3.31.77 \(\int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx\) [3077]

3.31.77.1 Optimal result

Integrand size = 26, antiderivative size = 283 \[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (b (3 d e-c f (1-m))-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\left (2 a b d f (1+m) (2 d e+c f m)-b^2 \left (2 d^2 e^2+4 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (2+3 m+m^2\right )\right ) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b e-a f)^2 (d e-c f)^3 m} \]

output

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

3.31.77.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (-\frac {2 d}{(e+f x)^2}-\frac {f (-a d f (2+m)+b (2 d e+c f m)) (c+d x)}{(b e-a f) (d e-c f) (e+f x)^2}-\frac {(b c-a d) \left (-2 a b d f (1+m) (2 d e+c f m)+b^2 \left (2 d^2 e^2+4 c d e f m+c^2 f^2 (-1+m) m\right )+a^2 d^2 f^2 \left (2+3 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (-d e+c f) (1+m) (c+d x)}\right )}{2 (b c-a d) (-d e+c f) m} \]

input

Integrate[((a + b*x)^m*(c + d*x)^(-1 - m))/(e + f*x)^3,x]
 

output

((a + b*x)^(1 + m)*((-2*d)/(e + f*x)^2 - (f*(-(a*d*f*(2 + m)) + b*(2*d*e + 
 c*f*m))*(c + d*x))/((b*e - a*f)*(d*e - c*f)*(e + f*x)^2) - ((b*c - a*d)*( 
-2*a*b*d*f*(1 + m)*(2*d*e + c*f*m) + b^2*(2*d^2*e^2 + 4*c*d*e*f*m + c^2*f^ 
2*(-1 + m)*m) + a^2*d^2*f^2*(2 + 3*m + m^2))*Hypergeometric2F1[2, 1 + m, 2 
 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^3*(-( 
d*e) + c*f)*(1 + m)*(c + d*x))))/(2*(b*c - a*d)*(-(d*e) + c*f)*m*(c + d*x) 
^m)
 

3.31.77.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {114, 25, 168, 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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

input

Int[((a + b*x)^m*(c + d*x)^(-1 - m))/(e + f*x)^3,x]
 

output

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

3.31.77.3.1 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]
 

3.31.77.4 Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-1-m}}{\left (f x +e \right )^{3}}d x\]

input

int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x)
 

output

int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x)
 

3.31.77.5 Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x, algorithm="fricas")
 

output

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

3.31.77.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=\text {Timed out} \]

input

integrate((b*x+a)**m*(d*x+c)**(-1-m)/(f*x+e)**3,x)
 

output

Timed out
 

3.31.77.7 Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x, algorithm="maxima")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3, x)
 

3.31.77.8 Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x, algorithm="giac")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3, x)
 

3.31.77.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^{m+1}} \,d x \]

input

int((a + b*x)^m/((e + f*x)^3*(c + d*x)^(m + 1)),x)
 

output

int((a + b*x)^m/((e + f*x)^3*(c + d*x)^(m + 1)), x)