Integrand size = 20, antiderivative size = 54 \[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=-\frac {5 (2+3 x)^{1+m}}{6 (1+m)}+\frac {11 (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )}{14 (1+m)} \] Output:
-5*(2+3*x)^(1+m)/(6+6*m)+11*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],4/7+6/7 *x)/(14+14*m)
Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=\frac {(2+3 x)^{1+m} \left (-35+33 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )\right )}{42 (1+m)} \] Input:
Integrate[((2 + 3*x)^m*(3 + 5*x))/(1 - 2*x),x]
Output:
((2 + 3*x)^(1 + m)*(-35 + 33*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3* x))/7]))/(42*(1 + m))
Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {90, 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.
\(\displaystyle \int \frac {(5 x+3) (3 x+2)^m}{1-2 x} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {11}{2} \int \frac {(3 x+2)^m}{1-2 x}dx-\frac {5 (3 x+2)^{m+1}}{6 (m+1)}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {11 (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2}{7} (3 x+2)\right )}{14 (m+1)}-\frac {5 (3 x+2)^{m+1}}{6 (m+1)}\) |
Input:
Int[((2 + 3*x)^m*(3 + 5*x))/(1 - 2*x),x]
Output:
(-5*(2 + 3*x)^(1 + m))/(6*(1 + m)) + (11*(2 + 3*x)^(1 + m)*Hypergeometric2 F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(14*(1 + m))
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]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
\[\int \frac {\left (2+3 x \right )^{m} \left (3+5 x \right )}{1-2 x}d x\]
Input:
int((2+3*x)^m*(3+5*x)/(1-2*x),x)
Output:
int((2+3*x)^m*(3+5*x)/(1-2*x),x)
\[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m} {\left (5 \, x + 3\right )}}{2 \, x - 1} \,d x } \] Input:
integrate((2+3*x)^m*(3+5*x)/(1-2*x),x, algorithm="fricas")
Output:
integral(-(3*x + 2)^m*(5*x + 3)/(2*x - 1), x)
\[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=- \int \frac {3 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac {5 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx \] Input:
integrate((2+3*x)**m*(3+5*x)/(1-2*x),x)
Output:
-Integral(3*(3*x + 2)**m/(2*x - 1), x) - Integral(5*x*(3*x + 2)**m/(2*x - 1), x)
\[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m} {\left (5 \, x + 3\right )}}{2 \, x - 1} \,d x } \] Input:
integrate((2+3*x)^m*(3+5*x)/(1-2*x),x, algorithm="maxima")
Output:
-integrate((3*x + 2)^m*(5*x + 3)/(2*x - 1), x)
\[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m} {\left (5 \, x + 3\right )}}{2 \, x - 1} \,d x } \] Input:
integrate((2+3*x)^m*(3+5*x)/(1-2*x),x, algorithm="giac")
Output:
integrate(-(3*x + 2)^m*(5*x + 3)/(2*x - 1), x)
Timed out. \[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=\int -\frac {{\left (3\,x+2\right )}^m\,\left (5\,x+3\right )}{2\,x-1} \,d x \] Input:
int(-((3*x + 2)^m*(5*x + 3))/(2*x - 1),x)
Output:
int(-((3*x + 2)^m*(5*x + 3))/(2*x - 1), x)
\[ \int \frac {(2+3 x)^m (3+5 x)}{1-2 x} \, dx=\frac {-15 \left (3 x +2\right )^{m} m x +12 \left (3 x +2\right )^{m} m +22 \left (3 x +2\right )^{m}-231 \left (\int \frac {\left (3 x +2\right )^{m} x}{6 x^{2}+x -2}d x \right ) m^{2}-231 \left (\int \frac {\left (3 x +2\right )^{m} x}{6 x^{2}+x -2}d x \right ) m}{6 m \left (m +1\right )} \] Input:
int((2+3*x)^m*(3+5*x)/(1-2*x),x)
Output:
( - 15*(3*x + 2)**m*m*x + 12*(3*x + 2)**m*m + 22*(3*x + 2)**m - 231*int((( 3*x + 2)**m*x)/(6*x**2 + x - 2),x)*m**2 - 231*int(((3*x + 2)**m*x)/(6*x**2 + x - 2),x)*m)/(6*m*(m + 1))