\(\int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx\) [1722]

Optimal result

Integrand size = 26, antiderivative size = 111 \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {2}{3} (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {49 (1+2 x)^{-m} (2+3 x)^{1+m}}{m}+\frac {2^{-1-m} \left (441-86 m+2 m^2\right ) (1+2 x)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (1+2 x))}{3 (1-m) m} \] Output:

2/3*(1+2*x)^(1-m)*(2+3*x)^(1+m)-49*(2+3*x)^(1+m)/m/((1+2*x)^m)+1/3*2^(-1-m 
)*(2*m^2-86*m+441)*(1+2*x)^(1-m)*hypergeom([-m, 1-m],[2-m],-3-6*x)/(1-m)/m
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {1}{6} (1+2 x)^{-m} \left (\frac {14 (-21+m) (2+3 x)^{1+m}}{m}+2 (2+3 x)^{1+m} (-5+4 x)-\frac {2^{-m} \left (441-86 m+2 m^2\right ) (1+2 x) \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3-6 x)}{(-1+m) m}\right ) \] Input:

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

Output:

((14*(-21 + m)*(2 + 3*x)^(1 + m))/m + 2*(2 + 3*x)^(1 + m)*(-5 + 4*x) - ((4 
41 - 86*m + 2*m^2)*(1 + 2*x)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x] 
)/(2^m*(-1 + m)*m))/(6*(1 + 2*x)^m)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {101, 27, 88, 79}

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.

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

\(\Big \downarrow \) 101

\(\displaystyle \frac {1}{12} \int 4 (2 x+1)^{-m-1} (3 x+2)^m (-5 m-2 (65-2 m) x+82)dx-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int (2 x+1)^{-m-1} (3 x+2)^m (-5 m-2 (65-2 m) x+82)dx-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {1}{3} \left (\frac {\left (2 m^2-86 m+441\right ) \int (2 x+1)^{-m} (3 x+2)^mdx}{m}-\frac {7 (21-m) (2 x+1)^{-m} (3 x+2)^{m+1}}{m}\right )-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {1}{3} \left (\frac {2^{-m-1} \left (2 m^2-86 m+441\right ) (2 x+1)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (2 x+1))}{(1-m) m}-\frac {7 (21-m) (2 x+1)^{-m} (3 x+2)^{m+1}}{m}\right )-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\)

Input:

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

Output:

-1/3*((5 - 4*x)*(2 + 3*x)^(1 + m))/(1 + 2*x)^m + ((-7*(21 - m)*(2 + 3*x)^( 
1 + m))/(m*(1 + 2*x)^m) + (2^(-1 - m)*(441 - 86*m + 2*m^2)*(1 + 2*x)^(1 - 
m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/((1 - m)*m))/3
 

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 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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((16*x^2 - 40*x + 25)*(3*x + 2)^m*(2*x + 1)^(-m - 1), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=25 \left (\int \frac {\left (3 x +2\right )^{m}}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )+16 \left (\int \frac {\left (3 x +2\right )^{m} x^{2}}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )-40 \left (\int \frac {\left (3 x +2\right )^{m} x}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right ) \] Input:

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

Output:

25*int((3*x + 2)**m/(2*(2*x + 1)**m*x + (2*x + 1)**m),x) + 16*int(((3*x + 
2)**m*x**2)/(2*(2*x + 1)**m*x + (2*x + 1)**m),x) - 40*int(((3*x + 2)**m*x) 
/(2*(2*x + 1)**m*x + (2*x + 1)**m),x)