\(\int \frac {(d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m}}{(f+g x)^3} \, dx\) [118]

Optimal result

Integrand size = 44, antiderivative size = 101 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\frac {c^2 d^2 (d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m} \operatorname {Hypergeometric2F1}\left (3,1-m,2-m,-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g)^3 (1-m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=-\frac {c^2 d^2 (d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m} \operatorname {Hypergeometric2F1}\left (3,1-m,2-m,\frac {g (a e+c d x)}{-c d f+a e g}\right )}{(c d f-a e g)^3 (-1+m)} \] Input:

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

Output:

-((c^2*d^2*(d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*Hypergeome 
tric2F1[3, 1 - m, 2 - m, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - 
a*e*g)^3*(-1 + m)))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1268, 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.

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

(c^2*d^2*(a*e + c*d*x)*(d + e*x)^m*Hypergeometric2F1[3, 1 - m, 2 - m, -((g 
*(a*e + c*d*x))/(c*d*f - a*e*g))])/((c*d*f - a*e*g)^3*(1 - m)*(a*d*e + (c* 
d^2 + a*e^2)*x + c*d*e*x^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 1268

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d 
 + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f 
 + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x+d)^m/(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, alg 
orithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((e*x+d)^m/(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x+d)^m/(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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