\(\int \frac {(d x)^m}{(a x^q+b x^r+c x^s)^2} \, dx\) [5]

Optimal result

Integrand size = 24, antiderivative size = 1 \[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^2} \, dx=0 \] Output:

0
 

Mathematica [F(-1)]

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

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

Output:

$Aborted
 

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.

\(\displaystyle \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^2} \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int \frac {x^{-2 s} (d x)^m}{\left (a x^{q-s}+b x^{r-s}+c\right )^2}dx\)

\(\Big \downarrow \) 30

\(\displaystyle x^{-m} (d x)^m \int \frac {x^{m-2 s}}{\left (a x^{q-s}+b x^{r-s}+c\right )^2}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle x^{-m} (d x)^m \int \frac {x^{m-2 s}}{\left (a x^{q-s}+b x^{r-s}+c\right )^2}dx\)

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x)^m/(2*a*b*x^q*x^r + a^2*x^(2*q) + b^2*x^(2*r) + c^2*x^(2*s) 
+ 2*(a*c*x^q + b*c*x^r)*x^s), x)
 

Sympy [F]

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

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

Output:

Integral((d*x)**m/(a*x**q + b*x**r + c*x**s)**2, x)
 

Maxima [F]

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

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

Output:

-d^m*x*x^m/(a^2*(q - s)*x^(2*q) + b^2*(r - s)*x^(2*r) + a*b*(q + r - 2*s)* 
e^(q*log(x) + r*log(x)) + (a*c*(q - s)*x^q + b*c*(r - s)*x^r)*x^s) + integ 
rate(((m*(q - s) - q^2 + s^2 + q - s)*a*d^m*e^(m*log(x) + q*log(x)) + (m*( 
r - s) - r^2 + s^2 + r - s)*b*d^m*e^(m*log(x) + r*log(x)))/((q^2 - 2*q*s + 
 s^2)*a^3*x^(3*q) + (r^2 - 2*r*s + s^2)*b^3*x^(3*r) + (q^2 + 2*q*(r - 2*s) 
 - 2*r*s + 3*s^2)*a^2*b*e^(2*q*log(x) + r*log(x)) + (2*q*(r - s) + r^2 - 4 
*r*s + 3*s^2)*a*b^2*e^(q*log(x) + 2*r*log(x)) + ((q^2 - 2*q*s + s^2)*a^2*c 
*x^(2*q) + (r^2 - 2*r*s + s^2)*b^2*c*x^(2*r) + 2*(q*(r - s) - r*s + s^2)*a 
*b*c*e^(q*log(x) + r*log(x)))*x^s), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d**m*int(x**m/(x**(2*q)*a**2 + 2*x**(q + r)*a*b + 2*x**(q + s)*a*c + x**(2 
*r)*b**2 + 2*x**(r + s)*b*c + x**(2*s)*c**2),x)