Integrand size = 26, antiderivative size = 132 \[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-1-m} (2+3 x)^{1+m} \left (2768-315 m+4 m^2-8 (43-m) (1+m) x\right )}{9 (1+m)}+\frac {2^{-m} \left (1323-128 m+2 m^2\right ) (1+2 x)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (1+2 x))}{9 m} \]
-1/3*(5-4*x)^2*(1+2*x)^(-1-m)*(2+3*x)^(1+m)-1/9*(1+2*x)^(-1-m)*(2+3*x)^(1+ m)*(2768-315*m+4*m^2-8*(43-m)*(1+m)*x)/(1+m)+1/9*(2*m^2-128*m+1323)*hyperg eom([-m, -m],[1-m],-3-6*x)/(2^m)/m/((1+2*x)^m)
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=\frac {2^{-m} (1+2 x)^{-1-m} \left (-2^m m (2+3 x)^{1+m} \left (2843-464 x+48 x^2+m^2 (4+8 x)+24 m \left (-10-19 x+2 x^2\right )\right )+\left (1323+1195 m-126 m^2+2 m^3\right ) (1+2 x) \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3-6 x)\right )}{9 m (1+m)} \]
((1 + 2*x)^(-1 - m)*(-(2^m*m*(2 + 3*x)^(1 + m)*(2843 - 464*x + 48*x^2 + m^ 2*(4 + 8*x) + 24*m*(-10 - 19*x + 2*x^2))) + (1323 + 1195*m - 126*m^2 + 2*m ^3)*(1 + 2*x)*Hypergeometric2F1[-m, -m, 1 - m, -3 - 6*x]))/(9*2^m*m*(1 + m ))
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {111, 27, 160, 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.
| \(\displaystyle \int (5-4 x)^3 (2 x+1)^{-m-2} (3 x+2)^m \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{12} \int 4 (5-4 x) (2 x+1)^{-m-2} (3 x+2)^m (-5 m-4 (43-m) x+54)dx-\frac {1}{3} (5-4 x)^2 (2 x+1)^{-m-1} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int (5-4 x) (2 x+1)^{-m-2} (3 x+2)^m (-5 m-4 (43-m) x+54)dx-\frac {1}{3} (5-4 x)^2 (2 x+1)^{-m-1} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 160 |
| \(\displaystyle \frac {1}{3} \left (-\frac {2}{3} \left (2 m^2-128 m+1323\right ) \int (2 x+1)^{-m-1} (3 x+2)^mdx-\frac {(3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right ) (2 x+1)^{-m-1}}{3 (m+1)}\right )-\frac {1}{3} (5-4 x)^2 (2 x+1)^{-m-1} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {1}{3} \left (\frac {2^{-m} \left (2 m^2-128 m+1323\right ) (2 x+1)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (2 x+1))}{3 m}-\frac {(2 x+1)^{-m-1} (3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right )}{3 (m+1)}\right )-\frac {1}{3} (5-4 x)^2 (2 x+1)^{-m-1} (3 x+2)^{m+1}\) |
-1/3*((5 - 4*x)^2*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m)) + (-1/3*((1 + 2*x) ^(-1 - m)*(2 + 3*x)^(1 + m)*(2768 - 315*m + 4*m^2 - 8*(43 - m)*(1 + m)*x)) /(1 + m) + ((1323 - 128*m + 2*m^2)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(3*2^m*m*(1 + 2*x)^m))/3
3.31.80.3.1 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^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* (f*g + e*h) - c*f*h*(m + 2)))/(b^2*d) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
\[\int \left (5-4 x \right )^{3} \left (1+2 x \right )^{-2-m} \left (2+3 x \right )^{m}d x\]
\[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 2} {\left (4 \, x - 5\right )}^{3} \,d x } \]
\[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=- \int \left (- 125 \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m}\right )\, dx - \int 300 x \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m}\, dx - \int \left (- 240 x^{2} \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m}\right )\, dx - \int 64 x^{3} \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m}\, dx \]
-Integral(-125*(2*x + 1)**(-m - 2)*(3*x + 2)**m, x) - Integral(300*x*(2*x + 1)**(-m - 2)*(3*x + 2)**m, x) - Integral(-240*x**2*(2*x + 1)**(-m - 2)*( 3*x + 2)**m, x) - Integral(64*x**3*(2*x + 1)**(-m - 2)*(3*x + 2)**m, x)
\[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 2} {\left (4 \, x - 5\right )}^{3} \,d x } \]
\[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 2} {\left (4 \, x - 5\right )}^{3} \,d x } \]
Timed out. \[ \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx=-\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^3}{{\left (2\,x+1\right )}^{m+2}} \,d x \]