3.31.96 \(\int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx\) [3096]

3.31.96.1 Optimal result

Integrand size = 26, antiderivative size = 333 \[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=-\frac {49 (15-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{9 (3+m)}+\frac {14}{9} (15-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}-\frac {2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac {196 (42-m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m)}+\frac {14 (15-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{9 (2+m) (3+m)}-\frac {28 (42-m) (29+4 m) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{3 (1+m) (2+m)}-\frac {14 (15-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{3 (3+m) \left (2+3 m+m^2\right )}+\frac {2^{3-m} (42-m) (1+2 x)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (1+2 x))}{3 m} \]

output

-49/9*(15-2*m)*(27+2*m)*(1+2*x)^(-3-m)*(2+3*x)^(1+m)/(3+m)+14/9*(15-2*m)*( 
5-4*x)*(1+2*x)^(-3-m)*(2+3*x)^(1+m)-2/3*(5-4*x)^3*(1+2*x)^(-3-m)*(2+3*x)^( 
1+m)+196/3*(42-m)*(1+2*x)^(-2-m)*(2+3*x)^(1+m)/(2+m)+14/9*(15-2*m)*(2*m^2+ 
52*m+579)*(1+2*x)^(-2-m)*(2+3*x)^(1+m)/(m^2+5*m+6)-28/3*(42-m)*(29+4*m)*(1 
+2*x)^(-1-m)*(2+3*x)^(1+m)/(m^2+3*m+2)-14/3*(15-2*m)*(2*m^2+52*m+579)*(1+2 
*x)^(-1-m)*(2+3*x)^(1+m)/(m^3+6*m^2+11*m+6)+1/3*2^(3-m)*(42-m)*hypergeom([ 
-m, -m],[1-m],-3-6*x)/m/((1+2*x)^m)
 

3.31.96.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48 \[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=\frac {2^{-m} (1+2 x)^{-3-m} \left (2^m (2+3 x)^{1+m} \left (32 m^3 (1+2 x)^2 (2+3 x)+18 \left (-28775-128102 x-150644 x^2+128 x^3\right )+m^2 \left (-7801+6416 x-41248 x^2+2304 x^3\right )+m \left (33867-61044 x-514752 x^2+4224 x^3\right )\right )-8 \left (-252-204 m-37 m^2+m^3\right ) (1+2 x)^2 \operatorname {Hypergeometric2F1}(-1-m,-1-m,-m,-3-6 x)\right )}{9 (1+m) (2+m) (3+m)} \]

input

Integrate[(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^m,x]
 

output

((1 + 2*x)^(-3 - m)*(2^m*(2 + 3*x)^(1 + m)*(32*m^3*(1 + 2*x)^2*(2 + 3*x) + 
 18*(-28775 - 128102*x - 150644*x^2 + 128*x^3) + m^2*(-7801 + 6416*x - 412 
48*x^2 + 2304*x^3) + m*(33867 - 61044*x - 514752*x^2 + 4224*x^3)) - 8*(-25 
2 - 204*m - 37*m^2 + m^3)*(1 + 2*x)^2*Hypergeometric2F1[-1 - m, -1 - m, -m 
, -3 - 6*x]))/(9*2^m*(1 + m)*(2 + m)*(3 + m))
 

3.31.96.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {111, 27, 177, 100, 27, 88, 79, 101, 27, 88, 55, 48}

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.

\(\displaystyle \int (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^m \, dx\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{6} \int -2 (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^m (10 m+8 (42-m) x+63)dx-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^m (10 m+8 (42-m) x+63)dx-\frac {2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}\)

