3.3088 \(\int (5-4 x)^4 (1+2 x)^{-3-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=188 \[ -\frac{2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac{7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac{1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac{1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]

[Out]

-((107 - 2*m)*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/9 - ((5 - 4*x)^3*(1 + 2*x)^(-2 - m)*(2 + 3*x)^
(1 + m))/3 + (7*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m)*(3*(4638 + 485*m + 108*m^2 - 2*m^3) + 2*(15209 + 1882*m -
530*m^2 + 8*m^3)*x))/(9*(2 + 3*m + m^2)) - (2^(2 - m)*(1323 - 85*m + m^2)*Hypergeometric2F1[-m, -m, 1 - m, -3
*(1 + 2*x)])/(9*m*(1 + 2*x)^m)

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Rubi [A]  time = 0.192771, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {100, 153, 145, 69} \[ -\frac{2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac{7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac{1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac{1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((107 - 2*m)*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/9 - ((5 - 4*x)^3*(1 + 2*x)^(-2 - m)*(2 + 3*x)^
(1 + m))/3 + (7*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m)*(3*(4638 + 485*m + 108*m^2 - 2*m^3) + 2*(15209 + 1882*m -
530*m^2 + 8*m^3)*x))/(9*(2 + 3*m + m^2)) - (2^(2 - m)*(1323 - 85*m + m^2)*Hypergeometric2F1[-m, -m, 1 - m, -3
*(1 + 2*x)])/(9*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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> 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)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(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] &&  !L
tQ[n, -2]))

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]))

Rubi steps

\begin{align*} \int (5-4 x)^4 (1+2 x)^{-3-m} (2+3 x)^m \, dx &=-\frac{1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac{1}{12} \int (5-4 x)^2 (1+2 x)^{-3-m} (2+3 x)^m (4 (26-5 m)-8 (107-2 m) x) \, dx\\ &=-\frac{1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac{1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac{1}{72} \int (5-4 x) (1+2 x)^{-3-m} (2+3 x)^m \left (-8 \left (3997+528 m-10 m^2\right )-64 \left (1323-85 m+m^2\right ) x\right ) \, dx\\ &=-\frac{1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac{1}{3} (5-4 x)^3 (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 (4638+485 m+108 m^2-2 m^3\right )+2 \left (15209+1882 m-530 m^2+8 m^3\right ) x\right )}{9 \left (2+3 m+m^2\right )}+\frac{1}{9} \left (8 \left (1323-85 m+m^2\right )\right ) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx\\ &=-\frac{1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac{1}{3} (5-4 x)^3 (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 (4638+485 m+108 m^2-2 m^3\right )+2 \left (15209+1882 m-530 m^2+8 m^3\right ) x\right )}{9 \left (2+3 m+m^2\right )}-\frac{2^{2-m} \left (1323-85 m+m^2\right ) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{9 m}\\ \end{align*}

Mathematica [A]  time = 0.230047, size = 153, normalized size = 0.81 \[ \frac{2^{-m} (2 x+1)^{-m-2} \left (2^m (3 x+2)^{m+1} \left (16 m^3 \left (6 x^2+7 x+2\right )+32 m^2 \left (18 x^3-219 x^2-274 x-80\right )+m \left (1728 x^3-21696 x^2+143632 x+12629\right )+6 \left (192 x^3-2432 x^2+118699 x+49177\right )\right )-4 \left (m^3-83 m^2+1153 m+2646\right ) (2 x+1) \, _2F_1(-m-1,-m-1;-m;-6 x-3)\right )}{27 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*x)^(-2 - m)*(2^m*(2 + 3*x)^(1 + m)*(16*m^3*(2 + 7*x + 6*x^2) + 32*m^2*(-80 - 274*x - 219*x^2 + 18*x^3)
+ 6*(49177 + 118699*x - 2432*x^2 + 192*x^3) + m*(12629 + 143632*x - 21696*x^2 + 1728*x^3)) - 4*(2646 + 1153*m
- 83*m^2 + m^3)*(1 + 2*x)*Hypergeometric2F1[-1 - m, -1 - m, -m, -3 - 6*x]))/(27*2^m*(1 + m)*(2 + 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 ) ^{-3-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)^(-3-m)*(2+3*x)^m,x)

[Out]

int((5-4*x)^4*(1+2*x)^(-3-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 - 3}{\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)^(-3-m)*(2+3*x)^m,x, algorithm="maxima")

[Out]

integrate((3*x + 2)^m*(2*x + 1)^(-m - 3)*(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 - 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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