3.3104 \(\int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=546 \[ -\frac{2^{3-m} (105-2 m) (2 x+1)^{-m-3} \, _2F_1(-m-3,-m-3;-m-2;-3 (2 x+1))}{81 (m+3)}-\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2) (m+3) (m+4)}-\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{27 \left (m^2+5 m+6\right )}+\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-2}}{9 \left (m^2+5 m+6\right )}+\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{(m+3) (m+4) \left (m^2+3 m+2\right )}+\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{9 \left (m^3+6 m^2+11 m+6\right )}-\frac{2}{3} (5-4 x)^4 (3 x+2)^{m+1} (2 x+1)^{-m-4}-\frac{7 (13-2 m) (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{3 (m+4)}+\frac{1127 (13-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+3) (m+4)}-\frac{322 (13-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+4)}+\frac{24334 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{81 (m+3)}-\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-3}}{27 (m+3)}+\frac{736 (105-2 m) (3 x+2)^{m+3} (2 x+1)^{-m-3}}{27 (m+3)} \]

[Out]

(-7*(13 - 2*m)*(5 - 4*x)^3*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (2*(5 - 4*x)^4*(1 + 2*x)^(-4 -
m)*(2 + 3*x)^(1 + m))/3 + (24334*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(81*(3 + m)) + (1127*(13 -
2*m)*(27 + 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(4 + m)) - (322*(13 - 2*m)*(5 - 4*x)*(1 + 2*x
)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (48668*(105 - 2*m)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(27*(6 +
5*m + m^2)) - (322*(13 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(3*(2 + m)*(3 + m)*(4
+ m)) + (322*(13 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/((3 + m)*(4 + m)*(2 + 3*m
+ m^2)) + (48668*(105 - 2*m)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(9*(6 + 11*m + 6*m^2 + m^3)) - (4232*(105 -
2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(2 + m))/(27*(3 + m)) + (4232*(105 - 2*m)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(2 +
m))/(9*(6 + 5*m + m^2)) + (736*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(3 + m))/(27*(3 + m)) - (2^(3 - m)*(1
05 - 2*m)*(1 + 2*x)^(-3 - m)*Hypergeometric2F1[-3 - m, -3 - m, -2 - m, -3*(1 + 2*x)])/(81*(3 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.446832, antiderivative size = 546, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {100, 159, 128, 45, 37, 69, 94, 90, 79} \[ -\frac{2^{3-m} (105-2 m) (2 x+1)^{-m-3} \, _2F_1(-m-3,-m-3;-m-2;-3 (2 x+1))}{81 (m+3)}-\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2) (m+3) (m+4)}-\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{27 \left (m^2+5 m+6\right )}+\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-2}}{9 \left (m^2+5 m+6\right )}+\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{(m+3) (m+4) \left (m^2+3 m+2\right )}+\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{9 \left (m^3+6 m^2+11 m+6\right )}-\frac{2}{3} (5-4 x)^4 (3 x+2)^{m+1} (2 x+1)^{-m-4}-\frac{7 (13-2 m) (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{3 (m+4)}+\frac{1127 (13-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+3) (m+4)}-\frac{322 (13-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+4)}+\frac{24334 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{81 (m+3)}-\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-3}}{27 (m+3)}+\frac{736 (105-2 m) (3 x+2)^{m+3} (2 x+1)^{-m-3}}{27 (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(5 - 4*x)^5*(1 + 2*x)^(-5 - m)*(2 + 3*x)^m,x]

[Out]

(-7*(13 - 2*m)*(5 - 4*x)^3*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (2*(5 - 4*x)^4*(1 + 2*x)^(-4 -
m)*(2 + 3*x)^(1 + m))/3 + (24334*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(81*(3 + m)) + (1127*(13 -
2*m)*(27 + 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(4 + m)) - (322*(13 - 2*m)*(5 - 4*x)*(1 + 2*x
)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (48668*(105 - 2*m)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(27*(6 +
5*m + m^2)) - (322*(13 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(3*(2 + m)*(3 + m)*(4
+ m)) + (322*(13 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/((3 + m)*(4 + m)*(2 + 3*m
+ m^2)) + (48668*(105 - 2*m)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(9*(6 + 11*m + 6*m^2 + m^3)) - (4232*(105 -
2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(2 + m))/(27*(3 + m)) + (4232*(105 - 2*m)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(2 +
m))/(9*(6 + 5*m + m^2)) + (736*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(3 + m))/(27*(3 + m)) - (2^(3 - m)*(1
05 - 2*m)*(1 + 2*x)^(-3 - m)*Hypergeometric2F1[-3 - m, -3 - m, -2 - m, -3*(1 + 2*x)])/(81*(3 + m))

