\(\int \frac {(d+e x)^m}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [387]

Optimal result

Integrand size = 35, antiderivative size = 64 \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^2 (d+e x)^{-2+m} \operatorname {Hypergeometric2F1}\left (3,-2+m,-1+m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^3 (2-m)} \] Output:

e^2*(e*x+d)^(-2+m)*hypergeom([3, -2+m],[-1+m],c*d*(e*x+d)/(-a*e^2+c*d^2))/ 
(-a*e^2+c*d^2)^3/(2-m)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^2 (d+e x)^{-2+m} \operatorname {Hypergeometric2F1}\left (3,-2+m,-1+m,-\frac {c d (d+e x)}{-c d^2+a e^2}\right )}{\left (-c d^2+a e^2\right )^3 (-2+m)} \] Input:

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

Output:

(e^2*(d + e*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*d*(d + e 
*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^3*(-2 + m))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 64, 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.

\(\displaystyle \int \frac {(d+e x)^m}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1121

\(\displaystyle \int \frac {(d+e x)^{m-3}}{(a e+c d x)^3}dx\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {e^2 (d+e x)^{m-2} \operatorname {Hypergeometric2F1}\left (3,m-2,m-1,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(2-m) \left (c d^2-a e^2\right )^3}\)

Input:

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

Output:

(e^2*(d + e*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, (c*d*(d + e*x 
))/(c*d^2 - a*e^2)])/((c*d^2 - a*e^2)^3*(2 - 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]

\[\int \frac {\left (e x +d \right )^{m}}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}d x\]

Input:

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x)
 

Output:

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x)
 

Fricas [F]

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

integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fric 
as")
 

Output:

integral((e*x + d)^m/(c^3*d^3*e^3*x^6 + a^3*d^3*e^3 + 3*(c^3*d^4*e^2 + a*c 
^2*d^2*e^4)*x^5 + 3*(c^3*d^5*e + 3*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x^4 + (c^3 
*d^6 + 9*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + a^3*e^6)*x^3 + 3*(a*c^2*d^5*e + 
 3*a^2*c*d^3*e^3 + a^3*d*e^5)*x^2 + 3*(a^2*c*d^4*e^2 + a^3*d^2*e^4)*x), x)
 

Sympy [F]

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

integrate((e*x+d)**m/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
 

Output:

Integral((d + e*x)**m/((d + e*x)**3*(a*e + c*d*x)**3), x)
 

Maxima [F]

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

integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,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)^3, x)
 

Giac [F]

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

integrate((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac 
")
 

Output:

integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^3} \,d x \] Input:

int((d + e*x)^m/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
 

Output:

int((d + e*x)^m/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\int \frac {\left (e x +d \right )^{m}}{c^{3} d^{3} e^{3} x^{6}+3 a \,c^{2} d^{2} e^{4} x^{5}+3 c^{3} d^{4} e^{2} x^{5}+3 a^{2} c d \,e^{5} x^{4}+9 a \,c^{2} d^{3} e^{3} x^{4}+3 c^{3} d^{5} e \,x^{4}+a^{3} e^{6} x^{3}+9 a^{2} c \,d^{2} e^{4} x^{3}+9 a \,c^{2} d^{4} e^{2} x^{3}+c^{3} d^{6} x^{3}+3 a^{3} d \,e^{5} x^{2}+9 a^{2} c \,d^{3} e^{3} x^{2}+3 a \,c^{2} d^{5} e \,x^{2}+3 a^{3} d^{2} e^{4} x +3 a^{2} c \,d^{4} e^{2} x +a^{3} d^{3} e^{3}}d x \] Input:

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
 

Output:

int((d + e*x)**m/(a**3*d**3*e**3 + 3*a**3*d**2*e**4*x + 3*a**3*d*e**5*x**2 
 + a**3*e**6*x**3 + 3*a**2*c*d**4*e**2*x + 9*a**2*c*d**3*e**3*x**2 + 9*a** 
2*c*d**2*e**4*x**3 + 3*a**2*c*d*e**5*x**4 + 3*a*c**2*d**5*e*x**2 + 9*a*c** 
2*d**4*e**2*x**3 + 9*a*c**2*d**3*e**3*x**4 + 3*a*c**2*d**2*e**4*x**5 + c** 
3*d**6*x**3 + 3*c**3*d**5*e*x**4 + 3*c**3*d**4*e**2*x**5 + c**3*d**3*e**3* 
x**6),x)