d*m)*x^3)*e^7 + (7*(B*c^3*d^2*m
^6 + 15*B*c^3*d^2*m^5 + 85*B*c^3*d^2*m^4 + 225*B*c^3*d^2*m^3 + 274*B*c^3*d^2*m^2 + 120*B*c^3*d^2*m)*x^6 + 6*((
3*B*b*c^2 + A*c^3)*d^2*m^6 + 18*(3*B*b*c^2 + A*c^3)*d^2*m^5 + 115*(3*B*b*c^2 + A*c^3)*d^2*m^4 + 330*(3*B*b*c^2
+ A*c^3)*d^2*m^3 + 424*(3*B*b*c^2 + A*c^3)*d^2*m^2 + 192*(3*B*b*c^2 + A*c^3)*d^2*m)*x^5 + 15*((B*b^2*c + A*b*
c^2)*d^2*m^6 + 21*(B*b^2*c + A*b*c^2)*d^2*m^5 + 157*(B*b^2*c + A*b*c^2)*d^2*m^4 + 507*(B*b^2*c + A*b*c^2)*d^2*
m^3 + 706*(B*b^2*c + A*b*c^2)*d^2*m^2 + 336*(B*b^2*c + A*b*c^2)*d^2*m)*x^4 + 4*((B*b^3 + 3*A*b^2*c)*d^2*m^6 +
24*(B*b^3 + 3*A*b^2*c)*d^2*m^5 + 211*(B*b^3 + 3*A*b^2*c)*d^2*m^4 + 816*(B*b^3 + 3*A*b^2*c)*d^2*m^3 + 1300*(B*b
^3 + 3*A*b^2*c)*d^2*m^2 + 672*(B*b^3 + 3*A*b^2*c)*d^2*m)*x^3 + 3*(A*b^3*d^2*m^6 + 27*A*b^3*d^2*m^5 + 277*A*b^3
*d^2*m^4 + 1317*A*b^3*d^2*m^3 + 2746*A*b^3*d^2*m^2 + 1680*A*b^3*d^2*m)*x^2)*e^6 - 6*(7*(B*c^3*d^3*m^5 + 10*B*c
^3*d^3*m^4 + 35*B*c^3*d^3*m^3 + 50*B*c^3*d^3*m^2 + 24*B*c^3*d^3*m)*x^5 + 5*((3*B*b*c^2 + A*c^3)*d^3*m^5 + 14*(
3*B*b*c^2 + A*c^3)*d^3*m^4 + 59*(3*B*b*c^2 + A*c^3)*d^3*m^3 + 94*(3*B*b*c^2 + A*c^3)*d^3*m^2 + 48*(3*B*b*c^2 +
A*c^3)*d^3*m)*x^4 + 10*((B*b^2*c + A*b*c^2)*d^3*m^5 + 18*(B*b^2*c + A*b*c^2)*d^3*m^4 + 103*(B*b^2*c + A*b*c^2
)*d^3*m^3 + 198*(B*b^2*c + A*b*c^2)*d^3*m^2 + 112*(B*b^2*c + A*b*c^2)*d^3*m)*x^3 + 2*((B*b^3 + 3*A*b^2*c)*d^3*
m^5 + 22*(B*b^3 + 3*A*b^2*c)*d^3*m^4 + 167*(B*b^3 + 3*A*b^2*c)*d^3*m^3 + 482*(B*b^3 + 3*A*b^2*c)*d^3*m^2 + 336
*(B*b^3 + 3*A*b^2*c)*d^3*m)*x^2 + (A*b^3*d^3*m^5 + 26*A*b^3*d^3*m^4 + 251*A*b^3*d^3*m^3 + 1066*A*b^3*d^3*m^2 +
1680*A*b^3*d^3*m)*x)*e^5 + 6*(A*b^3*d^4*m^4 + 26*A*b^3*d^4*m^3 + 251*A*b^3*d^4*m^2 + 1066*A*b^3*d^4*m + 1680*
A*b^3*d^4 + 35*(B*c^3*d^4*m^4 + 6*B*c^3*d^4*m^3 + 11*B*c^3*d^4*m^2 + 6*B*c^3*d^4*m)*x^4 + 20*((3*B*b*c^2 + A*c
^3)*d^4*m^4 + 11*(3*B*b*c^2 + A*c^3)*d^4*m^3 + 26*(3*B*b*c^2 + A*c^3)*d^4*m^2 + 16*(3*B*b*c^2 + A*c^3)*d^4*m)*
x^3 + 30*((B*b^2*c + A*b*c^2)*d^4*m^4 + 16*(B*b^2*c + A*b*c^2)*d^4*m^3 + 71*(B*b^2*c + A*b*c^2)*d^4*m^2 + 56*(
B*b^2*c + A*b*c^2)*d^4*m)*x^2 + 4*((B*b^3 + 3*A*b^2*c)*d^4*m^4 + 21*(B*b^3 + 3*A*b^2*c)*d^4*m^3 + 146*(B*b^3 +
3*A*b^2*c)*d^4*m^2 + 336*(B*b^3 + 3*A*b^2*c)*d^4*m)*x)*e^4 - 24*((B*b^3 + 3*A*b^2*c)*d^5*m^3 + 21*(B*b^3 + 3*
A*b^2*c)*d^5*m^2 + 146*(B*b^3 + 3*A*b^2*c)*d^5*m + 336*(B*b^3 + 3*A*b^2*c)*d^5 + 35*(B*c^3*d^5*m^3 + 3*B*c^3*d
^5*m^2 + 2*B*c^3*d^5*m)*x^3 + 15*((3*B*b*c^2 + A*c^3)*d^5*m^3 + 9*(3*B*b*c^2 + A*c^3)*d^5*m^2 + 8*(3*B*b*c^2 +
A*c^3)*d^5*m)*x^2 + 15*((B*b^2*c + A*b*c^2)*d^5*m^3 + 15*(B*b^2*c + A*b*c^2)*d^5*m^2 + 56*(B*b^2*c + A*b*c^2)
*d^5*m)*x)*e^3 + 360*((B*b^2*c + A*b*c^2)*d^6*m^2 + 15*(B*b^2*c + A*b*c^2)*d^6*m + 56*(B*b^2*c + A*b*c^2)*d^6
+ 7*(B*c^3*d^6*m^2 + B*c^3*d^6*m)*x^2 + 2*((3*B*b*c^2 + A*c^3)*d^6*m^2 + 8*(3*B*b*c^2 + A*c^3)*d^6*m)*x)*e^2 -
720*(7*B*c^3*d^7*m*x + (3*B*b*c^2 + A*c^3)*d^7*m + 8*(3*B*b*c^2 + A*c^3)*d^7)*e)*(x*e + d)^m*e^(-8)/(m^8 + 36
*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 59148 vs. \(2 (473) = 946\).
time = 13.33, size = 59148, normalized size = 122.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**m*(c*x**2+b*x)**3,x)

[Out]

Piecewise((d**m*(A*b**3*x**4/4 + 3*A*b**2*c*x**5/5 + A*b*c**2*x**6/2 + A*c**3*x**7/7 + B*b**3*x**5/5 + B*b**2*
c*x**6/2 + 3*B*b*c**2*x**7/7 + B*c**3*x**8/8), Eq(e, 0)), (-3*A*b**3*d**3*e**4/(420*d**7*e**8 + 2940*d**6*e**9
*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**1
4*x**6 + 420*e**15*x**7) - 21*A*b**3*d**2*e**5*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14
