3.3133 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^7} \, dx\)

Optimal. Leaf size=541 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (6 d e-c f (3-m))-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-7 m+12\right )-12 c d e f (3-m)+30 d^2 e^2\right )+b^3 \left (18 c^2 d e f^2 \left (m^2-7 m+12\right )-c^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-90 c d^2 e^2 f (3-m)+120 d^3 e^3\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{120 (m+1) (b e-a f)^7 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f \left (d e (7 m+12)-c f \left (-m^2+2 m+6\right )\right )+b^2 \left (c^2 f^2 \left (m^2-9 m+20\right )-2 c d e f (26-7 m)+38 d^2 e^2\right )\right )}{120 (e+f x)^4 (b e-a f)^3 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (b (8 d e-c f (5-m))-a d f (m+3))}{30 (e+f x)^5 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{6 (e+f x)^6 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(6*(b*e - a*f)*(d*e - c*f)*(e + f*x)^6) - (f*(b*(8*d*e - c*f*(5 - m))
- a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(30*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^5) - (f*(a^2*
d^2*f^2*(6 + 5*m + m^2) - 2*a*b*d*f*(d*e*(12 + 7*m) - c*f*(6 + 2*m - m^2)) + b^2*(38*d^2*e^2 - 2*c*d*e*f*(26 -
7*m) + c^2*f^2*(20 - 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(120*(b*e - a*f)^3*(d*e - c*f)^3*(e +
f*x)^4) + ((b*c - a*d)^3*(3*a^2*b*d^2*f^2*(6*d*e - c*f*(3 - 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)*(30*d^2*e^2 - 12*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)) + b^3*(120*d^3*e^3
- 90*c*d^2*e^2*f*(3 - m) + 18*c^2*d*e*f^2*(12 - 7*m + m^2) - c^3*f^3*(60 - 47*m + 12*m^2 - m^3)))*(a + b*x)^(
1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])
/(120*(b*e - a*f)^7*(d*e - c*f)^3*(1 + m))

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Rubi [A]  time = 1.02578, antiderivative size = 540, 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} \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (6 d e-c f (3-m))-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-7 m+12\right )-12 c d e f (3-m)+30 d^2 e^2\right )+b^3 \left (18 c^2 d e f^2 \left (m^2-7 m+12\right )-c^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-90 c d^2 e^2 f (3-m)+120 d^3 e^3\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{120 (m+1) (b e-a f)^7 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f \left (d e (7 m+12)-c f \left (-m^2+2 m+6\right )\right )+b^2 \left (c^2 f^2 \left (m^2-9 m+20\right )-2 c d e f (26-7 m)+38 d^2 e^2\right )\right )}{120 (e+f x)^4 (b e-a f)^3 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+3)-b c f (5-m)+8 b d e)}{30 (e+f x)^5 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{6 (e+f x)^6 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^7,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(6*(b*e - a*f)*(d*e - c*f)*(e + f*x)^6) - (f*(8*b*d*e - b*c*f*(5 - m)
- a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(30*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^5) - (f*(a^2*
d^2*f^2*(6 + 5*m + m^2) - 2*a*b*d*f*(d*e*(12 + 7*m) - c*f*(6 + 2*m - m^2)) + b^2*(38*d^2*e^2 - 2*c*d*e*f*(26 -
7*m) + c^2*f^2*(20 - 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(120*(b*e - a*f)^3*(d*e - c*f)^3*(e +
f*x)^4) + ((b*c - a*d)^3*(3*a^2*b*d^2*f^2*(6*d*e - c*f*(3 - 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)*(30*d^2*e^2 - 12*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)) + b^3*(120*d^3*e^3
- 90*c*d^2*e^2*f*(3 - m) + 18*c^2*d*e*f^2*(12 - 7*m + m^2) - c^3*f^3*(60 - 47*m + 12*m^2 - m^3)))*(a + b*x)^(
1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])
/(120*(b*e - a*f)^7*(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)^{2-m}}{(e+f x)^7} \, dx &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{6 (b e-a f) (d e-c f) (e+f x)^6}-\frac{\int \frac{(a+b x)^m (c+d x)^{2-m} (-b (6 d e-c f (5-m))+a d f (3+m)+2 b d f x)}{(e+f x)^6} \, dx}{6 (b e-a f) (d e-c f)}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{6 (b e-a f) (d e-c f) (e+f x)^6}-\frac{f (8 b d e-b c f (5-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{30 (b e-a f)^2 (d e-c f)^2 (e+f x)^5}+\frac{\int \frac{(a+b x)^m (c+d x)^{2-m} \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-a b d f \left (d e (21+13 m)-2 c f \left (6+2 m-m^2\right )\right )+b^2 \left (30 d^2 e^2-c d e f (47-13 m)+c^2 f^2 \left (20-9 m+m^2\right )\right )-b d f (8 b d e-b c f (5-m)-a d f (3+m)) x\right )}{(e+f x)^5} \, dx}{30 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{6 (b e-a f) (d e-c f) (e+f x)^6}-\frac{f (8 b d e-b c f (5-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{30 (b e-a f)^2 (d e-c f)^2 (e+f x)^5}-\frac{f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f \left (d e (12+7 m)-c f \left (6+2 m-m^2\right )\right )+b^2 \left (38 d^2 e^2-2 c d e f (26-7 m)+c^2 f^2 \left (20-9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{3-m}}{120 (b e-a f)^3 (d e-c f)^3 (e+f x)^4}-\frac{\int \frac{\left (-3 a^2 b d^2 f^2 (6 d e-c f (3-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 (30 d^2 e^2-12 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )-b^3 \left (120 d^3 e^3-90 c d^2 e^2 f (3-m)+18 c^2 d e f^2 \left (12-7 m+m^2\right )-c^3 f^3 \left (60-47 m+12 m^2-m^3\right )\right )\right ) (a+b x)^m (c+d x)^{2-m}}{(e+f x)^4} \, dx}{120 (b e-a f)^3 (d e-c f)^3}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{6 (b e-a f) (d e-c f) (e+f x)^6}-\frac{f (8 b d e-b c f (5-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{30 (b e-a f)^2 (d e-c f)^2 (e+f x)^5}-\frac{f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f \left (d e (12+7 m)-c f \left (6+2 m-m^2\right )\right )+b^2 \left (38 d^2 e^2-2 c d e f (26-7 m)+c^2 f^2 \left (20-9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{3-m}}{120 (b e-a f)^3 (d e-c f)^3 (e+f x)^4}+\frac{\left (3 a^2 b d^2 f^2 (6 d e-c f (3-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 (30 d^2 e^2-12 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )+b^3 \left (120 d^3 e^3-90 c d^2 e^2 f (3-m)+18 c^2 d e f^2 \left (12-7 m+m^2\right )-c^3 f^3 \left (60-47 m+12 m^2-m^3\right )\right )\right ) \int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^4} \, dx}{120 (b e-a f)^3 (d e-c f)^3}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{6 (b e-a f) (d e-c f) (e+f x)^6}-\frac{f (8 b d e-b c f (5-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{30 (b e-a f)^2 (d e-c f)^2 (e+f x)^5}-\frac{f \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f \left (d e (12+7 m)-c f \left (6+2 m-m^2\right )\right )+b^2 \left (38 d^2 e^2-2 c d e f (26-7 m)+c^2 f^2 \left (20-9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{3-m}}{120 (b e-a f)^3 (d e-c f)^3 (e+f x)^4}+\frac{(b c-a d)^3 \left (3 a^2 b d^2 f^2 (6 d e-c f (3-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 (30 d^2 e^2-12 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )+b^3 \left (120 d^3 e^3-90 c d^2 e^2 f (3-m)+18 c^2 d e f^2 \left (12-7 m+m^2\right )-c^3 f^3 \left (60-47 m+12 m^2-m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (4,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{120 (b e-a f)^7 (d e-c f)^3 (1+m)}\\ \end{align*}

