\(\int (d+e x)^{-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [402]

Optimal result

Integrand size = 37, antiderivative size = 91 \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {(d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2,2-p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.07 \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-1-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p ((a e+c d x) (d+e x))^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1139, 1138, 80, 79}

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.

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

\(\Big \downarrow \) 1139

\(\displaystyle (d+e x)^{-2 p} \left (\frac {e x}{d}+1\right )^{2 p} \int \left (\frac {e x}{d}+1\right )^{-2 p} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^pdx\)

\(\Big \downarrow \) 1138

\(\displaystyle (d+e x)^{-2 p} \left (\frac {e x}{d}+1\right )^p \left (a d e+c d^2 x\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \int \left (c x d^2+a e d\right )^p \left (\frac {e x}{d}+1\right )^{-p}dx\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

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

rule 1138

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] 
*(a/d + (c*x)/e)^FracPart[p]))   Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
&& (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer 
Q[4*p]))
 

rule 1139

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d^IntPart[m]*((d + e*x)^FracPart[m]/(1 + e*(x/d))^FracPart[m] 
)   Int[(1 + e*(x/d))^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IntegerQ[m] || GtQ[d, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p/(e*x + d)^(2*p), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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