700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 63*
A*b**3*d*e**6*x**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d*
*3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 105*A*b**3*e**7*x**3/(420*d**7*e*
*8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13
*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 12*A*b**2*c*d**4*e**3/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d
**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 42
0*e**15*x**7) - 84*A*b**2*c*d**3*e**4*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*
e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 252*A*b**2*c
*d**2*e**5*x**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*
e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 420*A*b**2*c*d*e**6*x**3/(420*d**7*e
**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**1
3*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 420*A*b**2*c*e**7*x**4/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820
*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 +
420*e**15*x**7) - 30*A*b*c**2*d**5*e**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*
e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 210*A*b*c**2
*d**4*e**3*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**
12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 630*A*b*c**2*d**3*e**4*x**2/(420*d**7*e
**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**1
3*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 1050*A*b*c**2*d**2*e**5*x**3/(420*d**7*e**8 + 2940*d**6*e**9*x
+ 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x
**6 + 420*e**15*x**7) - 1050*A*b*c**2*d*e**6*x**4/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 1
4700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 63
0*A*b*c**2*e**7*x**5/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*
d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 60*A*c**3*d**6*e/(420*d**7*e**8
+ 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x
**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 420*A*c**3*d**5*e**2*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d*
*5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420
*e**15*x**7) - 1260*A*c**3*d**4*e**3*x**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**
4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 2100*A*c**
3*d**3*e**4*x**3/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3
*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 2100*A*c**3*d**2*e**5*x**4/(420*d**
7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e
**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 1260*A*c**3*d*e**6*x**5/(420*d**7*e**8 + 2940*d**6*e**9*x +
8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**
6 + 420*e**15*x**7) - 420*A*c**3*e**7*x**6/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d*
*4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 4*B*b**3*
d**4*e**3/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*
x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 28*B*b**3*d**3*e**4*x/(420*d**7*e**8 + 294
0*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 +
2940*d*e**14*x**6 + 420*e**15*x**7) - 84*B*b**3*d**2*e**5*x**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e
**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**
15*x**7) - 140*B*b**3*d*e**6*x**3/(420*d**7*e**...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6663 vs. \(2 (495) = 990\).
