\(\int \frac {(d x)^m}{a x^q+b x^r+c x^s} \, dx\) [4]

Optimal result

Integrand size = 24, antiderivative size = 1 \[ \int \frac {(d x)^m}{a x^q+b x^r+c x^s} \, dx=0 \] Output:

0
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2028

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

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 2028

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

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{a \,x^{q}+b \,x^{r}+c \,x^{s}}d x\]

Input:

int((d*x)^m/(a*x^q+b*x^r+c*x^s),x)
 

Output:

int((d*x)^m/(a*x^q+b*x^r+c*x^s),x)
 

Fricas [F]

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

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

Output:

integral((d*x)^m/(a*x^q + b*x^r + c*x^s), x)
 

Sympy [F]

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

integrate((d*x)**m/(a*x**q+b*x**r+c*x**s),x)
 

Output:

Integral((d*x)**m/(a*x**q + b*x**r + c*x**s), x)
 

Maxima [F]

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

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

Output:

integrate((d*x)^m/(a*x^q + b*x^r + c*x^s), x)
 

Giac [F]

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

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

Output:

integrate((d*x)^m/(a*x^q + b*x^r + c*x^s), x)
 

Mupad [F(-1)]

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

int((d*x)^m/(a*x^q + b*x^r + c*x^s),x)
 

Output:

int((d*x)^m/(a*x^q + b*x^r + c*x^s), x)
 

Reduce [F]

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

int((d*x)^m/(a*x^q+b*x^r+c*x^s),x)
 

Output:

d**m*int(x**m/(x**q*a + x**r*b + x**s*c),x)