Mathematica [A]  time = 5.90877, size = 482, normalized size = 0.89 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-\frac{(e+f x)^2 \left (f (m+1) (c+d x)^4 (b e-a f)^4 \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f \left (c f \left (m^2-2 m-6\right )+d e (7 m+12)\right )+b^2 \left (c^2 f^2 \left (m^2-9 m+20\right )+2 c d e f (7 m-26)+38 d^2 e^2\right )\right )-(e+f x)^4 (b c-a d)^3 \left (3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (c f (m-3)+6 d e)-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-7 m+12\right )+12 c d e f (m-3)+30 d^2 e^2\right )+b^3 \left (18 c^2 d e f^2 \left (m^2-7 m+12\right )+c^3 f^3 \left (m^3-12 m^2+47 m-60\right )+90 c d^2 e^2 f (m-3)+120 d^3 e^3\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(m+1) (b e-a f)^6 (d e-c f)^2}+\frac{4 f (c+d x)^4 (e+f x) (a d f (m+3)-b (c f (m-5)+8 d e))}{(b e-a f) (d e-c f)}-20 f (c+d x)^4\right )}{120 (e+f x)^6 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^7,x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(-20*f*(c + d*x)^4 + (4*f*(-(b*(8*d*e + c*f*(-5 + m))) + a*d*f*(3 + m))*
(c + d*x)^4*(e + f*x))/((b*e - a*f)*(d*e - c*f)) - ((e + f*x)^2*(f*(b*e - a*f)^4*(1 + m)*(a^2*d^2*f^2*(6 + 5*m
+ m^2) + b^2*(38*d^2*e^2 + 2*c*d*e*f*(-26 + 7*m) + c^2*f^2*(20 - 9*m + m^2)) - 2*a*b*d*f*(d*e*(12 + 7*m) + c*
f*(-6 - 2*m + m^2)))*(c + d*x)^4 - (b*c - a*d)^3*(3*a^2*b*d^2*f^2*(6*d*e + c*f*(-3 + 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)*(30*d^2*e^2 + 12*c*d*e*f*(-3 + m) + c^2*f^2*(12 - 7*m
+ m^2)) + b^3*(120*d^3*e^3 + 90*c*d^2*e^2*f*(-3 + m) + 18*c^2*d*e*f^2*(12 - 7*m + m^2) + c^3*f^3*(-60 + 47*m -
12*m^2 + m^3)))*(e + f*x)^4*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)
)]))/((b*e - a*f)^6*(d*e - c*f)^2*(1 + m))))/(120*(b*e - a*f)*(d*e - c*f)*(e + f*x)^6)

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Maple [F]  time = 0.614, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{7}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,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 + 2}}{{\left (f x + e\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7, 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 + 2}}{f^{7} x^{7} + 7 \, e f^{6} x^{6} + 21 \, e^{2} f^{5} x^{5} + 35 \, e^{3} f^{4} x^{4} + 35 \, e^{4} f^{3} x^{3} + 21 \, e^{5} f^{2} x^{2} + 7 \, e^{6} f x + e^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^7*x^7 + 7*e*f^6*x^6 + 21*e^2*f^5*x^5 + 35*e^3*f^4*x^4 + 35*e^4*f^3*
x^3 + 21*e^5*f^2*x^2 + 7*e^6*f*x + e^7), 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)**(2-m)/(f*x+e)**7,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 + 2}}{{\left (f x + e\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7, x)