time = 0.77, size = 6663, normalized size = 13.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*B*c^3*m^7*x^8*e^8 + (x*e + d)^m*B*c^3*d*m^7*x^7*e^7 + 3*(x*e + d)^m*B*b*c^2*m^7*x^7*e^8 + (x*e +
d)^m*A*c^3*m^7*x^7*e^8 + 28*(x*e + d)^m*B*c^3*m^6*x^8*e^8 + 3*(x*e + d)^m*B*b*c^2*d*m^7*x^6*e^7 + (x*e + d)^m*
A*c^3*d*m^7*x^6*e^7 + 21*(x*e + d)^m*B*c^3*d*m^6*x^7*e^7 - 7*(x*e + d)^m*B*c^3*d^2*m^6*x^6*e^6 + 3*(x*e + d)^m
*B*b^2*c*m^7*x^6*e^8 + 3*(x*e + d)^m*A*b*c^2*m^7*x^6*e^8 + 87*(x*e + d)^m*B*b*c^2*m^6*x^7*e^8 + 29*(x*e + d)^m
*A*c^3*m^6*x^7*e^8 + 322*(x*e + d)^m*B*c^3*m^5*x^8*e^8 + 3*(x*e + d)^m*B*b^2*c*d*m^7*x^5*e^7 + 3*(x*e + d)^m*A
*b*c^2*d*m^7*x^5*e^7 + 69*(x*e + d)^m*B*b*c^2*d*m^6*x^6*e^7 + 23*(x*e + d)^m*A*c^3*d*m^6*x^6*e^7 + 175*(x*e +
d)^m*B*c^3*d*m^5*x^7*e^7 - 18*(x*e + d)^m*B*b*c^2*d^2*m^6*x^5*e^6 - 6*(x*e + d)^m*A*c^3*d^2*m^6*x^5*e^6 - 105*
(x*e + d)^m*B*c^3*d^2*m^5*x^6*e^6 + 42*(x*e + d)^m*B*c^3*d^3*m^5*x^5*e^5 + (x*e + d)^m*B*b^3*m^7*x^5*e^8 + 3*(
x*e + d)^m*A*b^2*c*m^7*x^5*e^8 + 90*(x*e + d)^m*B*b^2*c*m^6*x^6*e^8 + 90*(x*e + d)^m*A*b*c^2*m^6*x^6*e^8 + 102
9*(x*e + d)^m*B*b*c^2*m^5*x^7*e^8 + 343*(x*e + d)^m*A*c^3*m^5*x^7*e^8 + 1960*(x*e + d)^m*B*c^3*m^4*x^8*e^8 + (
x*e + d)^m*B*b^3*d*m^7*x^4*e^7 + 3*(x*e + d)^m*A*b^2*c*d*m^7*x^4*e^7 + 75*(x*e + d)^m*B*b^2*c*d*m^6*x^5*e^7 +
75*(x*e + d)^m*A*b*c^2*d*m^6*x^5*e^7 + 615*(x*e + d)^m*B*b*c^2*d*m^5*x^6*e^7 + 205*(x*e + d)^m*A*c^3*d*m^5*x^6
*e^7 + 735*(x*e + d)^m*B*c^3*d*m^4*x^7*e^7 - 15*(x*e + d)^m*B*b^2*c*d^2*m^6*x^4*e^6 - 15*(x*e + d)^m*A*b*c^2*d
^2*m^6*x^4*e^6 - 324*(x*e + d)^m*B*b*c^2*d^2*m^5*x^5*e^6 - 108*(x*e + d)^m*A*c^3*d^2*m^5*x^5*e^6 - 595*(x*e +
d)^m*B*c^3*d^2*m^4*x^6*e^6 + 90*(x*e + d)^m*B*b*c^2*d^3*m^5*x^4*e^5 + 30*(x*e + d)^m*A*c^3*d^3*m^5*x^4*e^5 + 4
