\(\int \frac {x (a+b x)^n}{(c+d x)^2} \, dx\) [563]

Optimal result

Integrand size = 16, antiderivative size = 95 \[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+n}}{b d n (c+d x)}+\frac {(a d-b c (1+n)) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d (b c-a d)^2 n (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+n} \left (\frac {c (-b c+a d)}{c+d x}+\frac {(-a d+b c (1+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{d (b c-a d)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

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

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x*(a + b*x)**n/(c + d*x)**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx=\text {too large to display} \] Input:

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

Output:

((a + b*x)**n*a*c + (a + b*x)**n*a*d*x - (a + b*x)**n*b*c*n*x + int(((a + 
b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 
2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + 
a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3) 
,x)*a**3*c*d**3*n + int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + 
a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2 
*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2*c**2* 
d*n*x**2 - b**2*c*d**2*n*x**3),x)*a**3*d**4*n*x - 2*int(((a + b*x)**n*x)/( 
a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d 
*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x** 
3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a**2*b*c 
**2*d**2*n**2 - 2*int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a* 
*2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n 
*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2*c**2*d* 
n*x**2 - b**2*c*d**2*n*x**3),x)*a**2*b*c**2*d**2*n - 2*int(((a + b*x)**n*x 
)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c** 
2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3* 
x**3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a**2* 
b*c*d**3*n**2*x - 2*int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + 
a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b*c*d...