3.31.86 \(\int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx\) [3086]

3.31.86.1 Optimal result

Integrand size = 26, antiderivative size = 432 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\frac {d \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)-c^2 f^2 \left (1-m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b c-a d) (b e-a f)^2 (d e-c f)^3 (1+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (b (4 d e-c f (1-m))-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {f \left (2 a b d f (2+m) (3 d e+c f m)-b^2 \left (6 d^2 e^2+6 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (6+5 m+m^2\right )\right ) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b e-a f)^2 (d e-c f)^4 m} \]

output

1/2*d*(a^2*d^2*f^2*(m^2+5*m+6)+b^2*(2*d^2*e^2+5*c*d*e*f*(1+m)-c^2*f^2*(-m^ 
2+1))-a*b*d*f*(d*e*(9+5*m)+c*f*(2*m^2+5*m+3)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m 
)/(-a*d+b*c)/(-a*f+b*e)^2/(-c*f+d*e)^3/(1+m)-1/2*f*(b*x+a)^(1+m)*(d*x+c)^( 
-1-m)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^2-1/2*f*(b*(4*d*e-c*f*(1-m))-a*d*f*(3+ 
m))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)-1/2*f*( 
2*a*b*d*f*(2+m)*(c*f*m+3*d*e)-b^2*(6*d^2*e^2+6*c*d*e*f*m-c^2*f^2*(1-m)*m)- 
a^2*d^2*f^2*(m^2+5*m+6))*(b*x+a)^m*hypergeom([1, -m],[1-m],(-a*f+b*e)*(d*x 
+c)/(-c*f+d*e)/(b*x+a))/(-a*f+b*e)^2/(-c*f+d*e)^4/m/((d*x+c)^m)
 

3.31.86.2 Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (f (b e-a f)^2 (d e-c f) (2 b d e+b c f (1+m)-a d f (3+m))+\frac {2 d (b e-a f)^3 (d e-c f)^2}{c+d x}+(e+f x) \left (f (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)+c^2 f^2 \left (-1+m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right )-\frac {(b c-a d) f \left (-2 a b d f (2+m) (3 d e+c f m)+b^2 \left (6 d^2 e^2+6 c d e f m+c^2 f^2 (-1+m) m\right )+a^2 d^2 f^2 \left (6+5 m+m^2\right )\right ) (e+f x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{c+d x}\right )\right )}{2 (b c-a d) (b e-a f)^3 (d e-c f)^2 (-d e+c f) (1+m) (e+f x)^2} \]

input

Integrate[((a + b*x)^m*(c + d*x)^(-2 - m))/(e + f*x)^3,x]
 

output

-1/2*((a + b*x)^(1 + m)*(f*(b*e - a*f)^2*(d*e - c*f)*(2*b*d*e + b*c*f*(1 + 
 m) - a*d*f*(3 + m)) + (2*d*(b*e - a*f)^3*(d*e - c*f)^2)/(c + d*x) + (e + 
f*x)*(f*(b*e - a*f)*(a^2*d^2*f^2*(6 + 5*m + m^2) + b^2*(2*d^2*e^2 + 5*c*d* 
e*f*(1 + m) + c^2*f^2*(-1 + m^2)) - a*b*d*f*(d*e*(9 + 5*m) + c*f*(3 + 5*m 
+ 2*m^2))) - ((b*c - a*d)*f*(-2*a*b*d*f*(2 + m)*(3*d*e + c*f*m) + b^2*(6*d 
^2*e^2 + 6*c*d*e*f*m + c^2*f^2*(-1 + m)*m) + a^2*d^2*f^2*(6 + 5*m + m^2))* 
(e + f*x)*Hypergeometric2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e 
 - a*f)*(c + d*x))])/(c + d*x))))/((b*c - a*d)*(b*e - a*f)^3*(d*e - c*f)^2 
*(-(d*e) + c*f)*(1 + m)*(c + d*x)^m*(e + f*x)^2)
 

