\(\int \frac {(d+e x)^m}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [388]

Optimal result

Integrand size = 35, antiderivative size = 65 \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {e^3 (d+e x)^{-3+m} \operatorname {Hypergeometric2F1}\left (4,-3+m,-2+m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^4 (3-m)} \] Output:

-e^3*(e*x+d)^(-3+m)*hypergeom([4, -3+m],[-2+m],c*d*(e*x+d)/(-a*e^2+c*d^2)) 
/(-a*e^2+c*d^2)^4/(3-m)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^3 (d+e x)^{-3+m} \operatorname {Hypergeometric2F1}\left (4,-3+m,-2+m,-\frac {c d (d+e x)}{-c d^2+a e^2}\right )}{\left (-c d^2+a e^2\right )^4 (-3+m)} \] Input:

Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
 

Output:

(e^3*(d + e*x)^(-3 + m)*Hypergeometric2F1[4, -3 + m, -2 + m, -((c*d*(d + e 
*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^4*(-3 + m))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 78}

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.

\(\displaystyle \int \frac {(d+e x)^m}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\)

\(\Big \downarrow \) 1121

\(\displaystyle \int \frac {(d+e x)^{m-4}}{(a e+c d x)^4}dx\)

\(\Big \downarrow \) 78

\(\displaystyle -\frac {e^3 (d+e x)^{m-3} \operatorname {Hypergeometric2F1}\left (4,m-3,m-2,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(3-m) \left (c d^2-a e^2\right )^4}\)

Input:

Int[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
 

Output:

-((e^3*(d + e*x)^(-3 + m)*Hypergeometric2F1[4, -3 + m, -2 + m, (c*d*(d + e 
*x))/(c*d^2 - a*e^2)])/((c*d^2 - a*e^2)^4*(3 - m)))
 

Defintions of rubi rules used

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 1121

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] 
/; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int 
egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
 

Maple [F]

\[\int \frac {\left (e x +d \right )^{m}}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{4}}d x\]

Input:

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x)
 

Output:

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x)
 

Fricas [F]

\[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{4}} \,d x } \] Input:

integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fric 
as")
 

Output:

integral((e*x + d)^m/(c^4*d^4*e^4*x^8 + a^4*d^4*e^4 + 4*(c^4*d^5*e^3 + a*c 
^3*d^3*e^5)*x^7 + 2*(3*c^4*d^6*e^2 + 8*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)* 
x^6 + 4*(c^4*d^7*e + 6*a*c^3*d^5*e^3 + 6*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x^ 
5 + (c^4*d^8 + 16*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 16*a^3*c*d^2*e^6 + 
a^4*e^8)*x^4 + 4*(a*c^3*d^7*e + 6*a^2*c^2*d^5*e^3 + 6*a^3*c*d^3*e^5 + a^4* 
d*e^7)*x^3 + 2*(3*a^2*c^2*d^6*e^2 + 8*a^3*c*d^4*e^4 + 3*a^4*d^2*e^6)*x^2 + 
 4*(a^3*c*d^5*e^3 + a^4*d^3*e^5)*x), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((e*x+d)**m/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{4}} \,d x } \] Input:

integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxi 
ma")
 

Output:

integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4, x)
 

Giac [F]

\[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{4}} \,d x } \] Input:

integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac 
")
 

Output:

integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^4} \,d x \] Input:

int((d + e*x)^m/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
 

Output:

int((d + e*x)^m/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4, x)
 

Reduce [F]

\[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\int \frac {\left (e x +d \right )^{m}}{c^{4} d^{4} e^{4} x^{8}+4 a \,c^{3} d^{3} e^{5} x^{7}+4 c^{4} d^{5} e^{3} x^{7}+6 a^{2} c^{2} d^{2} e^{6} x^{6}+16 a \,c^{3} d^{4} e^{4} x^{6}+6 c^{4} d^{6} e^{2} x^{6}+4 a^{3} c d \,e^{7} x^{5}+24 a^{2} c^{2} d^{3} e^{5} x^{5}+24 a \,c^{3} d^{5} e^{3} x^{5}+4 c^{4} d^{7} e \,x^{5}+a^{4} e^{8} x^{4}+16 a^{3} c \,d^{2} e^{6} x^{4}+36 a^{2} c^{2} d^{4} e^{4} x^{4}+16 a \,c^{3} d^{6} e^{2} x^{4}+c^{4} d^{8} x^{4}+4 a^{4} d \,e^{7} x^{3}+24 a^{3} c \,d^{3} e^{5} x^{3}+24 a^{2} c^{2} d^{5} e^{3} x^{3}+4 a \,c^{3} d^{7} e \,x^{3}+6 a^{4} d^{2} e^{6} x^{2}+16 a^{3} c \,d^{4} e^{4} x^{2}+6 a^{2} c^{2} d^{6} e^{2} x^{2}+4 a^{4} d^{3} e^{5} x +4 a^{3} c \,d^{5} e^{3} x +a^{4} d^{4} e^{4}}d x \] Input:

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
 

Output:

int((d + e*x)**m/(a**4*d**4*e**4 + 4*a**4*d**3*e**5*x + 6*a**4*d**2*e**6*x 
**2 + 4*a**4*d*e**7*x**3 + a**4*e**8*x**4 + 4*a**3*c*d**5*e**3*x + 16*a**3 
*c*d**4*e**4*x**2 + 24*a**3*c*d**3*e**5*x**3 + 16*a**3*c*d**2*e**6*x**4 + 
4*a**3*c*d*e**7*x**5 + 6*a**2*c**2*d**6*e**2*x**2 + 24*a**2*c**2*d**5*e**3 
*x**3 + 36*a**2*c**2*d**4*e**4*x**4 + 24*a**2*c**2*d**3*e**5*x**5 + 6*a**2 
*c**2*d**2*e**6*x**6 + 4*a*c**3*d**7*e*x**3 + 16*a*c**3*d**6*e**2*x**4 + 2 
4*a*c**3*d**5*e**3*x**5 + 16*a*c**3*d**4*e**4*x**6 + 4*a*c**3*d**3*e**5*x* 
*7 + c**4*d**8*x**4 + 4*c**4*d**7*e*x**5 + 6*c**4*d**6*e**2*x**6 + 4*c**4* 
d**5*e**3*x**7 + c**4*d**4*e**4*x**8),x)