Integrand size = 26, antiderivative size = 139 \[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=-\frac {2}{3} (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (186-m+2 m^2\right )+2 \left (677+102 m-8 m^2\right ) x\right )}{3 \left (2+3 m+m^2\right )}-\frac {2^{1-m} (63-2 m) (1+2 x)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (1+2 x))}{3 m} \]
-2/3*(5-4*x)^2*(1+2*x)^(-2-m)*(2+3*x)^(1+m)+7/3*(1+2*x)^(-2-m)*(2+3*x)^(1+ m)*(6*m^2-3*m+558+2*(-8*m^2+102*m+677)*x)/(m^2+3*m+2)-1/3*2^(1-m)*(63-2*m) *hypergeom([-m, -m],[1-m],-3-6*x)/m/((1+2*x)^m)
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=\frac {2^{-m} (1+2 x)^{-2-m} \left (-2^m m (2+3 x)^{1+m} \left (-3806-9638 x+64 x^2+8 (m+2 m x)^2+3 m \left (57-556 x+32 x^2\right )\right )+2 \left (-126-185 m-57 m^2+2 m^3\right ) (1+2 x)^2 \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3-6 x)\right )}{3 m (1+m) (2+m)} \]
((1 + 2*x)^(-2 - m)*(-(2^m*m*(2 + 3*x)^(1 + m)*(-3806 - 9638*x + 64*x^2 + 8*(m + 2*m*x)^2 + 3*m*(57 - 556*x + 32*x^2))) + 2*(-126 - 185*m - 57*m^2 + 2*m^3)*(1 + 2*x)^2*Hypergeometric2F1[-m, -m, 1 - m, -3 - 6*x]))/(3*2^m*m* (1 + m)*(2 + m))
Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {111, 27, 162, 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.
| \(\displaystyle \int (5-4 x)^3 (2 x+1)^{-m-3} (3 x+2)^m \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{6} \int -2 (5-4 x) (2 x+1)^{-m-3} (3 x+2)^m (10 m+4 (63-2 m) x+7)dx-\frac {2}{3} (5-4 x)^2 (2 x+1)^{-m-2} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} \int (5-4 x) (2 x+1)^{-m-3} (3 x+2)^m (10 m+4 (63-2 m) x+7)dx-\frac {2}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{3} \left (4 (63-2 m) \int (2 x+1)^{-m-1} (3 x+2)^mdx+\frac {7 (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right ) (2 x+1)^{-m-2}}{m^2+3 m+2}\right )-\frac {2}{3} (5-4 x)^2 (2 x+1)^{-m-2} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {1}{3} \left (\frac {7 (2 x+1)^{-m-2} (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right )}{m^2+3 m+2}-\frac {2^{1-m} (63-2 m) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{m}\right )-\frac {2}{3} (5-4 x)^2 (2 x+1)^{-m-2} (3 x+2)^{m+1}\) |
(-2*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/3 + ((7*(1 + 2*x)^(- 2 - m)*(2 + 3*x)^(1 + m)*(3*(186 - m + 2*m^2) + 2*(677 + 102*m - 8*m^2)*x) )/(2 + 3*m + m^2) - (2^(1 - m)*(63 - 2*m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(m*(1 + 2*x)^m))/3
3.31.89.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
\[\int \left (5-4 x \right )^{3} \left (1+2 x \right )^{-3-m} \left (2+3 x \right )^{m}d x\]
\[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{3} \,d x } \]
\[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=- \int \left (- 125 \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m}\right )\, dx - \int 300 x \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m}\, dx - \int \left (- 240 x^{2} \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m}\right )\, dx - \int 64 x^{3} \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m}\, dx \]
-Integral(-125*(2*x + 1)**(-m - 3)*(3*x + 2)**m, x) - Integral(300*x*(2*x + 1)**(-m - 3)*(3*x + 2)**m, x) - Integral(-240*x**2*(2*x + 1)**(-m - 3)*( 3*x + 2)**m, x) - Integral(64*x**3*(2*x + 1)**(-m - 3)*(3*x + 2)**m, x)
\[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{3} \,d x } \]
\[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{3} \,d x } \]
Timed out. \[ \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx=-\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^3}{{\left (2\,x+1\right )}^{m+3}} \,d x \]