\(\int \frac {(e x)^m (c+d x)}{a x^n+b x^{1+n}} \, dx\) [367]

Optimal result

Integrand size = 26, antiderivative size = 82 \[ \int \frac {(e x)^m (c+d x)}{a x^n+b x^{1+n}} \, dx=\frac {d x^{1-n} (e x)^m}{b (1+m-n)}+\frac {(b c-a d) x^{1-n} (e x)^m \operatorname {Hypergeometric2F1}\left (1,1+m-n,2+m-n,-\frac {b x}{a}\right )}{a b (1+m-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74 \[ \int \frac {(e x)^m (c+d x)}{a x^n+b x^{1+n}} \, dx=\frac {x^{1-n} (e x)^m \left (a d+(b c-a d) \operatorname {Hypergeometric2F1}\left (1,1+m-n,2+m-n,-\frac {b x}{a}\right )\right )}{a b (1+m-n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2027, 30, 90, 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.

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

((e*x)^m*((d*x^(1 + m - n))/(b*(1 + m - n)) + ((b*c - a*d)*x^(1 + m - n)*H 
ypergeometric2F1[1, 1 + m - n, 2 + m - n, -((b*x)/a)])/(a*b*(1 + m - 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 90

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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