3.31.86.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {114, 25, 168, 25, 172, 27, 141}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b x)^m (c+d x)^{-m-2}}{(e+f x)^3} \, dx\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {\int -\frac {(a+b x)^m (c+d x)^{-m-2} (2 b d e-b c f (1-m)-a d f (m+3)-2 b d f x)}{(e+f x)^2}dx}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(a+b x)^m (c+d x)^{-m-2} (2 b d e-b c f (1-m)-a d f (m+3)-2 b d f x)}{(e+f x)^2}dx}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {\int -\frac {(a+b x)^m (c+d x)^{-m-2} \left (\left (2 d^2 e^2+c d f (5 m+1) e-c^2 f^2 (1-m) m\right ) b^2-a d f (2 c f m (m+2)+d e (5 m+9)) b-d f (4 b d e-b c f (1-m)-a d f (m+3)) x b+a^2 d^2 f^2 \left (m^2+5 m+6\right )\right )}{e+f x}dx}{(b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)-b c f (1-m)+4 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {(a+b x)^m (c+d x)^{-m-2} \left (\left (2 d^2 e^2+c d f (5 m+1) e-c^2 f^2 (1-m) m\right ) b^2-a d f (2 c f m (m+2)+d e (5 m+9)) b-d f (4 b d e-b c f (1-m)-a d f (m+3)) x b+a^2 d^2 f^2 \left (m^2+5 m+6\right )\right )}{e+f x}dx}{(b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)-b c f (1-m)+4 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\frac {\frac {\int \frac {(b c-a d) f (m+1) \left (-\left (\left (6 d^2 e^2+6 c d f m e-c^2 f^2 (1-m) m\right ) b^2\right )+2 a d f (m+2) (3 d e+c f m) b-a^2 d^2 f^2 \left (m^2+5 m+6\right )\right ) (a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{(m+1) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+5 m+3\right )+d e (5 m+9)\right )+b^2 \left (-c^2 f^2 \left (1-m^2\right )+5 c d e f (m+1)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}}{(b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)-b c f (1-m)+4 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {f \left (-a^2 d^2 f^2 \left (m^2+5 m+6\right )+2 a b d f (m+2) (c f m+3 d e)-\left (b^2 \left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )\right )\right ) \int \frac {(a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{d e-c f}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+5 m+3\right )+d e (5 m+9)\right )+b^2 \left (-c^2 f^2 \left (1-m^2\right )+5 c d e f (m+1)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}}{(b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)-b c f (1-m)+4 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {\frac {\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+5 m+3\right )+d e (5 m+9)\right )+b^2 \left (-c^2 f^2 \left (1-m^2\right )+5 c d e f (m+1)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}-\frac {f (a+b x)^m (c+d x)^{-m} \left (-a^2 d^2 f^2 \left (m^2+5 m+6\right )+2 a b d f (m+2) (c f m+3 d e)-\left (b^2 \left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (d e-c f)^2}}{(b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)-b c f (1-m)+4 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\)

input

Int[((a + b*x)^m*(c + d*x)^(-2 - m))/(e + f*x)^3,x]
 

output

-1/2*(f*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*e - a*f)*(d*e - c*f)*(e 
+ f*x)^2) + (-((f*(4*b*d*e - b*c*f*(1 - m) - a*d*f*(3 + m))*(a + b*x)^(1 + 
 m)*(c + d*x)^(-1 - m))/((b*e - a*f)*(d*e - c*f)*(e + f*x))) + ((d*(a^2*d^ 
2*f^2*(6 + 5*m + m^2) + b^2*(2*d^2*e^2 + 5*c*d*e*f*(1 + m) - c^2*f^2*(1 - 
m^2)) - a*b*d*f*(d*e*(9 + 5*m) + c*f*(3 + 5*m + 2*m^2)))*(a + b*x)^(1 + m) 
*(c + d*x)^(-1 - m))/((b*c - a*d)*(d*e - c*f)*(1 + m)) - (f*(2*a*b*d*f*(2 
+ m)*(3*d*e + c*f*m) - b^2*(6*d^2*e^2 + 6*c*d*e*f*m - c^2*f^2*(1 - m)*m) - 
 a^2*d^2*f^2*(6 + 5*m + m^2))*(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)^2*m*(c + d* 
x)^m))/((b*e - a*f)*(d*e - c*f)))/(2*(b*e - a*f)*(d*e - c*f))
 

3.31.86.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 141

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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, -1]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + Simp[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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

3.31.86.4 Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-2-m}}{\left (f x +e \right )^{3}}d x\]

input

int((b*x+a)^m*(d*x+c)^(-2-m)/(f*x+e)^3,x)
 

output

int((b*x+a)^m*(d*x+c)^(-2-m)/(f*x+e)^3,x)
 

3.31.86.5 Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-2-m)/(f*x+e)^3,x, algorithm="fricas")
 

output

integral((b*x + a)^m*(d*x + c)^(-m - 2)/(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x 
 + e^3), x)
 

3.31.86.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\text {Timed out} \]

input

integrate((b*x+a)**m*(d*x+c)**(-2-m)/(f*x+e)**3,x)
 

output

Timed out
 

3.31.86.7 Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-2-m)/(f*x+e)^3,x, algorithm="maxima")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 2)/(f*x + e)^3, x)
 

3.31.86.8 Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-2-m)/(f*x+e)^3,x, algorithm="giac")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 2)/(f*x + e)^3, x)
 

3.31.86.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^{m+2}} \,d x \]

input

int((a + b*x)^m/((e + f*x)^3*(c + d*x)^(m + 2)),x)
 

output

int((a + b*x)^m/((e + f*x)^3*(c + d*x)^(m + 2)), x)