\(\int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx\) [1721]

Optimal result

Integrand size = 26, antiderivative size = 145 \[ \int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {2}{27} (193-2 m) (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {8}{9} (1+2 x)^{2-m} (2+3 x)^{1+m}-\frac {343 (1+2 x)^{-m} (2+3 x)^{1+m}}{m}+\frac {2^{-1-m} \left (27783-8324 m+390 m^2-4 m^3\right ) (1+2 x)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (1+2 x))}{27 (1-m) m} \] Output:

2/27*(193-2*m)*(1+2*x)^(1-m)*(2+3*x)^(1+m)-8/9*(1+2*x)^(2-m)*(2+3*x)^(1+m) 
-343*(2+3*x)^(1+m)/m/((1+2*x)^m)+1/27*2^(-1-m)*(-4*m^3+390*m^2-8324*m+2778 
3)*(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 = 106, normalized size of antiderivative = 0.73 \[ \int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {2^{-1-m} (1+2 x)^{-m} \left (-\left ((-1+m) (4+6 x)^{1+m} \left (9261+m^2 (4+8 x)+m \left (-362-676 x+96 x^2\right )\right )\right )+\left (-27783+8324 m-390 m^2+4 m^3\right ) (1+2 x) \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3-6 x)\right )}{27 (-1+m) m} \] Input:

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

Output:

(2^(-1 - m)*(-((-1 + m)*(4 + 6*x)^(1 + m)*(9261 + m^2*(4 + 8*x) + m*(-362 
- 676*x + 96*x^2))) + (-27783 + 8324*m - 390*m^2 + 4*m^3)*(1 + 2*x)*Hyperg 
eometric2F1[1 - m, -m, 2 - m, -3 - 6*x]))/(27*(-1 + m)*m*(1 + 2*x)^m)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {111, 27, 163, 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.

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

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{18} \int 2 (5-4 x) (2 x+1)^{-m-1} (3 x+2)^m (-10 m-4 (109-2 m) x+223)dx-\frac {2}{9} (5-4 x)^2 (2 x+1)^{-m} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int (5-4 x) (2 x+1)^{-m-1} (3 x+2)^m (-10 m-4 (109-2 m) x+223)dx-\frac {2}{9} (5-4 x)^2 (2 x+1)^{-m} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 163

\(\displaystyle \frac {1}{9} \left (\frac {\left (-4 m^3+390 m^2-8324 m+27783\right ) \int (2 x+1)^{-m} (3 x+2)^mdx}{3 m}-\frac {(2 x+1)^{-m} (3 x+2)^{m+1} \left (4 m^2-4 (109-2 m) m x-512 m+9261\right )}{3 m}\right )-\frac {2}{9} (5-4 x)^2 (2 x+1)^{-m} (3 x+2)^{m+1}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {1}{9} \left (\frac {2^{-m-1} \left (-4 m^3+390 m^2-8324 m+27783\right ) (2 x+1)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (2 x+1))}{3 (1-m) m}-\frac {(2 x+1)^{-m} (3 x+2)^{m+1} \left (4 m^2-4 (109-2 m) m x-512 m+9261\right )}{3 m}\right )-\frac {2}{9} (5-4 x)^2 (2 x+1)^{-m} (3 x+2)^{m+1}\)

Input:

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

Output:

(-2*(5 - 4*x)^2*(2 + 3*x)^(1 + m))/(9*(1 + 2*x)^m) + (-1/3*((2 + 3*x)^(1 + 
 m)*(9261 - 512*m + 4*m^2 - 4*(109 - 2*m)*m*x))/(m*(1 + 2*x)^m) + (2^(-1 - 
 m)*(27783 - 8324*m + 390*m^2 - 4*m^3)*(1 + 2*x)^(1 - m)*Hypergeometric2F1 
[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(3*(1 - m)*m))/9
 

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 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 163

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(64*x^3 - 240*x^2 + 300*x - 125)*(3*x + 2)^m*(2*x + 1)^(-m - 1), 
 x)
 

Sympy [F]

\[ \int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx=- \int \left (- 125 \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m}\right )\, dx - \int 300 x \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m}\, dx - \int \left (- 240 x^{2} \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m}\right )\, dx - \int 64 x^{3} \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m}\, dx \] Input:

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

Output:

-Integral(-125*(2*x + 1)**(-m - 1)*(3*x + 2)**m, x) - Integral(300*x*(2*x 
+ 1)**(-m - 1)*(3*x + 2)**m, x) - Integral(-240*x**2*(2*x + 1)**(-m - 1)*( 
3*x + 2)**m, x) - Integral(64*x**3*(2*x + 1)**(-m - 1)*(3*x + 2)**m, x)
 

Maxima [F]

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

integrate((5-4*x)^3*(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)^3, x)
 

Giac [F]

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

integrate((5-4*x)^3*(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)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx=125 \left (\int \frac {\left (3 x +2\right )^{m}}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )-64 \left (\int \frac {\left (3 x +2\right )^{m} x^{3}}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )+240 \left (\int \frac {\left (3 x +2\right )^{m} x^{2}}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )-300 \left (\int \frac {\left (3 x +2\right )^{m} x}{2 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right ) \] Input:

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

Output:

125*int((3*x + 2)**m/(2*(2*x + 1)**m*x + (2*x + 1)**m),x) - 64*int(((3*x + 
 2)**m*x**3)/(2*(2*x + 1)**m*x + (2*x + 1)**m),x) + 240*int(((3*x + 2)**m* 
x**2)/(2*(2*x + 1)**m*x + (2*x + 1)**m),x) - 300*int(((3*x + 2)**m*x)/(2*( 
2*x + 1)**m*x + (2*x + 1)**m),x)