3.3110 \(\int \frac{(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx\)

Optimal. Leaf size=557 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (m+3) (m+4) (b c-a d)^3 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (c^2 d e f^2 \left (m^2+11 m+46\right )-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4 (d e-c f)^4}-\frac{f^4 (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (d e-c f)^5}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)+b (3 d e-c f (m+7)))}{(m+3) (m+4) (b c-a d)^2 (d e-c f)^2} \]

[Out]

(d*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/((b*c - a*d)*(d*e - c*f)*(4 + m)) + (d*(a*d*f*(4 + m) + b*(3*d*e - c*
f*(7 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((b*c - a*d)^2*(d*e - c*f)^2*(3 + m)*(4 + m)) + (d*(a^2*d^2*
f^2*(12 + 7*m + m^2) + 2*a*b*d*f*(4 + m)*(d*e - c*f*(4 + m)) + b^2*(6*d^2*e^2 - 2*c*d*e*f*(10 + m) + c^2*f^2*(
26 + 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)^3*(d*e - c*f)^3*(2 + m)*(3 + m)*(4 + m))
+ (d*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3) + a^2*b*d^2*f^2*(12 + 7*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2*d*f
*(4 + m)*(2*d^2*e^2 - 2*c*d*e*f*(5 + m) + c^2*f^2*(26 + 17*m + 3*m^2)) + b^3*(6*d^3*e^3 - 2*c*d^2*e^2*f*(13 +
m) + c^2*d*e*f^2*(46 + 11*m + m^2) - c^3*f^3*(50 + 35*m + 10*m^2 + m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)
)/((b*c - a*d)^4*(d*e - c*f)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (f^4*(a + b*x)^m*Hypergeometric2F1[1, -m, 1
- m, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((d*e - c*f)^5*m*(c + d*x)^m)

________________________________________________________________________________________

Rubi [A]  time = 1.20735, antiderivative size = 569, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {129, 155, 12, 131} \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (m+3) (m+4) (b c-a d)^3 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (c^2 d e f^2 \left (m^2+11 m+46\right )-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4 (d e-c f)^4}+\frac{f^4 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (m+4) (b c-a d)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^m*(c + d*x)^(-5 - m))/(e + f*x),x]

[Out]

(d*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/((b*c - a*d)*(d*e - c*f)*(4 + m)) + (d*(3*b*d*e + a*d*f*(4 + m) - b*c
*f*(7 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((b*c - a*d)^2*(d*e - c*f)^2*(3 + m)*(4 + m)) + (d*(a^2*d^2*
f^2*(12 + 7*m + m^2) + 2*a*b*d*f*(4 + m)*(d*e - c*f*(4 + m)) + b^2*(6*d^2*e^2 - 2*c*d*e*f*(10 + m) + c^2*f^2*(
26 + 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)^3*(d*e - c*f)^3*(2 + m)*(3 + m)*(4 + m))
+ (d*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3) + a^2*b*d^2*f^2*(12 + 7*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2*d*f
*(4 + m)*(2*d^2*e^2 - 2*c*d*e*f*(5 + m) + c^2*f^2*(26 + 17*m + 3*m^2)) + b^3*(6*d^3*e^3 - 2*c*d^2*e^2*f*(13 +
m) + c^2*d*e*f^2*(46 + 11*m + m^2) - c^3*f^3*(50 + 35*m + 10*m^2 + m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)
)/((b*c - a*d)^4*(d*e - c*f)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) + (f^4*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hy
pergeometric2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)*(d*e - c*f)^4*
(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 155

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] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

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)^{-5-m}}{e+f x} \, dx &=\frac{d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac{\int \frac{(a+b x)^m (c+d x)^{-4-m} (3 b d e-b c f (4+m)+a d f (4+m)+3 b d f x)}{e+f x} \, dx}{(b c-a d) (d e-c f) (4+m)}\\ &=\frac{d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac{d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac{\int \frac{(a+b x)^m (c+d x)^{-3-m} \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (3+m))+b^2 \left (6 d^2 e^2-2 c d e f (7+m)+c^2 f^2 \left (12+7 m+m^2\right )\right )+2 b d f (3 b d e+a d f (4+m)-b c f (7+m)) x\right )}{e+f x} \, dx}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}\\ &=\frac{d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac{d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac{\int \frac{(a+b x)^m (c+d x)^{-2-m} \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-3 c f (2+m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (4+m)+3 c^2 f^2 \left (6+5 m+m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (10+m)+c^2 d e f^2 \left (26+9 m+m^2\right )-c^3 f^3 \left (24+26 m+9 m^2+m^3\right )\right )+b d f \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) x\right )}{e+f x} \, dx}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}\\ &=\frac{d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac{d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac{d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}+\frac{\int \frac{(b c-a d)^4 f^4 (1+m) (2+m) (3+m) (4+m) (a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}\\ &=\frac{d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac{d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac{d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}+\frac{f^4 \int \frac{(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(d e-c f)^4}\\ &=\frac{d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac{d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac{d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}+\frac{f^4 (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (1,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f) (d e-c f)^4 (1+m)}\\ \end{align*}

Mathematica [A]  time = 2.35318, size = 525, normalized size = 0.94 \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left (-\frac{(c+d x)^2 \left ((c+d x) \left (f^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) \left (-(b c-a d)^4\right ) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )-d (m+1) (b e-a f) \left (a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (c^2 d e f^2 \left (m^2+11 m+46\right )-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )\right )+d (m+1)^2 (b c-a d) (b e-a f) (d e-c f) \left (-a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (c f (m+4)-d e)+b^2 \left (-\left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )\right )\right )}{(m+1)^2 (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^3}-\frac{d (c+d x) (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (b c-a d) (c f-d e)}+d\right )}{(m+4) (b c-a d) (c f-d e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^m*(c + d*x)^(-5 - m))/(e + f*x),x]

[Out]

-(((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(d - (d*(3*b*d*e + a*d*f*(4 + m) - b*c*f*(7 + m))*(c + d*x))/((b*c - a
*d)*(-(d*e) + c*f)*(3 + m)) - ((c + d*x)^2*(d*(b*c - a*d)*(b*e - a*f)*(d*e - c*f)*(1 + m)^2*(-(a^2*d^2*f^2*(12
+ 7*m + m^2)) + 2*a*b*d*f*(4 + m)*(-(d*e) + c*f*(4 + m)) - b^2*(6*d^2*e^2 - 2*c*d*e*f*(10 + m) + c^2*f^2*(26
+ 9*m + m^2))) + (c + d*x)*(-(d*(b*e - a*f)*(1 + m)*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3) + a^2*b*d^2*f^2*(12
+ 7*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2*d*f*(4 + m)*(2*d^2*e^2 - 2*c*d*e*f*(5 + m) + c^2*f^2*(26 + 17*m +
3*m^2)) + b^3*(6*d^3*e^3 - 2*c*d^2*e^2*f*(13 + m) + c^2*d*e*f^2*(46 + 11*m + m^2) - c^3*f^3*(50 + 35*m + 10*m^
2 + m^3)))) - (b*c - a*d)^4*f^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*Hypergeometric2F1[1, 1 + m, 2 + m, ((d*e -
c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])))/((b*c - a*d)^3*(b*e - a*f)*(d*e - c*f)^3*(1 + m)^2*(2 + m)*(3 + m
))))/((b*c - a*d)*(-(d*e) + c*f)*(4 + m)))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-5-m}}{fx+e}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),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 - 5}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), 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 - 5}}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), 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)**(-5-m)/(f*x+e),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 - 5}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), x)