Integrand size = 26, antiderivative size = 121 \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=-\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {2^{-1-m} \left (441-86 m+2 m^2\right ) (1+2 x)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (1+2 x))}{3 (1-m) m} \]
-7/3*(21-m)*(2+3*x)^(1+m)/m/((1+2*x)^m)-1/3*(5-4*x)*(2+3*x)^(1+m)/((1+2*x) ^m)+1/3*2^(-1-m)*(2*m^2-86*m+441)*(1+2*x)^(1-m)*hypergeom([-m, 1-m],[2-m], -3-6*x)/(1-m)/m
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79 \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {1}{6} (1+2 x)^{-m} \left (\frac {14 (-21+m) (2+3 x)^{1+m}}{m}+2 (2+3 x)^{1+m} (-5+4 x)-\frac {2^{-m} \left (441-86 m+2 m^2\right ) (1+2 x) \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3-6 x)}{(-1+m) m}\right ) \]
((14*(-21 + m)*(2 + 3*x)^(1 + m))/m + 2*(2 + 3*x)^(1 + m)*(-5 + 4*x) - ((4 41 - 86*m + 2*m^2)*(1 + 2*x)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x] )/(2^m*(-1 + m)*m))/(6*(1 + 2*x)^m)
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {101, 27, 88, 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.
\(\displaystyle \int (5-4 x)^2 (2 x+1)^{-m-1} (3 x+2)^m \, dx\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {1}{12} \int 4 (2 x+1)^{-m-1} (3 x+2)^m (-5 m-2 (65-2 m) x+82)dx-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int (2 x+1)^{-m-1} (3 x+2)^m (-5 m-2 (65-2 m) x+82)dx-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {1}{3} \left (\frac {\left (2 m^2-86 m+441\right ) \int (2 x+1)^{-m} (3 x+2)^mdx}{m}-\frac {7 (21-m) (2 x+1)^{-m} (3 x+2)^{m+1}}{m}\right )-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {1}{3} \left (\frac {2^{-m-1} \left (2 m^2-86 m+441\right ) (2 x+1)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (2 x+1))}{(1-m) m}-\frac {7 (21-m) (2 x+1)^{-m} (3 x+2)^{m+1}}{m}\right )-\frac {1}{3} (5-4 x) (2 x+1)^{-m} (3 x+2)^{m+1}\) |
-1/3*((5 - 4*x)*(2 + 3*x)^(1 + m))/(1 + 2*x)^m + ((-7*(21 - m)*(2 + 3*x)^( 1 + m))/(m*(1 + 2*x)^m) + (2^(-1 - m)*(441 - 86*m + 2*m^2)*(1 + 2*x)^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/((1 - m)*m))/3
3.31.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))
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)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
\[\int \left (5-4 x \right )^{2} \left (1+2 x \right )^{-1-m} \left (2+3 x \right )^{m}d x\]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 1} {\left (4 \, x - 5\right )}^{2} \,d x } \]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m} \left (4 x - 5\right )^{2}\, dx \]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 1} {\left (4 \, x - 5\right )}^{2} \,d x } \]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 1} {\left (4 \, x - 5\right )}^{2} \,d x } \]
Timed out. \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^2}{{\left (2\,x+1\right )}^{m+1}} \,d x \]