\(\int \frac {(d x)^m}{(a x^q+b x^r+c x^s)^3} \, dx\) [6]

Optimal result

Integrand size = 24, antiderivative size = 1 \[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=0 \] Output:

0
 

Mathematica [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=\text {\$Aborted} \] Input:

Integrate[(d*x)^m/(a*x^q + b*x^r + c*x^s)^3,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.

\(\displaystyle \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int \frac {x^{-3 s} (d x)^m}{\left (a x^{q-s}+b x^{r-s}+c\right )^3}dx\)

\(\Big \downarrow \) 30

\(\displaystyle x^{-m} (d x)^m \int \frac {x^{m-3 s}}{\left (a x^{q-s}+b x^{r-s}+c\right )^3}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle x^{-m} (d x)^m \int \frac {x^{m-3 s}}{\left (a x^{q-s}+b x^{r-s}+c\right )^3}dx\)

Input:

Int[(d*x)^m/(a*x^q + b*x^r + c*x^s)^3,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]

\[\int \frac {\left (d x \right )^{m}}{\left (a \,x^{q}+b \,x^{r}+c \,x^{s}\right )^{3}}d x\]

Input:

int((d*x)^m/(a*x^q+b*x^r+c*x^s)^3,x)
 

Output:

int((d*x)^m/(a*x^q+b*x^r+c*x^s)^3,x)
 

Fricas [F]

\[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (a x^{q} + b x^{r} + c x^{s}\right )}^{3}} \,d x } \] Input:

integrate((d*x)^m/(a*x^q+b*x^r+c*x^s)^3,x, algorithm="fricas")
 

Output:

integral((d*x)^m/(3*a*b^2*x^q*x^(2*r) + 3*a^2*b*x^(2*q)*x^r + a^3*x^(3*q) 
+ b^3*x^(3*r) + c^3*x^(3*s) + 3*(a*c^2*x^q + b*c^2*x^r)*x^(2*s) + 3*(2*a*b 
*c*x^q*x^r + a^2*c*x^(2*q) + b^2*c*x^(2*r))*x^s), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x)**m/(a*x**q+b*x**r+c*x**s)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (a x^{q} + b x^{r} + c x^{s}\right )}^{3}} \,d x } \] Input:

integrate((d*x)^m/(a*x^q+b*x^r+c*x^s)^3,x, algorithm="maxima")
 

Output:

-1/2*((m*(q - s) - q*(3*s - 1) + 3*s^2 - s)*a^2*d^m*x*e^(m*log(x) + 2*q*lo 
g(x)) + (m*(q + r - 2*s) - q^2 + q*(2*r - 3*s + 1) - r^2 - r*(3*s - 1) + 6 
*s^2 - 2*s)*a*b*d^m*x*e^(m*log(x) + q*log(x) + r*log(x)) + (m*(r - s) - r* 
(3*s - 1) + 3*s^2 - s)*b^2*d^m*x*e^(m*log(x) + 2*r*log(x)) + ((m*(q - s) - 
 q^2 - q*(s - 1) + 2*s^2 - s)*a*c*d^m*x*e^(m*log(x) + q*log(x)) + (m*(r - 
s) - r^2 - r*(s - 1) + 2*s^2 - s)*b*c*d^m*x*e^(m*log(x) + r*log(x)))*x^s)/ 
((q^3 - 3*q^2*s + 3*q*s^2 - s^3)*a^5*x^(5*q) + (r^3 - 3*r^2*s + 3*r*s^2 - 
s^3)*b^5*x^(5*r) + (2*q^3 + 3*q^2*(r - 3*s) + 3*r*s^2 - 5*s^3 - 6*(r*s - 2 
*s^2)*q)*a^4*b*e^(4*q*log(x) + r*log(x)) + (q^3 + 3*q^2*(2*r - 3*s) - 3*r^ 
2*s + 12*r*s^2 - 10*s^3 + 3*(r^2 - 6*r*s + 6*s^2)*q)*a^3*b^2*e^(3*q*log(x) 
 + 2*r*log(x)) + (3*q^2*(r - s) + r^3 - 9*r^2*s + 18*r*s^2 - 10*s^3 + 6*(r 
^2 - 3*r*s + 2*s^2)*q)*a^2*b^3*e^(2*q*log(x) + 3*r*log(x)) + (2*r^3 - 9*r^ 
2*s + 12*r*s^2 - 5*s^3 + 3*(r^2 - 2*r*s + s^2)*q)*a*b^4*e^(q*log(x) + 4*r* 
log(x)) + ((q^3 - 3*q^2*s + 3*q*s^2 - s^3)*a^3*c^2*x^(3*q) + (r^3 - 3*r^2* 
s + 3*r*s^2 - s^3)*b^3*c^2*x^(3*r) + 3*(q^2*(r - s) + r*s^2 - s^3 - 2*(r*s 
 - s^2)*q)*a^2*b*c^2*e^(2*q*log(x) + r*log(x)) - 3*(r^2*s - 2*r*s^2 + s^3 
- (r^2 - 2*r*s + s^2)*q)*a*b^2*c^2*e^(q*log(x) + 2*r*log(x)))*x^(2*s) + 2* 
((q^3 - 3*q^2*s + 3*q*s^2 - s^3)*a^4*c*x^(4*q) + (r^3 - 3*r^2*s + 3*r*s^2 
- s^3)*b^4*c*x^(4*r) + (q^3 + 3*q^2*(r - 2*s) + 3*r*s^2 - 4*s^3 - 3*(2*r*s 
 - 3*s^2)*q)*a^3*b*c*e^(3*q*log(x) + r*log(x)) + 3*(q^2*(r - s) - r^2*s...
 

Giac [F]

\[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (a x^{q} + b x^{r} + c x^{s}\right )}^{3}} \,d x } \] Input:

integrate((d*x)^m/(a*x^q+b*x^r+c*x^s)^3,x, algorithm="giac")
 

Output:

integrate((d*x)^m/(a*x^q + b*x^r + c*x^s)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a\,x^q+b\,x^r+c\,x^s\right )}^3} \,d x \] Input:

int((d*x)^m/(a*x^q + b*x^r + c*x^s)^3,x)
 

Output:

int((d*x)^m/(a*x^q + b*x^r + c*x^s)^3, x)
 

Reduce [F]

\[ \int \frac {(d x)^m}{\left (a x^q+b x^r+c x^s\right )^3} \, dx=d^{m} \left (\int \frac {x^{m}}{x^{3 q} a^{3}+3 x^{2 q +r} a^{2} b +3 x^{2 q +s} a^{2} c +3 x^{q +2 r} a \,b^{2}+6 x^{q +r +s} a b c +3 x^{q +2 s} a \,c^{2}+x^{3 r} b^{3}+3 x^{2 r +s} b^{2} c +3 x^{r +2 s} b \,c^{2}+x^{3 s} c^{3}}d x \right ) \] Input:

int((d*x)^m/(a*x^q+b*x^r+c*x^s)^3,x)
 

Output:

d**m*int(x**m/(x**(3*q)*a**3 + 3*x**(2*q + r)*a**2*b + 3*x**(2*q + s)*a**2 
*c + 3*x**(q + 2*r)*a*b**2 + 6*x**(q + r + s)*a*b*c + 3*x**(q + 2*s)*a*c** 
2 + x**(3*r)*b**3 + 3*x**(2*r + s)*b**2*c + 3*x**(r + 2*s)*b*c**2 + x**(3* 
s)*c**3),x)