Integrand size = 20, antiderivative size = 52 \[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=-\frac {2 (2+3 x)^{1+m}}{15 (1+m)}-\frac {11 (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x))}{5 (1+m)} \] Output:
-2*(2+3*x)^(1+m)/(15+15*m)-11*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],10+15 *x)/(5+5*m)
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=-\frac {(2+3 x)^{1+m} (2+33 \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x)))}{15 (1+m)} \] Input:
Integrate[((1 - 2*x)*(2 + 3*x)^m)/(3 + 5*x),x]
Output:
-1/15*((2 + 3*x)^(1 + m)*(2 + 33*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)]))/(1 + m)
Time = 0.16 (sec) , antiderivative size = 52, 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.
\(\displaystyle \int \frac {(1-2 x) (3 x+2)^m}{5 x+3} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {11}{5} \int \frac {(3 x+2)^m}{5 x+3}dx-\frac {2 (3 x+2)^{m+1}}{15 (m+1)}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle -\frac {11 (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,5 (3 x+2))}{5 (m+1)}-\frac {2 (3 x+2)^{m+1}}{15 (m+1)}\) |
Input:
Int[((1 - 2*x)*(2 + 3*x)^m)/(3 + 5*x),x]
Output:
(-2*(2 + 3*x)^(1 + m))/(15*(1 + m)) - (11*(2 + 3*x)^(1 + m)*Hypergeometric 2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(5*(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]
\[\int \frac {\left (1-2 x \right ) \left (2+3 x \right )^{m}}{3+5 x}d x\]
Input:
int((1-2*x)*(2+3*x)^m/(3+5*x),x)
Output:
int((1-2*x)*(2+3*x)^m/(3+5*x),x)
\[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m} {\left (2 \, x - 1\right )}}{5 \, x + 3} \,d x } \] Input:
integrate((1-2*x)*(2+3*x)^m/(3+5*x),x, algorithm="fricas")
Output:
integral(-(3*x + 2)^m*(2*x - 1)/(5*x + 3), x)
\[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=- \int \left (- \frac {\left (3 x + 2\right )^{m}}{5 x + 3}\right )\, dx - \int \frac {2 x \left (3 x + 2\right )^{m}}{5 x + 3}\, dx \] Input:
integrate((1-2*x)*(2+3*x)**m/(3+5*x),x)
Output:
-Integral(-(3*x + 2)**m/(5*x + 3), x) - Integral(2*x*(3*x + 2)**m/(5*x + 3 ), x)
\[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m} {\left (2 \, x - 1\right )}}{5 \, x + 3} \,d x } \] Input:
integrate((1-2*x)*(2+3*x)^m/(3+5*x),x, algorithm="maxima")
Output:
-integrate((3*x + 2)^m*(2*x - 1)/(5*x + 3), x)
\[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m} {\left (2 \, x - 1\right )}}{5 \, x + 3} \,d x } \] Input:
integrate((1-2*x)*(2+3*x)^m/(3+5*x),x, algorithm="giac")
Output:
integrate(-(3*x + 2)^m*(2*x - 1)/(5*x + 3), x)
Timed out. \[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=-\int \frac {\left (2\,x-1\right )\,{\left (3\,x+2\right )}^m}{5\,x+3} \,d x \] Input:
int(-((2*x - 1)*(3*x + 2)^m)/(5*x + 3),x)
Output:
-int(((2*x - 1)*(3*x + 2)^m)/(5*x + 3), x)
\[ \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx=\frac {-18 \left (3 x +2\right )^{m} m x +10 \left (3 x +2\right )^{m} m +22 \left (3 x +2\right )^{m}-33 \left (\int \frac {\left (3 x +2\right )^{m} x}{15 x^{2}+19 x +6}d x \right ) m^{2}-33 \left (\int \frac {\left (3 x +2\right )^{m} x}{15 x^{2}+19 x +6}d x \right ) m}{45 m \left (m +1\right )} \] Input:
int((1-2*x)*(2+3*x)^m/(3+5*x),x)
Output:
( - 18*(3*x + 2)**m*m*x + 10*(3*x + 2)**m*m + 22*(3*x + 2)**m - 33*int(((3 *x + 2)**m*x)/(15*x**2 + 19*x + 6),x)*m**2 - 33*int(((3*x + 2)**m*x)/(15*x **2 + 19*x + 6),x)*m)/(45*m*(m + 1))