3.3096 \(\int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=333 \[ \frac{2^{3-m} (42-m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 (m+2) (m+3)}-\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+3) \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}-\frac{49 (15-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{9 (m+3)}+\frac{14}{9} (15-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}+\frac{196 (42-m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2)}-\frac{28 (42-m) (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+1) (m+2)} \]

[Out]

(-49*(15 - 2*m)*(27 + 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(9*(3 + m)) + (14*(15 - 2*m)*(5 - 4*x)*(1 + 2
*x)^(-3 - m)*(2 + 3*x)^(1 + m))/9 - (2*(5 - 4*x)^3*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + (196*(42 - m)*(1
+ 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(3*(2 + m)) + (14*(15 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-2 - m)*(2 + 3
*x)^(1 + m))/(9*(2 + m)*(3 + m)) - (28*(42 - m)*(29 + 4*m)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3*(1 + m)*(2
+ m)) - (14*(15 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(2 + 3*m + m^2))
+ (2^(3 - m)*(42 - m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(3*m*(1 + 2*x)^m)

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Rubi [A]  time = 0.310347, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {100, 159, 89, 79, 69, 90, 45, 37} \[ \frac{2^{3-m} (42-m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 (m+2) (m+3)}-\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+3) \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}-\frac{49 (15-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{9 (m+3)}+\frac{14}{9} (15-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}+\frac{196 (42-m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2)}-\frac{28 (42-m) (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-49*(15 - 2*m)*(27 + 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(9*(3 + m)) + (14*(15 - 2*m)*(5 - 4*x)*(1 + 2
*x)^(-3 - m)*(2 + 3*x)^(1 + m))/9 - (2*(5 - 4*x)^3*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + (196*(42 - m)*(1
+ 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(3*(2 + m)) + (14*(15 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-2 - m)*(2 + 3
*x)^(1 + m))/(9*(2 + m)*(3 + m)) - (28*(42 - m)*(29 + 4*m)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3*(1 + m)*(2
+ m)) - (14*(15 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(2 + 3*m + m^2))
+ (2^(3 - m)*(42 - m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(3*m*(1 + 2*x)^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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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]

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 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 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]

Rubi steps

\begin{align*} \int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx &=-\frac{2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac{1}{6} \int (5-4 x)^2 (1+2 x)^{-4-m} (2+3 x)^m (-2 (63+10 m)-16 (42-m) x) \, dx\\ &=-\frac{2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac{1}{3} (7 (15-2 m)) \int (5-4 x)^2 (1+2 x)^{-4-m} (2+3 x)^m \, dx-\frac{1}{3} (4 (42-m)) \int (5-4 x)^2 (1+2 x)^{-3-m} (2+3 x)^m \, dx\\ &=\frac{14}{9} (15-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}-\frac{2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac{196 (42-m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m)}-\frac{1}{18} (7 (15-2 m)) \int (1+2 x)^{-4-m} (2+3 x)^m (-2 (181+10 m)+16 (2+m) x) \, dx-\frac{(42-m) \int (1+2 x)^{-2-m} (2+3 x)^m (-12 (65+8 m)+32 (2+m) x) \, dx}{3 (2+m)}\\ &=-\frac{49 (15-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{9 (3+m)}+\frac{14}{9} (15-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}-\frac{2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac{196 (42-m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m)}-\frac{28 (42-m) (29+4 m) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{3 (1+m) (2+m)}-\frac{1}{3} (16 (42-m)) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx-\frac{\left (14 (15-2 m) \left (579+52 m+2 m^2\right )\right ) \int (1+2 x)^{-3-m} (2+3 x)^m \, dx}{9 (3+m)}\\ &=-\frac{49 (15-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{9 (3+m)}+\frac{14}{9} (15-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}-\frac{2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac{196 (42-m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m)}+\frac{14 (15-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{9 (2+m) (3+m)}-\frac{28 (42-m) (29+4 m) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{3 (1+m) (2+m)}+\frac{2^{3-m} (42-m) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{3 m}+\frac{\left (14 (15-2 m) \left (579+52 m+2 m^2\right )\right ) \int (1+2 x)^{-2-m} (2+3 x)^m \, dx}{3 (2+m) (3+m)}\\ &=-\frac{49 (15-2 m) (27+2 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{9 (3+m)}+\frac{14}{9} (15-2 m) (5-4 x) (1+2 x)^{-3-m} (2+3 x)^{1+m}-\frac{2}{3} (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^{1+m}+\frac{196 (42-m) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{3 (2+m)}+\frac{14 (15-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{9 (2+m) (3+m)}-\frac{28 (42-m) (29+4 m) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{3 (1+m) (2+m)}-\frac{14 (15-2 m) \left (579+52 m+2 m^2\right ) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{3 (1+m) (2+m) (3+m)}+\frac{2^{3-m} (42-m) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{3 m}\\ \end{align*}

Mathematica [A]  time = 0.182186, size = 161, normalized size = 0.48 \[ \frac{2^{-m} (2 x+1)^{-m-3} \left (2^m (3 x+2)^{m+1} \left (m^2 \left (2304 x^3-41248 x^2+6416 x-7801\right )+32 m^3 (2 x+1)^2 (3 x+2)+m \left (4224 x^3-514752 x^2-61044 x+33867\right )+18 \left (128 x^3-150644 x^2-128102 x-28775\right )\right )-8 \left (m^3-37 m^2-204 m-252\right ) (2 x+1)^2 \, _2F_1(-m-1,-m-1;-m;-6 x-3)\right )}{9 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*x)^(-3 - m)*(2^m*(2 + 3*x)^(1 + m)*(32*m^3*(1 + 2*x)^2*(2 + 3*x) + 18*(-28775 - 128102*x - 150644*x^2
+ 128*x^3) + m^2*(-7801 + 6416*x - 41248*x^2 + 2304*x^3) + m*(33867 - 61044*x - 514752*x^2 + 4224*x^3)) - 8*(-
252 - 204*m - 37*m^2 + m^3)*(1 + 2*x)^2*Hypergeometric2F1[-1 - m, -1 - m, -m, -3 - 6*x]))/(9*2^m*(1 + m)*(2 +
m)*(3 + m))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( 5-4\,x \right ) ^{4} \left ( 1+2\,x \right ) ^{-4-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)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x)

[Out]

int((5-4*x)^4*(1+2*x)^(-4-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 - 4}{\left (4 \, x - 5\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (256 \, x^{4} - 1280 \, x^{3} + 2400 \, x^{2} - 2000 \, x + 625\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((256*x^4 - 1280*x^3 + 2400*x^2 - 2000*x + 625)*(3*x + 2)^m*(2*x + 1)^(-m - 4), 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)**4*(1+2*x)**(-4-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 - 4}{\left (4 \, x - 5\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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