\(\int \frac {(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx\) [241]

Optimal result

Integrand size = 20, antiderivative size = 65 \[ \int \frac {(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx=-\frac {2 (2+3 x)^{1+m}}{15 m (3+5 x)}-\frac {(2-33 m) (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(2,1+m,2+m,5 (2+3 x))}{5 m (1+m)} \] Output:

-2/15*(2+3*x)^(1+m)/m/(3+5*x)-1/5*(2-33*m)*(2+3*x)^(1+m)*hypergeom([2, 1+m 
],[2+m],10+15*x)/m/(1+m)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx=-\frac {(2+3 x)^{1+m} (11 (1+m)+(-2+33 m) (3+5 x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x)))}{5 (1+m) (3+5 x)} \] Input:

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

Output:

-1/5*((2 + 3*x)^(1 + m)*(11*(1 + m) + (-2 + 33*m)*(3 + 5*x)*Hypergeometric 
2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)]))/((1 + m)*(3 + 5*x))
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {87, 78}

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.

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

\(\Big \downarrow \) 87

\(\displaystyle -\frac {1}{5} (2-33 m) \int \frac {(3 x+2)^m}{5 x+3}dx-\frac {11 (3 x+2)^{m+1}}{5 (5 x+3)}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {(2-33 m) (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,5 (3 x+2))}{5 (m+1)}-\frac {11 (3 x+2)^{m+1}}{5 (5 x+3)}\)

Input:

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

Output:

(-11*(2 + 3*x)^(1 + m))/(5*(3 + 5*x)) + ((2 - 33*m)*(2 + 3*x)^(1 + m)*Hype 
rgeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(5*(1 + m))
 

Defintions of rubi rules used

rule 78

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

rule 87

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)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(3*x + 2)^m*(2*x - 1)/(25*x^2 + 30*x + 9), x)
 

Sympy [F]

\[ \int \frac {(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx=- \int \left (- \frac {\left (3 x + 2\right )^{m}}{25 x^{2} + 30 x + 9}\right )\, dx - \int \frac {2 x \left (3 x + 2\right )^{m}}{25 x^{2} + 30 x + 9}\, dx \] Input:

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

Output:

-Integral(-(3*x + 2)**m/(25*x**2 + 30*x + 9), x) - Integral(2*x*(3*x + 2)* 
*m/(25*x**2 + 30*x + 9), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx=\frac {-18 \left (3 x +2\right )^{m} m x +10 \left (3 x +2\right )^{m} m +20 \left (3 x +2\right )^{m} x +12 \left (3 x +2\right )^{m}-1485 \left (\int \frac {\left (3 x +2\right )^{m} x}{675 m \,x^{3}+1260 m \,x^{2}-750 x^{3}+783 m x -1400 x^{2}+162 m -870 x -180}d x \right ) m^{3} x -891 \left (\int \frac {\left (3 x +2\right )^{m} x}{675 m \,x^{3}+1260 m \,x^{2}-750 x^{3}+783 m x -1400 x^{2}+162 m -870 x -180}d x \right ) m^{3}+1740 \left (\int \frac {\left (3 x +2\right )^{m} x}{675 m \,x^{3}+1260 m \,x^{2}-750 x^{3}+783 m x -1400 x^{2}+162 m -870 x -180}d x \right ) m^{2} x +1044 \left (\int \frac {\left (3 x +2\right )^{m} x}{675 m \,x^{3}+1260 m \,x^{2}-750 x^{3}+783 m x -1400 x^{2}+162 m -870 x -180}d x \right ) m^{2}-100 \left (\int \frac {\left (3 x +2\right )^{m} x}{675 m \,x^{3}+1260 m \,x^{2}-750 x^{3}+783 m x -1400 x^{2}+162 m -870 x -180}d x \right ) m x -60 \left (\int \frac {\left (3 x +2\right )^{m} x}{675 m \,x^{3}+1260 m \,x^{2}-750 x^{3}+783 m x -1400 x^{2}+162 m -870 x -180}d x \right ) m}{5 m \left (45 m x +27 m -50 x -30\right )} \] Input:

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

Output:

( - 18*(3*x + 2)**m*m*x + 10*(3*x + 2)**m*m + 20*(3*x + 2)**m*x + 12*(3*x 
+ 2)**m - 1485*int(((3*x + 2)**m*x)/(675*m*x**3 + 1260*m*x**2 + 783*m*x + 
162*m - 750*x**3 - 1400*x**2 - 870*x - 180),x)*m**3*x - 891*int(((3*x + 2) 
**m*x)/(675*m*x**3 + 1260*m*x**2 + 783*m*x + 162*m - 750*x**3 - 1400*x**2 
- 870*x - 180),x)*m**3 + 1740*int(((3*x + 2)**m*x)/(675*m*x**3 + 1260*m*x* 
*2 + 783*m*x + 162*m - 750*x**3 - 1400*x**2 - 870*x - 180),x)*m**2*x + 104 
4*int(((3*x + 2)**m*x)/(675*m*x**3 + 1260*m*x**2 + 783*m*x + 162*m - 750*x 
**3 - 1400*x**2 - 870*x - 180),x)*m**2 - 100*int(((3*x + 2)**m*x)/(675*m*x 
**3 + 1260*m*x**2 + 783*m*x + 162*m - 750*x**3 - 1400*x**2 - 870*x - 180), 
x)*m*x - 60*int(((3*x + 2)**m*x)/(675*m*x**3 + 1260*m*x**2 + 783*m*x + 162 
*m - 750*x**3 - 1400*x**2 - 870*x - 180),x)*m)/(5*m*(45*m*x + 27*m - 50*x 
- 30))