20*(x*e + d)^m*B*c^3*d^3*m^4*x^5*e^5 - 210*(x*e + d)^m*B*c^3*d^4*m^4*x^4*e^4 + (x*e + d)^m*A*b^3*m^7*x^4*e^8 +
31*(x*e + d)^m*B*b^3*m^6*x^5*e^8 + 93*(x*e + d)^m*A*b^2*c*m^6*x^5*e^8 + 1098*(x*e + d)^m*B*b^2*c*m^5*x^6*e^8
+ 1098*(x*e + d)^m*A*b*c^2*m^5*x^6*e^8 + 6405*(x*e + d)^m*B*b*c^2*m^4*x^7*e^8 + 2135*(x*e + d)^m*A*c^3*m^4*x^7
*e^8 + 6769*(x*e + d)^m*B*c^3*m^3*x^8*e^8 + (x*e + d)^m*A*b^3*d*m^7*x^3*e^7 + 27*(x*e + d)^m*B*b^3*d*m^6*x^4*e
^7 + 81*(x*e + d)^m*A*b^2*c*d*m^6*x^4*e^7 + 723*(x*e + d)^m*B*b^2*c*d*m^5*x^5*e^7 + 723*(x*e + d)^m*A*b*c^2*d*
m^5*x^5*e^7 + 2715*(x*e + d)^m*B*b*c^2*d*m^4*x^6*e^7 + 905*(x*e + d)^m*A*c^3*d*m^4*x^6*e^7 + 1624*(x*e + d)^m*
B*c^3*d*m^3*x^7*e^7 - 4*(x*e + d)^m*B*b^3*d^2*m^6*x^3*e^6 - 12*(x*e + d)^m*A*b^2*c*d^2*m^6*x^3*e^6 - 315*(x*e
+ d)^m*B*b^2*c*d^2*m^5*x^4*e^6 - 315*(x*e + d)^m*A*b*c^2*d^2*m^5*x^4*e^6 - 2070*(x*e + d)^m*B*b*c^2*d^2*m^4*x^
5*e^6 - 690*(x*e + d)^m*A*c^3*d^2*m^4*x^5*e^6 - 1575*(x*e + d)^m*B*c^3*d^2*m^3*x^6*e^6 + 60*(x*e + d)^m*B*b^2*
c*d^3*m^5*x^3*e^5 + 60*(x*e + d)^m*A*b*c^2*d^3*m^5*x^3*e^5 + 1260*(x*e + d)^m*B*b*c^2*d^3*m^4*x^4*e^5 + 420*(x
*e + d)^m*A*c^3*d^3*m^4*x^4*e^5 + 1470*(x*e + d)^m*B*c^3*d^3*m^3*x^5*e^5 - 360*(x*e + d)^m*B*b*c^2*d^4*m^4*x^3
*e^4 - 120*(x*e + d)^m*A*c^3*d^4*m^4*x^3*e^4 - 1260*(x*e + d)^m*B*c^3*d^4*m^3*x^4*e^4 + 840*(x*e + d)^m*B*c^3*
d^5*m^3*x^3*e^3 + 32*(x*e + d)^m*A*b^3*m^6*x^4*e^8 + 391*(x*e + d)^m*B*b^3*m^5*x^5*e^8 + 1173*(x*e + d)^m*A*b^
2*c*m^5*x^5*e^8 + 7020*(x*e + d)^m*B*b^2*c*m^4*x^6*e^8 + 7020*(x*e + d)^m*A*b*c^2*m^4*x^6*e^8 + 22512*(x*e + d
)^m*B*b*c^2*m^3*x^7*e^8 + 7504*(x*e + d)^m*A*c^3*m^3*x^7*e^8 + 13132*(x*e + d)^m*B*c^3*m^2*x^8*e^8 + 29*(x*e +
