\(\int \frac {(e x)^m (c+d x)}{(a x^n+b x^{1+n})^2} \, dx\) [368]

Optimal result

Integrand size = 26, antiderivative size = 94 \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\frac {d x^{1-2 n} (e x)^m}{b (m-2 n) (a+b x)}+\frac {\left (\frac {c}{1+m-2 n}-\frac {a d}{b m-2 b n}\right ) x^{1-2 n} (e x)^m \operatorname {Hypergeometric2F1}\left (2,1+m-2 n,2+m-2 n,-\frac {b x}{a}\right )}{a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\frac {x^{1-2 n} (e x)^m \left (\frac {a (b c-a d)}{a+b x}+\frac {(-b c (m-2 n)+a d (1+m-2 n)) \operatorname {Hypergeometric2F1}\left (1,1+m-2 n,2+m-2 n,-\frac {b x}{a}\right )}{1+m-2 n}\right )}{a^2 b} \] Input:

Integrate[((e*x)^m*(c + d*x))/(a*x^n + b*x^(1 + n))^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2027, 30, 87, 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.

\(\displaystyle \int \frac {(c+d x) (e x)^m}{\left (a x^n+b x^{n+1}\right )^2} \, dx\)

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 74

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

Input:

Int[((e*x)^m*(c + d*x))/(a*x^n + b*x^(1 + n))^2,x]
 

Output:

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

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

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 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]))))
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )}{\left (a \,x^{n}+b \,x^{1+n}\right )^{2}}d x\]

Input:

int((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^2,x)
 

Output:

int((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^2,x)
 

Fricas [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{n + 1} + a x^{n}\right )}^{2}} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int \frac {\left (e x\right )^{m} \left (c + d x\right )}{\left (a x^{n} + b x^{n + 1}\right )^{2}}\, dx \] Input:

integrate((e*x)**m*(d*x+c)/(a*x**n+b*x**(1+n))**2,x)
 

Output:

Integral((e*x)**m*(c + d*x)/(a*x**n + b*x**(n + 1))**2, x)
 

Maxima [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{n + 1} + a x^{n}\right )}^{2}} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^2,x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(e*x)^m/(b*x^(n + 1) + a*x^n)^2, x)
 

Giac [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{n + 1} + a x^{n}\right )}^{2}} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)*(e*x)^m/(b*x^(n + 1) + a*x^n)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (c+d\,x\right )}{{\left (a\,x^n+b\,x^{n+1}\right )}^2} \,d x \] Input:

int(((e*x)^m*(c + d*x))/(a*x^n + b*x^(n + 1))^2,x)
 

Output:

int(((e*x)^m*(c + d*x))/(a*x^n + b*x^(n + 1))^2, x)
 

Reduce [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\text {too large to display} \] Input:

int((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^2,x)
 

Output:

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