\(\int \frac {(a+b x)^m}{(c+d x) (e+f x)^2} \, dx\) [1685]

Optimal result

Integrand size = 22, antiderivative size = 187 \[ \int \frac {(a+b x)^m}{(c+d x) (e+f x)^2} \, dx=-\frac {f (a+b x)^{1+m}}{(b e-a f) (d e-c f) (e+f x)}+\frac {d^2 (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (d e-c f)^2 (1+m)}+\frac {f (a d f-b d e (1-m)-b c f m) (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f)^2 (d e-c f)^2 (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^m}{(c+d x) (e+f x)^2} \, dx=\frac {(a+b x)^{1+m} \left (-\frac {f}{e+f x}-\frac {d^2 (b e-a f) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{(b c-a d) (-d e+c f) (1+m)}+\frac {f (a d f+b d e (-1+m)-b c f m) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {f (a+b x)}{-b e+a f}\right )}{(b e-a f) (d e-c f) (1+m)}\right )}{(b e-a f) (d e-c f)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {114, 174, 78}

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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

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 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^m}{(c+d x) (e+f x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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