d)^m*A*b^3*d*m^6*x^3*e^7 + 283*(x*e + d)^m*B*b^3*d*m^5*x^4*e^7 + 849*(x*e + d)^m*A*b^2*c*d*m^5*x^4*e^7 + 3405
*(x*e + d)^m*B*b^2*c*d*m^4*x^5*e^7 + 3405*(x*e + d)^m*A*b*c^2*d*m^4*x^5*e^7 + 6222*(x*e + d)^m*B*b*c^2*d*m^3*x
^6*e^7 + 2074*(x*e + d)^m*A*c^3*d*m^3*x^6*e^7 + 1764*(x*e + d)^m*B*c^3*d*m^2*x^7*e^7 - 3*(x*e + d)^m*A*b^3*d^2
*m^6*x^2*e^6 - 96*(x*e + d)^m*B*b^3*d^2*m^5*x^3*e^6 - 288*(x*e + d)^m*A*b^2*c*d^2*m^5*x^3*e^6 - 2355*(x*e + d)
^m*B*b^2*c*d^2*m^4*x^4*e^6 - 2355*(x*e + d)^m*A*b*c^2*d^2*m^4*x^4*e^6 - 5940*(x*e + d)^m*B*b*c^2*d^2*m^3*x^5*e
^6 - 1980*(x*e + d)^m*A*c^3*d^2*m^3*x^5*e^6 - 1918*(x*e + d)^m*B*c^3*d^2*m^2*x^6*e^6 + 12*(x*e + d)^m*B*b^3*d^
3*m^5*x^2*e^5 + 36*(x*e + d)^m*A*b^2*c*d^3*m^5*x^2*e^5 + 1080*(x*e + d)^m*B*b^2*c*d^3*m^4*x^3*e^5 + 1080*(x*e
+ d)^m*A*b*c^2*d^3*m^4*x^3*e^5 + 5310*(x*e + d)^m*B*b*c^2*d^3*m^3*x^4*e^5 + 1770*(x*e + d)^m*A*c^3*d^3*m^3*x^4
*e^5 + 2100*(x*e + d)^m*B*c^3*d^3*m^2*x^5*e^5 - 180*(x*e + d)^m*B*b^2*c*d^4*m^4*x^2*e^4 - 180*(x*e + d)^m*A*b*
c^2*d^4*m^4*x^2*e^4 - 3960*(x*e + d)^m*B*b*c^2*d^4*m^3*x^3*e^4 - 1320*(x*e + d)^m*A*c^3*d^4*m^3*x^3*e^4 - 2310
*(x*e + d)^m*B*c^3*d^4*m^2*x^4*e^4 + 1080*(x*e + d)^m*B*b*c^2*d^5*m^3*x^2*e^3 + 360*(x*e + d)^m*A*c^3*d^5*m^3*
x^2*e^3 + 2520*(x*e + d)^m*B*c^3*d^5*m^2*x^3*e^3 - 2520*(x*e + d)^m*B*c^3*d^6*m^2*x^2*e^2 + 418*(x*e + d)^m*A*
b^3*m^5*x^4*e^8 + 2581*(x*e + d)^m*B*b^3*m^4*x^5*e^8 + 7743*(x*e + d)^m*A*b^2*c*m^4*x^5*e^8 + 25227*(x*e + d)^
m*B*b^2*c*m^3*x^6*e^8 + 25227*(x*e + d)^m*A*b*c^2*m^3*x^6*e^8 + 44268*(x*e + d)^m*B*b*c^2*m^2*x^7*e^8 + 14756*
(x*e + d)^m*A*c^3*m^2*x^7*e^8 + 13068*(x*e + d)^m*B*c^3*m*x^8*e^8 + 331*(x*e + d)^m*A*b^3*d*m^5*x^3*e^7 + 1449
*(x*e + d)^m*B*b^3*d*m^4*x^4*e^7 + 4347*(x*e + ...

