\(\int (e x)^m (c+d x)^n (a+b x^2)^2 \, dx\) [279]

Optimal result

Integrand size = 22, antiderivative size = 378 \[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=-\frac {b c (2+m) \left (b c^2 \left (12+7 m+m^2\right )+2 a d^2 \left (20+m^2+9 n+n^2+m (9+2 n)\right )\right ) (e x)^{1+m} (c+d x)^{1+n}}{d^4 e (2+m+n) (3+m+n) (4+m+n) (5+m+n)}+\frac {b \left (b c^2 \left (12+7 m+m^2\right )+2 a d^2 \left (20+m^2+9 n+n^2+m (9+2 n)\right )\right ) (e x)^{2+m} (c+d x)^{1+n}}{d^3 e^2 (3+m+n) (4+m+n) (5+m+n)}-\frac {b^2 c (4+m) (e x)^{3+m} (c+d x)^{1+n}}{d^2 e^3 (4+m+n) (5+m+n)}+\frac {b^2 (e x)^{4+m} (c+d x)^{1+n}}{d e^4 (5+m+n)}+\frac {\left (\frac {a^2}{1+m}+\frac {b c^2 (2+m) \left (b c^2 \left (12+7 m+m^2\right )+2 a d^2 \left (20+m^2+9 n+n^2+m (9+2 n)\right )\right )}{d^4 (2+m+n) (3+m+n) (4+m+n) (5+m+n)}\right ) (e x)^{1+m} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{e} \] Output:

-b*c*(2+m)*(b*c^2*(m^2+7*m+12)+2*a*d^2*(20+m^2+9*n+n^2+m*(9+2*n)))*(e*x)^( 
1+m)*(d*x+c)^(1+n)/d^4/e/(2+m+n)/(3+m+n)/(4+m+n)/(5+m+n)+b*(b*c^2*(m^2+7*m 
+12)+2*a*d^2*(20+m^2+9*n+n^2+m*(9+2*n)))*(e*x)^(2+m)*(d*x+c)^(1+n)/d^3/e^2 
/(3+m+n)/(4+m+n)/(5+m+n)-b^2*c*(4+m)*(e*x)^(3+m)*(d*x+c)^(1+n)/d^2/e^3/(4+ 
m+n)/(5+m+n)+b^2*(e*x)^(4+m)*(d*x+c)^(1+n)/d/e^4/(5+m+n)+(a^2/(1+m)+b*c^2* 
(2+m)*(b*c^2*(m^2+7*m+12)+2*a*d^2*(20+m^2+9*n+n^2+m*(9+2*n)))/d^4/(2+m+n)/ 
(3+m+n)/(4+m+n)/(5+m+n))*(e*x)^(1+m)*(d*x+c)^n*hypergeom([-n, 1+m],[2+m],- 
d*x/c)/e/(((d*x+c)/c)^n)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.73 \[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {x (e x)^m (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (b^2 c^4 \operatorname {Hypergeometric2F1}\left (1+m,-4-n,2+m,-\frac {d x}{c}\right )-4 b^2 c^4 \operatorname {Hypergeometric2F1}\left (1+m,-3-n,2+m,-\frac {d x}{c}\right )+6 b^2 c^4 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,-\frac {d x}{c}\right )+2 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,-\frac {d x}{c}\right )-4 b^2 c^4 \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,-\frac {d x}{c}\right )-4 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,-\frac {d x}{c}\right )+b^2 c^4 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )+2 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )+a^2 d^4 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )\right )}{d^4 (1+m)} \] Input:

Integrate[(e*x)^m*(c + d*x)^n*(a + b*x^2)^2,x]
 

Output:

(x*(e*x)^m*(c + d*x)^n*(b^2*c^4*Hypergeometric2F1[1 + m, -4 - n, 2 + m, -( 
(d*x)/c)] - 4*b^2*c^4*Hypergeometric2F1[1 + m, -3 - n, 2 + m, -((d*x)/c)] 
+ 6*b^2*c^4*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((d*x)/c)] + 2*a*b*c^ 
2*d^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((d*x)/c)] - 4*b^2*c^4*Hype 
rgeometric2F1[1 + m, -1 - n, 2 + m, -((d*x)/c)] - 4*a*b*c^2*d^2*Hypergeome 
tric2F1[1 + m, -1 - n, 2 + m, -((d*x)/c)] + b^2*c^4*Hypergeometric2F1[1 + 
m, -n, 2 + m, -((d*x)/c)] + 2*a*b*c^2*d^2*Hypergeometric2F1[1 + m, -n, 2 + 
 m, -((d*x)/c)] + a^2*d^4*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)]) 
)/(d^4*(1 + m)*(1 + (d*x)/c)^n)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {521, 2125, 27, 521, 27, 90, 76, 74}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a+b x^2\right )^2 (e x)^m (c+d x)^n \, dx\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int (e x)^m (c+d x)^n \left (-b^2 c (m+4) x^3 e^4+2 a b d (m+n+5) x^2 e^4+a^2 d (m+n+5) e^4\right )dx}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int e^7 (e x)^m (c+d x)^n \left (a^2 (m+n+4) (m+n+5) d^2+b \left (b \left (m^2+7 m+12\right ) c^2+2 a d^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x^2\right )dx}{d e^3 (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \int (e x)^m (c+d x)^n \left (a^2 (m+n+4) (m+n+5) d^2+b \left (b \left (m^2+7 m+12\right ) c^2+2 a d^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x^2\right )dx}{d (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\frac {e^4 \left (\frac {\int e^2 (e x)^m (c+d x)^n \left (a^2 d^3 (m+n+3) (m+n+4) (m+n+5)-b c (m+2) \left (b \left (m^2+7 m+12\right ) c^2+2 a d^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x\right )dx}{d e^2 (m+n+3)}+\frac {b (e x)^{m+2} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e^2 (m+n+3)}\right )}{d (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \left (\frac {\int (e x)^m (c+d x)^n \left (a^2 d^3 (m+n+3) (m+n+4) (m+n+5)-b c (m+2) \left (b \left (m^2+7 m+12\right ) c^2+2 a d^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x\right )dx}{d (m+n+3)}+\frac {b (e x)^{m+2} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e^2 (m+n+3)}\right )}{d (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {\left (a^2 d^4 (m+n+3) (m+n+4) (m+n+5)+\frac {b c^2 (m+1) (m+2) \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{m+n+2}\right ) \int (e x)^m (c+d x)^ndx}{d}-\frac {b c (m+2) (e x)^{m+1} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e (m+n+2)}}{d (m+n+3)}+\frac {b (e x)^{m+2} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e^2 (m+n+3)}\right )}{d (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 76

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (a^2 d^4 (m+n+3) (m+n+4) (m+n+5)+\frac {b c^2 (m+1) (m+2) \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{m+n+2}\right ) \int (e x)^m \left (\frac {d x}{c}+1\right )^ndx}{d}-\frac {b c (m+2) (e x)^{m+1} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e (m+n+2)}}{d (m+n+3)}+\frac {b (e x)^{m+2} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e^2 (m+n+3)}\right )}{d (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (a^2 d^4 (m+n+3) (m+n+4) (m+n+5)+\frac {b c^2 (m+1) (m+2) \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{m+n+2}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d x}{c}\right )}{d e (m+1)}-\frac {b c (m+2) (e x)^{m+1} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e (m+n+2)}}{d (m+n+3)}+\frac {b (e x)^{m+2} (c+d x)^{n+1} \left (2 a d^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+b c^2 \left (m^2+7 m+12\right )\right )}{d e^2 (m+n+3)}\right )}{d (m+n+4)}-\frac {b^2 c e (m+4) (e x)^{m+3} (c+d x)^{n+1}}{d (m+n+4)}}{d e^4 (m+n+5)}+\frac {b^2 (e x)^{m+4} (c+d x)^{n+1}}{d e^4 (m+n+5)}\)

Input:

Int[(e*x)^m*(c + d*x)^n*(a + b*x^2)^2,x]
 

Output:

(b^2*(e*x)^(4 + m)*(c + d*x)^(1 + n))/(d*e^4*(5 + m + n)) + (-((b^2*c*e*(4 
 + m)*(e*x)^(3 + m)*(c + d*x)^(1 + n))/(d*(4 + m + n))) + (e^4*((b*(b*c^2* 
(12 + 7*m + m^2) + 2*a*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)))*(e*x)^(2 
+ m)*(c + d*x)^(1 + n))/(d*e^2*(3 + m + n)) + (-((b*c*(2 + m)*(b*c^2*(12 + 
 7*m + m^2) + 2*a*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)))*(e*x)^(1 + m)* 
(c + d*x)^(1 + n))/(d*e*(2 + m + n))) + ((a^2*d^4*(3 + m + n)*(4 + m + n)* 
(5 + m + n) + (b*c^2*(1 + m)*(2 + m)*(b*c^2*(12 + 7*m + m^2) + 2*a*d^2*(20 
 + m^2 + 9*n + n^2 + m*(9 + 2*n))))/(2 + m + n))*(e*x)^(1 + m)*(c + d*x)^n 
*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(d*e*(1 + m)*(1 + (d*x)/ 
c)^n))/(d*(3 + m + n))))/(d*(4 + m + n)))/(d*e^4*(5 + m + n))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 90

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

rule 521

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> Simp[b^p*(e*x)^(m + 2*p)*((c + d*x)^(n + 1)/(d*e^(2*p)*(m + n 
 + 2*p + 1))), x] + Simp[1/(d*e^(2*p)*(m + n + 2*p + 1))   Int[(e*x)^m*(c + 
 d*x)^n*ExpandToSum[d*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x^2)^p - b^p*(e*x)^ 
(2*p)) - b^p*(e*c)*(m + 2*p)*(e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] &&  !IntegerQ[m] &&  !I 
ntegerQ[n]
 

rule 2125

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x 
)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( 
m + n + q + 1))   Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q 
+ 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) 
^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, 
 x] && PolyQ[Px, x]
 

