3.3121 \(\int \frac{(a+b x)^m (c+d x)^{1-m}}{(e+f x)^6} \, dx\)

Optimal. Leaf size=542 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \left (3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (5 d e-c f (2-m))-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-5 m+6\right )-10 c d e f (2-m)+20 d^2 e^2\right )+b^3 \left (15 c^2 d e f^2 \left (m^2-5 m+6\right )-c^3 f^3 \left (-m^3+9 m^2-26 m+24\right )-60 c d^2 e^2 f (2-m)+60 d^3 e^3\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{60 (m+1) (b e-a f)^6 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (3 d e (4 m+7)-c f \left (-2 m^2+2 m+9\right )\right )+b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )-3 c d e f (11-4 m)+27 d^2 e^2\right )\right )}{60 (e+f x)^3 (b e-a f)^3 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} (b (7 d e-c f (4-m))-a d f (m+3))}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(5*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5) - (f*(b*(7*d*e - c*f*(4 - m))
- a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(20*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^4) - (f*(a^2*
d^2*f^2*(6 + 5*m + m^2) - a*b*d*f*(3*d*e*(7 + 4*m) - c*f*(9 + 2*m - 2*m^2)) + b^2*(27*d^2*e^2 - 3*c*d*e*f*(11
- 4*m) + c^2*f^2*(12 - 7*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(60*(b*e - a*f)^3*(d*e - c*f)^3*(e +
f*x)^3) + ((b*c - a*d)^2*(3*a^2*b*d^2*f^2*(5*d*e - c*f*(2 - m))*(2 + 3*m + m^2) - a^3*d^3*f^3*(6 + 11*m + 6*m^
2 + m^3) - 3*a*b^2*d*f*(1 + m)*(20*d^2*e^2 - 10*c*d*e*f*(2 - m) + c^2*f^2*(6 - 5*m + m^2)) + b^3*(60*d^3*e^3 -
60*c*d^2*e^2*f*(2 - m) + 15*c^2*d*e*f^2*(6 - 5*m + m^2) - c^3*f^3*(24 - 26*m + 9*m^2 - m^3)))*(a + b*x)^(1 +
m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[3, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(60
*(b*e - a*f)^6*(d*e - c*f)^3*(1 + m))

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Rubi [A]  time = 0.936091, antiderivative size = 541, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {129, 151, 12, 131} \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \left (3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (5 d e-c f (2-m))-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-5 m+6\right )-10 c d e f (2-m)+20 d^2 e^2\right )+b^3 \left (15 c^2 d e f^2 \left (m^2-5 m+6\right )-c^3 f^3 \left (-m^3+9 m^2-26 m+24\right )-60 c d^2 e^2 f (2-m)+60 d^3 e^3\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{60 (m+1) (b e-a f)^6 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (3 d e (4 m+7)-c f \left (-2 m^2+2 m+9\right )\right )+b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )-3 c d e f (11-4 m)+27 d^2 e^2\right )\right )}{60 (e+f x)^3 (b e-a f)^3 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} (-a d f (m+3)-b c f (4-m)+7 b d e)}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^6,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(5*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5) - (f*(7*b*d*e - b*c*f*(4 - m)
- a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(20*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^4) - (f*(a^2*
d^2*f^2*(6 + 5*m + m^2) - a*b*d*f*(3*d*e*(7 + 4*m) - c*f*(9 + 2*m - 2*m^2)) + b^2*(27*d^2*e^2 - 3*c*d*e*f*(11
- 4*m) + c^2*f^2*(12 - 7*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(60*(b*e - a*f)^3*(d*e - c*f)^3*(e +
f*x)^3) + ((b*c - a*d)^2*(3*a^2*b*d^2*f^2*(5*d*e - c*f*(2 - m))*(2 + 3*m + m^2) - a^3*d^3*f^3*(6 + 11*m + 6*m^
2 + m^3) - 3*a*b^2*d*f*(1 + m)*(20*d^2*e^2 - 10*c*d*e*f*(2 - m) + c^2*f^2*(6 - 5*m + m^2)) + b^3*(60*d^3*e^3 -
60*c*d^2*e^2*f*(2 - m) + 15*c^2*d*e*f^2*(6 - 5*m + m^2) - c^3*f^3*(24 - 26*m + 9*m^2 - m^3)))*(a + b*x)^(1 +
m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[3, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(60
*(b*e - a*f)^6*(d*e - c*f)^3*(1 + m))