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Mupad [B]
time = 2.88, size = 2500, normalized size = 5.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(A + B*x)*(d + e*x)^m,x)

[Out]

(B*c^3*x^8*(d + e*x)^m*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))/(109584*m
+ 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - ((d + e*x)^m*(5040*B*c^3*d
^8 - 5760*A*c^3*d^7*e + 10080*A*b^3*d^4*e^4 - 8064*B*b^3*d^5*e^3 + 20160*A*b*c^2*d^6*e^2 - 24192*A*b^2*c*d^5*e
^3 + 20160*B*b^2*c*d^6*e^2 + 6396*A*b^3*d^4*e^4*m - 3504*B*b^3*d^5*e^3*m + 1506*A*b^3*d^4*e^4*m^2 + 156*A*b^3*
d^4*e^4*m^3 + 6*A*b^3*d^4*e^4*m^4 - 504*B*b^3*d^5*e^3*m^2 - 24*B*b^3*d^5*e^3*m^3 - 17280*B*b*c^2*d^7*e - 720*A
*c^3*d^7*e*m + 360*A*b*c^2*d^6*e^2*m^2 - 1512*A*b^2*c*d^5*e^3*m^2 - 72*A*b^2*c*d^5*e^3*m^3 + 360*B*b^2*c*d^6*e
^2*m^2 - 2160*B*b*c^2*d^7*e*m + 5400*A*b*c^2*d^6*e^2*m - 10512*A*b^2*c*d^5*e^3*m + 5400*B*b^2*c*d^6*e^2*m))/(e
^8*(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (x^5*(d + e*
x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)*(336*B*b^3*e^3 + 1008*A*b^2*c*e^3 + 146*B*b^3*e^3*m + 42*B*c^3*d^3*m
+ 21*B*b^3*e^3*m^2 + B*b^3*e^3*m^3 + 63*A*b^2*c*e^3*m^2 + 3*A*b^2*c*e^3*m^3 - 6*A*c^3*d^2*e*m^2 + 438*A*b^2*c*
e^3*m - 48*A*c^3*d^2*e*m + 168*A*b*c^2*d*e^2*m - 144*B*b*c^2*d^2*e*m + 168*B*b^2*c*d*e^2*m + 45*A*b*c^2*d*e^2*
m^2 + 3*A*b*c^2*d*e^2*m^3 - 18*B*b*c^2*d^2*e*m^2 + 45*B*b^2*c*d*e^2*m^2 + 3*B*b^2*c*d*e^2*m^3))/(e^3*(109584*m
+ 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (x^4*(d + e*x)^m*(11*m +
6*m^2 + m^3 + 6)*(1680*A*b^3*e^4 + 1066*A*b^3*e^4*m - 210*B*c^3*d^4*m + 251*A*b^3*e^4*m^2 + 26*A*b^3*e^4*m^3
+ A*b^3*e^4*m^4 + 30*A*c^3*d^3*e*m^2 + 146*B*b^3*d*e^3*m^2 + 21*B*b^3*d*e^3*m^3 + B*b^3*d*e^3*m^4 + 240*A*c^3*
d^3*e*m + 336*B*b^3*d*e^3*m - 225*A*b*c^2*d^2*e^2*m^2 - 15*A*b*c^2*d^2*e^2*m^3 - 225*B*b^2*c*d^2*e^2*m^2 - 15*
B*b^2*c*d^2*e^2*m^3 + 1008*A*b^2*c*d*e^3*m + 720*B*b*c^2*d^3*e*m - 840*A*b*c^2*d^2*e^2*m + 438*A*b^2*c*d*e^3*m
^2 + 63*A*b^2*c*d*e^3*m^3 + 3*A*b^2*c*d*e^3*m^4 - 840*B*b^2*c*d^2*e^2*m + 90*B*b*c^2*d^3*e*m^2))/(e^4*(109584*
m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (6*d^3*m*x*(d + e*x)^m*
(1680*A*b^3*e^4 + 840*B*c^3*d^4 - 960*A*c^3*d^3*e - 1344*B*b^3*d*e^3 + 1066*A*b^3*e^4*m + 251*A*b^3*e^4*m^2 +
26*A*b^3*e^4*m^3 + A*b^3*e^4*m^4 + 3360*A*b*c^2*d^2*e^2 + 3360*B*b^2*c*d^2*e^2 - 84*B*b^3*d*e^3*m^2 - 4*B*b^3*