Maple [F]

\[\int \left (e x \right )^{m} \left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{2}d x\]

Input:

int((e*x)^m*(d*x+c)^n*(b*x^2+a)^2,x)
 

Output:

int((e*x)^m*(d*x+c)^n*(b*x^2+a)^2,x)
 

Fricas [F]

\[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{n} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)^n*(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x + c)^n*(e*x)^m, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 17.61 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.34 \[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {a^{2} c^{n} e^{m} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 a b c^{n} e^{m} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} + \frac {b^{2} c^{n} e^{m} x^{m + 5} \Gamma \left (m + 5\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 5 \\ m + 6 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 6\right )} \] Input:

integrate((e*x)**m*(d*x+c)**n*(b*x**2+a)**2,x)
 

Output:

a**2*c**n*e**m*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*ex 
p_polar(I*pi)/c)/gamma(m + 2) + 2*a*b*c**n*e**m*x**(m + 3)*gamma(m + 3)*hy 
per((-n, m + 3), (m + 4,), d*x*exp_polar(I*pi)/c)/gamma(m + 4) + b**2*c**n 
*e**m*x**(m + 5)*gamma(m + 5)*hyper((-n, m + 5), (m + 6,), d*x*exp_polar(I 
*pi)/c)/gamma(m + 6)
 

Maxima [F]

\[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{n} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)^n*(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^2*(d*x + c)^n*(e*x)^m, x)
 

Giac [F]

\[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{n} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)^n*(b*x^2+a)^2,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^2*(d*x + c)^n*(e*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\int {\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^n \,d x \] Input:

int((e*x)^m*(a + b*x^2)^2*(c + d*x)^n,x)
 

Output:

int((e*x)^m*(a + b*x^2)^2*(c + d*x)^n, x)
 

Reduce [F]

\[ \int (e x)^m (c+d x)^n \left (a+b x^2\right )^2 \, dx=\text {too large to display} \] Input:

int((e*x)^m*(d*x+c)^n*(b*x^2+a)^2,x)
 

Output:

(e**m*(x**m*(c + d*x)**n*a**2*c*d**4*m**4*n + 4*x**m*(c + d*x)**n*a**2*c*d 
**4*m**3*n**2 + 14*x**m*(c + d*x)**n*a**2*c*d**4*m**3*n + 6*x**m*(c + d*x) 
**n*a**2*c*d**4*m**2*n**3 + 42*x**m*(c + d*x)**n*a**2*c*d**4*m**2*n**2 + 7 
1*x**m*(c + d*x)**n*a**2*c*d**4*m**2*n + 4*x**m*(c + d*x)**n*a**2*c*d**4*m 
*n**4 + 42*x**m*(c + d*x)**n*a**2*c*d**4*m*n**3 + 142*x**m*(c + d*x)**n*a* 
*2*c*d**4*m*n**2 + 154*x**m*(c + d*x)**n*a**2*c*d**4*m*n + x**m*(c + d*x)* 
*n*a**2*c*d**4*n**5 + 14*x**m*(c + d*x)**n*a**2*c*d**4*n**4 + 71*x**m*(c + 
 d*x)**n*a**2*c*d**4*n**3 + 154*x**m*(c + d*x)**n*a**2*c*d**4*n**2 + 120*x 
**m*(c + d*x)**n*a**2*c*d**4*n + x**m*(c + d*x)**n*a**2*d**5*m**5*x + 5*x* 
*m*(c + d*x)**n*a**2*d**5*m**4*n*x + 14*x**m*(c + d*x)**n*a**2*d**5*m**4*x 
 + 10*x**m*(c + d*x)**n*a**2*d**5*m**3*n**2*x + 56*x**m*(c + d*x)**n*a**2* 
d**5*m**3*n*x + 71*x**m*(c + d*x)**n*a**2*d**5*m**3*x + 10*x**m*(c + d*x)* 
*n*a**2*d**5*m**2*n**3*x + 84*x**m*(c + d*x)**n*a**2*d**5*m**2*n**2*x + 21 
3*x**m*(c + d*x)**n*a**2*d**5*m**2*n*x + 154*x**m*(c + d*x)**n*a**2*d**5*m 
**2*x + 5*x**m*(c + d*x)**n*a**2*d**5*m*n**4*x + 56*x**m*(c + d*x)**n*a**2 
*d**5*m*n**3*x + 213*x**m*(c + d*x)**n*a**2*d**5*m*n**2*x + 308*x**m*(c + 
d*x)**n*a**2*d**5*m*n*x + 120*x**m*(c + d*x)**n*a**2*d**5*m*x + x**m*(c + 
d*x)**n*a**2*d**5*n**5*x + 14*x**m*(c + d*x)**n*a**2*d**5*n**4*x + 71*x**m 
*(c + d*x)**n*a**2*d**5*n**3*x + 154*x**m*(c + d*x)**n*a**2*d**5*n**2*x + 
120*x**m*(c + d*x)**n*a**2*d**5*n*x + 2*x**m*(c + d*x)**n*a*b*c**3*d**2...