Rule 129

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{(a+b x)^m (c+d x)^{1-m}}{(e+f x)^6} \, dx &=-\frac{f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{\int \frac{(a+b x)^m (c+d x)^{1-m} (-b (5 d e-c f (4-m))+a d f (3+m)+2 b d f x)}{(e+f x)^5} \, dx}{5 (b e-a f) (d e-c f)}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}+\frac{\int \frac{(a+b x)^m (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (d e (18+11 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (20 d^2 e^2-c d e f (29-11 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )-b d f (7 b d e-b c f (4-m)-a d f (3+m)) x\right )}{(e+f x)^4} \, dx}{20 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac{f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}-\frac{\int \frac{\left (-3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )+a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )+3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )-b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) (a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx}{60 (b e-a f)^3 (d e-c f)^3}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac{f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}+\frac{\left (3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )-a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )-3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )+b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) \int \frac{(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx}{60 (b e-a f)^3 (d e-c f)^3}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{2-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (7 b d e-b c f (4-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac{f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (3 d e (7+4 m)-c f \left (9+2 m-2 m^2\right )\right )+b^2 \left (27 d^2 e^2-3 c d e f (11-4 m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{2-m}}{60 (b e-a f)^3 (d e-c f)^3 (e+f x)^3}+\frac{(b c-a d)^2 \left (3 a^2 b d^2 f^2 (5 d e-c f (2-m)) \left (2+3 m+m^2\right )-a^3 d^3 f^3 \left (6+11 m+6 m^2+m^3\right )-3 a b^2 d f (1+m) \left (20 d^2 e^2-10 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )+b^3 \left (60 d^3 e^3-60 c d^2 e^2 f (2-m)+15 c^2 d e f^2 \left (6-5 m+m^2\right )-c^3 f^3 \left (24-26 m+9 m^2-m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{60 (b e-a f)^6 (d e-c f)^3 (1+m)}\\ \end{align*}

Mathematica [A]  time = 3.37094, size = 484, normalized size = 0.89 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-\frac{(e+f x)^2 \left (f (m+1) (c+d x)^3 (b e-a f)^3 \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f \left (c f \left (-2 m^2+2 m+9\right )-3 d e (4 m+7)\right )+b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )+3 c d e f (4 m-11)+27 d^2 e^2\right )\right )-(e+f x)^3 (b c-a d)^2 \left (3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (c f (m-2)+5 d e)-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-5 m+6\right )+10 c d e f (m-2)+20 d^2 e^2\right )+b^3 \left (15 c^2 d e f^2 \left (m^2-5 m+6\right )+c^3 f^3 \left (m^3-9 m^2+26 m-24\right )+60 c d^2 e^2 f (m-2)+60 d^3 e^3\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(m+1) (b e-a f)^5 (d e-c f)^2}+\frac{3 f (c+d x)^3 (e+f x) (a d f (m+3)-b (c f (m-4)+7 d e))}{(b e-a f) (d e-c f)}-12 f (c+d x)^3\right )}{60 (e+f x)^5 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^6,x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(-12*f*(c + d*x)^3 + (3*f*(-(b*(7*d*e + c*f*(-4 + m))) + a*d*f*(3 + m))*
(c + d*x)^3*(e + f*x))/((b*e - a*f)*(d*e - c*f)) - ((e + f*x)^2*(f*(b*e - a*f)^3*(1 + m)*(a^2*d^2*f^2*(6 + 5*m
+ m^2) + a*b*d*f*(-3*d*e*(7 + 4*m) + c*f*(9 + 2*m - 2*m^2)) + b^2*(27*d^2*e^2 + 3*c*d*e*f*(-11 + 4*m) + c^2*f
^2*(12 - 7*m + m^2)))*(c + d*x)^3 - (b*c - a*d)^2*(3*a^2*b*d^2*f^2*(5*d*e + c*f*(-2 + m))*(2 + 3*m + m^2) - a^
3*d^3*f^3*(6 + 11*m + 6*m^2 + m^3) - 3*a*b^2*d*f*(1 + m)*(20*d^2*e^2 + 10*c*d*e*f*(-2 + m) + c^2*f^2*(6 - 5*m
+ m^2)) + b^3*(60*d^3*e^3 + 60*c*d^2*e^2*f*(-2 + m) + 15*c^2*d*e*f^2*(6 - 5*m + m^2) + c^3*f^3*(-24 + 26*m - 9
*m^2 + m^3)))*(e + f*x)^3*Hypergeometric2F1[3, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])
)/((b*e - a*f)^5*(d*e - c*f)^2*(1 + m))))/(60*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5)

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Maple [F]  time = 0.384, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m}}{ \left ( fx+e \right ) ^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^6,x)

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^6,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^6,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{f^{6} x^{6} + 6 \, e f^{5} x^{5} + 15 \, e^{2} f^{4} x^{4} + 20 \, e^{3} f^{3} x^{3} + 15 \, e^{4} f^{2} x^{2} + 6 \, e^{5} f x + e^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^6,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 1)/(f^6*x^6 + 6*e*f^5*x^5 + 15*e^2*f^4*x^4 + 20*e^3*f^3*x^3 + 15*e^4*f^2*
x^2 + 6*e^5*f*x + e^6), 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((b*x+a)**m*(d*x+c)**(1-m)/(f*x+e)**6,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^6,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^6, x)