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(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]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(a + b*x)^m*(
c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (SumSimplerQ[m, 1] || ( !SumS
implerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rule 128

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (
ILtQ[m, 0] && ILtQ[n, 0]))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[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]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(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] && SumSimplerQ[p, 1]

Rubi steps

\begin{align*} \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx &=-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}+\frac{1}{6} \int (5-4 x)^3 (1+2 x)^{-5-m} (2+3 x)^m (-2 (119+10 m)-8 (105-2 m) x) \, dx\\ &=-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}+\frac{1}{3} (7 (13-2 m)) \int (5-4 x)^3 (1+2 x)^{-5-m} (2+3 x)^m \, dx-\frac{1}{3} (2 (105-2 m)) \int (5-4 x)^3 (1+2 x)^{-4-m} (2+3 x)^m \, dx\\ &=-\frac{7 (13-2 m) (5-4 x)^3 (1+2 x)^{-4-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}-\frac{1}{3} (2 (105-2 m)) \int \left (\frac{12167}{27} (1+2 x)^{-4-m} (2+3 x)^m-\frac{2116}{9} (1+2 x)^{-4-m} (2+3 x)^{1+m}+\frac{368}{9} (1+2 x)^{-4-m} (2+3 x)^{2+m}-\frac{64}{27} (1+2 x)^{-4-m} (2+3 x)^{3+m}\right ) \, dx-\frac{(161 (13-2 m)) \int (5-4 x)^2 (1+2 x)^{-4-m} (2+3 x)^m \, dx}{4+m}\\ &=-\frac{7 (13-2 m) (5-4 x)^3 (1+2 x)^{-4-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}-\frac{322 (13-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (4+m)}+\frac{1}{81} (128 (105-2 m)) \int (1+2 x)^{-4-m} (2+3 x)^{3+m} \, dx-\frac{1}{27} (736 (105-2 m)) \int (1+2 x)^{-4-m} (2+3 x)^{2+m} \, dx+\frac{1}{27} (4232 (105-2 m)) \int (1+2 x)^{-4-m} (2+3 x)^{1+m} \, dx-\frac{1}{81} (24334 (105-2 m)) \int (1+2 x)^{-4-m} (2+3 x)^m \, dx+\frac{(161 (13-2 m)) \int (1+2 x)^{-4-m} (2+3 x)^m (-2 (181+10 m)+16 (2+m) x) \, dx}{6 (4+m)}\\ &=-\frac{7 (13-2 m) (5-4 x)^3 (1+2 x)^{-4-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}+\frac{24334 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{81 (3+m)}+\frac{1127 (13-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (3+m) (4+m)}-\frac{322 (13-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{4232 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{2+m}}{27 (3+m)}+\frac{736 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{3+m}}{27 (3+m)}-\frac{2^{3-m} (105-2 m) (1+2 x)^{-3-m} \, _2F_1(-3-m,-3-m;-2-m;-3 (1+2 x))}{81 (3+m)}-\frac{(4232 (105-2 m)) \int (1+2 x)^{-3-m} (2+3 x)^{1+m} \, dx}{9 (3+m)}+\frac{(48668 (105-2 m)) \int (1+2 x)^{-3-m} (2+3 x)^m \, dx}{27 (3+m)}+\frac{\left (322 (13-2 m) \left (579+52 m+2 m^2\right )\right ) \int (1+2 x)^{-3-m} (2+3 x)^m \, dx}{3 (3+m) (4+m)}\\ &=-\frac{7 (13-2 m) (5-4 x)^3 (1+2 x)^{-4-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}+\frac{24334 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{81 (3+m)}+\frac{1127 (13-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (3+m) (4+m)}-\frac{322 (13-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{48668 (105-2 m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{27 (2+m) (3+m)}-\frac{322 (13-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m) (3+m) (4+m)}-\frac{4232 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{2+m}}{27 (3+m)}+\frac{4232 (105-2 m) (1+2 x)^{-2-m} (2+3 x)^{2+m}}{9 (2+m) (3+m)}+\frac{736 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{3+m}}{27 (3+m)}-\frac{2^{3-m} (105-2 m) (1+2 x)^{-3-m} \, _2F_1(-3-m,-3-m;-2-m;-3 (1+2 x))}{81 (3+m)}-\frac{(48668 (105-2 m)) \int (1+2 x)^{-2-m} (2+3 x)^m \, dx}{9 (2+m) (3+m)}-\frac{\left (322 (13-2 m) \left (579+52 m+2 m^2\right )\right ) \int (1+2 x)^{-2-m} (2+3 x)^m \, dx}{(2+m) (3+m) (4+m)}\\ &=-\frac{7 (13-2 m) (5-4 x)^3 (1+2 x)^{-4-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{2}{3} (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^{1+m}+\frac{24334 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{81 (3+m)}+\frac{1127 (13-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (3+m) (4+m)}-\frac{322 (13-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{3 (4+m)}-\frac{48668 (105-2 m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{27 (2+m) (3+m)}-\frac{322 (13-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m) (3+m) (4+m)}+\frac{48668 (105-2 m) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{9 (1+m) (2+m) (3+m)}+\frac{322 (13-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{(1+m) (2+m) (3+m) (4+m)}-\frac{4232 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{2+m}}{27 (3+m)}+\frac{4232 (105-2 m) (1+2 x)^{-2-m} (2+3 x)^{2+m}}{9 (2+m) (3+m)}+\frac{736 (105-2 m) (1+2 x)^{-3-m} (2+3 x)^{3+m}}{27 (3+m)}-\frac{2^{3-m} (105-2 m) (1+2 x)^{-3-m} \, _2F_1(-3-m,-3-m;-2-m;-3 (1+2 x))}{81 (3+m)}\\ \end{align*}