\(\Big \downarrow \) 177

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^mdx-4 (42-m) \int (5-4 x)^2 (2 x+1)^{-m-3} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^mdx-4 (42-m) \left (\frac {\int -4 (2 x+1)^{-m-2} (3 x+2)^m (3 (8 m+65)-8 (m+2) x)dx}{4 (m+2)}-\frac {49 (2 x+1)^{-m-2} (3 x+2)^{m+1}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^mdx-4 (42-m) \left (-\frac {\int (2 x+1)^{-m-2} (3 x+2)^m (3 (8 m+65)-8 (m+2) x)dx}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^mdx-4 (42-m) \left (-\frac {-4 (m+2) \int (2 x+1)^{-m-1} (3 x+2)^mdx-\frac {7 (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^mdx-4 (42-m) \left (-\frac {\frac {2^{1-m} (m+2) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}-\frac {7 (4 m+29) (2 x+1)^{-m-1} (3 x+2)^{m+1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \left (\frac {2}{3} (5-4 x) (2 x+1)^{-m-3} (3 x+2)^{m+1}-\frac {1}{6} \int -2 (2 x+1)^{-m-4} (3 x+2)^m (10 m-8 (m+2) x+181)dx\right )-4 (42-m) \left (-\frac {\frac {2^{1-m} (m+2) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}-\frac {7 (4 m+29) (2 x+1)^{-m-1} (3 x+2)^{m+1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \left (\frac {1}{3} \int (2 x+1)^{-m-4} (3 x+2)^m (10 m-8 (m+2) x+181)dx+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )-4 (42-m) \left (-\frac {\frac {2^{1-m} (m+2) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}-\frac {7 (4 m+29) (2 x+1)^{-m-1} (3 x+2)^{m+1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \int (2 x+1)^{-m-3} (3 x+2)^mdx}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )-4 (42-m) \left (-\frac {\frac {2^{1-m} (m+2) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}-\frac {7 (4 m+29) (2 x+1)^{-m-1} (3 x+2)^{m+1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \left (-\frac {3 \int (2 x+1)^{-m-2} (3 x+2)^mdx}{m+2}-\frac {(3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )-4 (42-m) \left (-\frac {\frac {2^{1-m} (m+2) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}-\frac {7 (4 m+29) (2 x+1)^{-m-1} (3 x+2)^{m+1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {1}{3} \left (7 (15-2 m) \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \left (\frac {3 (2 x+1)^{-m-1} (3 x+2)^{m+1}}{(m+1) (m+2)}-\frac {(2 x+1)^{-m-2} (3 x+2)^{m+1}}{m+2}\right )}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )-4 (42-m) \left (-\frac {\frac {2^{1-m} (m+2) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}-\frac {7 (4 m+29) (2 x+1)^{-m-1} (3 x+2)^{m+1}}{m+1}}{m+2}-\frac {49 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )\right )-\frac {2}{3} (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^{m+1}\)

input

Int[(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^m,x]
 

output

(-2*(5 - 4*x)^3*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + (7*(15 - 2*m)*(( 
2*(5 - 4*x)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + ((-7*(27 + 2*m)*(1 + 
 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3 + m) - (2*(579 + 52*m + 2*m^2)*(-(((1 
 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(2 + m)) + (3*(1 + 2*x)^(-1 - m)*(2 + 
3*x)^(1 + m))/((1 + m)*(2 + m))))/(3 + m))/3) - 4*(42 - m)*((-49*(1 + 2*x) 
^(-2 - m)*(2 + 3*x)^(1 + m))/(2 + m) - ((-7*(29 + 4*m)*(1 + 2*x)^(-1 - m)* 
(2 + 3*x)^(1 + m))/(1 + m) + (2^(1 - m)*(2 + m)*Hypergeometric2F1[-m, -m, 
1 - m, -3*(1 + 2*x)])/(m*(1 + 2*x)^m))/(2 + m)))/3
 

3.31.96.3.1 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 48

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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

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] - Simp[d*(S 
implify[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] && 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] ||  !SumSimp 
lerQ[n, 1])
 

rule 79

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

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] && SumSimpl 
erQ[p, 1]
 

rule 100

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[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 111

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

rule 177

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b   Int[(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b   Int[(a + b*x)^m*(c + d*x)^ 
n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su 
mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))
 

3.31.96.4 Maple [F]

\[\int \left (5-4 x \right )^{4} \left (1+2 x \right )^{-4-m} \left (2+3 x \right )^{m}d x\]

input

int((5-4*x)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x)
 

output

int((5-4*x)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x)
 

3.31.96.5 Fricas [F]

\[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 4} {\left (4 \, x - 5\right )}^{4} \,d x } \]

input

integrate((5-4*x)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x, algorithm="fricas")
 

output

integral((256*x^4 - 1280*x^3 + 2400*x^2 - 2000*x + 625)*(3*x + 2)^m*(2*x + 
 1)^(-m - 4), x)
 

3.31.96.6 Sympy [F]

\[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=\int \left (2 x + 1\right )^{- m - 4} \left (3 x + 2\right )^{m} \left (4 x - 5\right )^{4}\, dx \]

input

integrate((5-4*x)**4*(1+2*x)**(-4-m)*(2+3*x)**m,x)
 

output

Integral((2*x + 1)**(-m - 4)*(3*x + 2)**m*(4*x - 5)**4, x)
 

3.31.96.7 Maxima [F]

\[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 4} {\left (4 \, x - 5\right )}^{4} \,d x } \]

input

integrate((5-4*x)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x, algorithm="maxima")
 

output

integrate((3*x + 2)^m*(2*x + 1)^(-m - 4)*(4*x - 5)^4, x)
 

3.31.96.8 Giac [F]

\[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 4} {\left (4 \, x - 5\right )}^{4} \,d x } \]

input

integrate((5-4*x)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x, algorithm="giac")
 

output

integrate((3*x + 2)^m*(2*x + 1)^(-m - 4)*(4*x - 5)^4, x)
 

3.31.96.9 Mupad [F(-1)]

Timed out. \[ \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx=\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^4}{{\left (2\,x+1\right )}^{m+4}} \,d x \]

input

int(((3*x + 2)^m*(4*x - 5)^4)/(2*x + 1)^(m + 4),x)
 

output

int(((3*x + 2)^m*(4*x - 5)^4)/(2*x + 1)^(m + 4), x)