d*e^3*m^3 - 4032*A*b^2*c*d*e^3 - 2880*B*b*c^2*d^3*e - 120*A*c^3*d^3*e*m - 584*B*b^3*d*e^3*m + 60*A*b*c^2*d^2*e
^2*m^2 + 60*B*b^2*c*d^2*e^2*m^2 - 1752*A*b^2*c*d*e^3*m - 360*B*b*c^2*d^3*e*m + 900*A*b*c^2*d^2*e^2*m - 252*A*b
^2*c*d*e^3*m^2 - 12*A*b^2*c*d*e^3*m^3 + 900*B*b^2*c*d^2*e^2*m))/(e^7*(109584*m + 118124*m^2 + 67284*m^3 + 2244
9*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (c^2*x^7*(d + e*x)^m*(8*A*c*e + 24*B*b*e + A*c*e*m + 3*B
*b*e*m + B*c*d*m)*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(e*(109584*m + 118124*m^2 + 67
284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (c*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85*m
^3 + 15*m^4 + m^5 + 120)*(168*B*b^2*e^2 + 168*A*b*c*e^2 + 45*B*b^2*e^2*m - 7*B*c^2*d^2*m + 3*B*b^2*e^2*m^2 + 4
5*A*b*c*e^2*m + 8*A*c^2*d*e*m + 3*A*b*c*e^2*m^2 + A*c^2*d*e*m^2 + 24*B*b*c*d*e*m + 3*B*b*c*d*e*m^2))/(e^2*(109
584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (d*m*x^3*(d + e*x)^
m*(3*m + m^2 + 2)*(1680*A*b^3*e^4 + 840*B*c^3*d^4 - 960*A*c^3*d^3*e - 1344*B*b^3*d*e^3 + 1066*A*b^3*e^4*m + 25
1*A*b^3*e^4*m^2 + 26*A*b^3*e^4*m^3 + A*b^3*e^4*m^4 + 3360*A*b*c^2*d^2*e^2 + 3360*B*b^2*c*d^2*e^2 - 84*B*b^3*d*
e^3*m^2 - 4*B*b^3*d*e^3*m^3 - 4032*A*b^2*c*d*e^3 - 2880*B*b*c^2*d^3*e - 120*A*c^3*d^3*e*m - 584*B*b^3*d*e^3*m
+ 60*A*b*c^2*d^2*e^2*m^2 + 60*B*b^2*c*d^2*e^2*m^2 - 1752*A*b^2*c*d*e^3*m - 360*B*b*c^2*d^3*e*m + 900*A*b*c^2*d
^2*e^2*m - 252*A*b^2*c*d*e^3*m^2 - 12*A*b^2*c*d*e^3*m^3 + 900*B*b^2*c*d^2*e^2*m))/(e^5*(109584*m + 118124*m^2
+ 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) - (3*d^2*m*x^2*(m + 1)*(d + e*x)^m*(1680
*A*b^3*e^4 + 840*B*c^3*d^4 - 960*A*c^3*d^3*e - 1344*B*b^3*d*e^3 + 1066*A*b^3*e^4*m + 251*A*b^3*e^4*m^2 + 26*A*
b^3*e^4*m^3 + A*b^3*e^4*m^4 + 3360*A*b*c^2*d^2*e^2 + 3360*B*b^2*c*d^2*e^2 - 84*B*b^3*d*e^3*m^2 - 4*B*b^3*d*e^3
*m^3 - 4032*A*b^2*c*d*e^3 - 2880*B*b*c^2*d^3*e - 120*A*c^3*d^3*e*m - 584*B*b^3*d*e^3*m + 60*A*b*c^2*d^2*e^2*m^
2 + 60*B*b^2*c*d^2*e^2*m^2 - 1752*A*b^2*c*d*e^3*m - 360*B*b*c^2*d^3*e*m + 900*A*b*c^2*d^2*e^2*m - 252*A*b^2*c*
d*e^3*m^2 - 12*A*b^2*c*d*e^3*m^3 + 900*B*b^2*c*d^2*e^2*m))/(e^6*(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4
+ 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320))

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