Mathematica [A]  time = 10.469, size = 274, normalized size = 0.5 \[ \frac{2^{4-m} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}-\frac{560 (-6 x-3)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+1;m+2;6 x+4)}{m+1}-\frac{13720\ 3^{m+2} (-2 x-1)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+3;m+2;6 x+4)}{m+1}-\frac{16807\ 3^{m+4} (-2 x-1)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+5;m+2;6 x+4)}{m+1}-\frac{12005\ 3^{m+3} (-2 x-1)^m (4 x+2)^{-m} (6 x+4)^{m+1} \, _2F_1(m+1,m+4;m+2;6 x+4)}{m+1}+\frac{3920 (3 x+2)^{m+1} (2 x+1)^{-m-1}}{m+1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(5 - 4*x)^5*(1 + 2*x)^(-5 - m)*(2 + 3*x)^m,x]

[Out]

(3920*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(1 + m) + (2^(4 - m)*(1 + 2*x)^(1 - m)*Hypergeometric2F1[1 - m, -m
, 2 - m, -3 - 6*x])/(-1 + m) - (560*(-3 - 6*x)^m*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1 + m, 1 + m, 2 + m, 4 +
6*x])/((1 + m)*(1 + 2*x)^m) - (13720*3^(2 + m)*(-1 - 2*x)^m*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1 + m, 3 + m,
2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m) - (12005*3^(3 + m)*(-1 - 2*x)^m*(4 + 6*x)^(1 + m)*Hypergeometric2F1[1 +
m, 4 + m, 2 + m, 4 + 6*x])/((1 + m)*(2 + 4*x)^m) - (16807*3^(4 + m)*(-1 - 2*x)^m*(2 + 3*x)^(1 + m)*Hypergeome
tric2F1[1 + m, 5 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m)

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int \left ( 5-4\,x \right ) ^{5} \left ( 1+2\,x \right ) ^{-5-m} \left ( 2+3\,x \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x)

[Out]

int((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 5}{\left (4 \, x - 5\right )}^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x, algorithm="maxima")

[Out]

-integrate((3*x + 2)^m*(2*x + 1)^(-m - 5)*(4*x - 5)^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (1024 \, x^{5} - 6400 \, x^{4} + 16000 \, x^{3} - 20000 \, x^{2} + 12500 \, x - 3125\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 5}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x, algorithm="fricas")

[Out]

integral(-(1024*x^5 - 6400*x^4 + 16000*x^3 - 20000*x^2 + 12500*x - 3125)*(3*x + 2)^m*(2*x + 1)^(-m - 5), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)**5*(1+2*x)**(-5-m)*(2+3*x)**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 5}{\left (4 \, x - 5\right )}^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x, algorithm="giac")

[Out]

integrate(-(3*x + 2)^m*(2*x + 1)^(-m - 5)*(4*x - 5)^5, x)