\(\int \frac {(2+3 x)^m (3+5 x)^3}{1-2 x} \, dx\) [1678]

Optimal result

Integrand size = 22, antiderivative size = 90 \[ \int \frac {(2+3 x)^m (3+5 x)^3}{1-2 x} \, dx=-\frac {5135 (2+3 x)^{1+m}}{216 (1+m)}-\frac {725 (2+3 x)^{2+m}}{108 (2+m)}-\frac {125 (2+3 x)^{3+m}}{54 (3+m)}+\frac {1331 (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )}{56 (1+m)} \] Output:

-5135*(2+3*x)^(1+m)/(216+216*m)-725*(2+3*x)^(2+m)/(216+108*m)-125*(2+3*x)^ 
(3+m)/(162+54*m)+1331*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],4/7+6/7*x)/(5 
6+56*m)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^m (3+5 x)^3}{1-2 x} \, dx=\frac {(2+3 x)^{1+m} \left (-\frac {35945}{1+m}-\frac {10150 (2+3 x)}{2+m}-\frac {3500 (2+3 x)^2}{3+m}+\frac {35937 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )}{1+m}\right )}{1512} \] Input:

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

Output:

((2 + 3*x)^(1 + m)*(-35945/(1 + m) - (10150*(2 + 3*x))/(2 + m) - (3500*(2 
+ 3*x)^2)/(3 + m) + (35937*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x) 
)/7])/(1 + m)))/1512
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 2009}

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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {1331 (3 x+2)^m}{8 (1-2 x)}-\frac {5135}{72} (3 x+2)^m-\frac {725}{36} (3 x+2)^{m+1}-\frac {125}{18} (3 x+2)^{m+2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1331 (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2}{7} (3 x+2)\right )}{56 (m+1)}-\frac {5135 (3 x+2)^{m+1}}{216 (m+1)}-\frac {725 (3 x+2)^{m+2}}{108 (m+2)}-\frac {125 (3 x+2)^{m+3}}{54 (m+3)}\)

Input:

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

Output:

(-5135*(2 + 3*x)^(1 + m))/(216*(1 + m)) - (725*(2 + 3*x)^(2 + m))/(108*(2 
+ m)) - (125*(2 + 3*x)^(3 + m))/(54*(3 + m)) + (1331*(2 + 3*x)^(1 + m)*Hyp 
ergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(56*(1 + m))
 

Defintions of rubi rules used

rule 99

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(125*x^3 + 225*x^2 + 135*x + 27)*(3*x + 2)^m/(2*x - 1), x)
 

Sympy [F]

\[ \int \frac {(2+3 x)^m (3+5 x)^3}{1-2 x} \, dx=- \int \frac {27 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac {135 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac {225 x^{2} \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac {125 x^{3} \left (3 x + 2\right )^{m}}{2 x - 1}\, dx \] Input:

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

Output:

-Integral(27*(3*x + 2)**m/(2*x - 1), x) - Integral(135*x*(3*x + 2)**m/(2*x 
 - 1), x) - Integral(225*x**2*(3*x + 2)**m/(2*x - 1), x) - Integral(125*x* 
*3*(3*x + 2)**m/(2*x - 1), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(2+3 x)^m (3+5 x)^3}{1-2 x} \, dx=\frac {-13500 \left (3 x +2\right )^{m} m^{3} x^{3}-40050 \left (3 x +2\right )^{m} m^{3} x^{2}-50805 \left (3 x +2\right )^{m} m^{3} x +3888 \left (3 x +2\right )^{m} m^{3}-40500 \left (3 x +2\right )^{m} m^{2} x^{3}-133200 \left (3 x +2\right )^{m} m^{2} x^{2}-200625 \left (3 x +2\right )^{m} m^{2} x +57198 \left (3 x +2\right )^{m} m^{2}-27000 \left (3 x +2\right )^{m} m \,x^{3}-93150 \left (3 x +2\right )^{m} m \,x^{2}-180630 \left (3 x +2\right )^{m} m x +176518 \left (3 x +2\right )^{m} m +143748 \left (3 x +2\right )^{m}-251559 \left (\int \frac {\left (3 x +2\right )^{m} x}{6 x^{2}+x -2}d x \right ) m^{4}-1509354 \left (\int \frac {\left (3 x +2\right )^{m} x}{6 x^{2}+x -2}d x \right ) m^{3}-2767149 \left (\int \frac {\left (3 x +2\right )^{m} x}{6 x^{2}+x -2}d x \right ) m^{2}-1509354 \left (\int \frac {\left (3 x +2\right )^{m} x}{6 x^{2}+x -2}d x \right ) m}{216 m \left (m^{3}+6 m^{2}+11 m +6\right )} \] Input:

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

Output:

( - 13500*(3*x + 2)**m*m**3*x**3 - 40050*(3*x + 2)**m*m**3*x**2 - 50805*(3 
*x + 2)**m*m**3*x + 3888*(3*x + 2)**m*m**3 - 40500*(3*x + 2)**m*m**2*x**3 
- 133200*(3*x + 2)**m*m**2*x**2 - 200625*(3*x + 2)**m*m**2*x + 57198*(3*x 
+ 2)**m*m**2 - 27000*(3*x + 2)**m*m*x**3 - 93150*(3*x + 2)**m*m*x**2 - 180 
630*(3*x + 2)**m*m*x + 176518*(3*x + 2)**m*m + 143748*(3*x + 2)**m - 25155 
9*int(((3*x + 2)**m*x)/(6*x**2 + x - 2),x)*m**4 - 1509354*int(((3*x + 2)** 
m*x)/(6*x**2 + x - 2),x)*m**3 - 2767149*int(((3*x + 2)**m*x)/(6*x**2 + x - 
 2),x)*m**2 - 1509354*int(((3*x + 2)**m*x)/(6*x**2 + x - 2),x)*m)/(216*m*( 
m**3 + 6*m**2 + 11*m + 6))