\(\int \frac {x^m}{(a+b x) (c+d x)^2} \, dx\) [533]

Optimal result

Integrand size = 18, antiderivative size = 125 \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=-\frac {d x^{1+m}}{c (b c-a d) (c+d x)}+\frac {b^2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a (b c-a d)^2 (1+m)}-\frac {d (b c (1-m)+a d m) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{c^2 (b c-a d)^2 (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\frac {x^{1+m} \left (b^2 c^2 (c+d x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+a d \left (-c (b c-a d) (1+m)+(b c (-1+m)-a d m) (c+d x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )\right )\right )}{a c^2 (b c-a d)^2 (1+m) (c+d x)} \] Input:

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

Output:

(x^(1 + m)*(b^2*c^2*(c + d*x)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a 
)] + a*d*(-(c*(b*c - a*d)*(1 + m)) + (b*c*(-1 + m) - a*d*m)*(c + d*x)*Hype 
rgeometric2F1[1, 1 + m, 2 + m, -((d*x)/c)])))/(a*c^2*(b*c - a*d)^2*(1 + m) 
*(c + d*x))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {114, 25, 174, 74}

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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

-((d*x^(1 + m))/(c*(b*c - a*d)*(c + d*x))) + ((b^2*c*x^(1 + m)*Hypergeomet 
ric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(a*(b*c - a*d)*(1 + m)) - (d*(a*d*m + 
 b*(c - c*m))*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*x)/c)])/(c 
*(b*c - a*d)*(1 + m)))/(c*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

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

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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