\(\int \frac {(d+e x)^m}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [386]

Optimal result

Integrand size = 35, antiderivative size = 61 \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {e (d+e x)^{-1+m} \operatorname {Hypergeometric2F1}\left (2,-1+m,m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^2 (1-m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e (d+e x)^{-1+m} \operatorname {Hypergeometric2F1}\left (2,-1+m,m,-\frac {c d (d+e x)}{-c d^2+a e^2}\right )}{\left (-c d^2+a e^2\right )^2 (-1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 61, 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.

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

\(\Big \downarrow \) 1121

\(\displaystyle \int \frac {(d+e x)^{m-2}}{(a e+c d x)^2}dx\)

\(\Big \downarrow \) 78

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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