\(\int (d+e x)^{-5-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [397]

Optimal result

Integrand size = 39, antiderivative size = 288 \[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (4+p)}+\frac {6 c^2 d^2 (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^3 (2+p) (3+p) (4+p)}+\frac {6 c^3 d^3 (d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^4 (1+p) (2+p) (3+p) (4+p)}+\frac {3 c d (d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (3+p) (4+p)} \] Output:

(e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)/(-a*e^2+c*d^2)/(4 
+p)+6*c^2*d^2*(e*x+d)^(-3-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)/(-a 
*e^2+c*d^2)^3/(2+p)/(3+p)/(4+p)+6*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2 
)^(p+1)/(-a*e^2+c*d^2)^4/(p+1)/(2+p)/(3+p)/(4+p)/((e*x+d)^(2*p+2))+3*c*d*( 
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)/(-a*e^2+c*d^2)^2/(3+p)/(4+p)/((e*x+ 
d)^(4+2*p))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.75 \[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-5-2 p} ((a e+c d x) (d+e x))^{1+p} \left (-a^3 e^6 \left (6+11 p+6 p^2+p^3\right )+3 a^2 c d e^4 \left (2+3 p+p^2\right ) (d (4+p)+e x)-3 a c^2 d^2 e^2 (1+p) \left (d^2 \left (12+7 p+p^2\right )+2 d e (4+p) x+2 e^2 x^2\right )+c^3 d^3 \left (d^3 \left (24+26 p+9 p^2+p^3\right )+3 d^2 e \left (12+7 p+p^2\right ) x+6 d e^2 (4+p) x^2+6 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^4 (1+p) (2+p) (3+p) (4+p)} \] Input:

Integrate[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x 
]
 

Output:

((d + e*x)^(-5 - 2*p)*((a*e + c*d*x)*(d + e*x))^(1 + p)*(-(a^3*e^6*(6 + 11 
*p + 6*p^2 + p^3)) + 3*a^2*c*d*e^4*(2 + 3*p + p^2)*(d*(4 + p) + e*x) - 3*a 
*c^2*d^2*e^2*(1 + p)*(d^2*(12 + 7*p + p^2) + 2*d*e*(4 + p)*x + 2*e^2*x^2) 
+ c^3*d^3*(d^3*(24 + 26*p + 9*p^2 + p^3) + 3*d^2*e*(12 + 7*p + p^2)*x + 6* 
d*e^2*(4 + p)*x^2 + 6*e^3*x^3)))/((c*d^2 - a*e^2)^4*(1 + p)*(2 + p)*(3 + p 
)*(4 + p))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1129, 1129, 1129, 1123}

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.

\(\displaystyle \int (d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, dx\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {3 c d \int (d+e x)^{-2 (p+2)} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^pdx}{(p+4) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {3 c d \left (\frac {2 c d \int (d+e x)^{-2 p-3} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^pdx}{(p+3) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}\right )}{(p+4) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {3 c d \left (\frac {2 c d \left (\frac {c d \int (d+e x)^{-2 (p+1)} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^pdx}{(p+2) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) \left (c d^2-a e^2\right )}\right )}{(p+3) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}\right )}{(p+4) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac {3 c d \left (\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}+\frac {2 c d \left (\frac {(d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) \left (c d^2-a e^2\right )}+\frac {c d (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) \left (c d^2-a e^2\right )^2}\right )}{(p+3) \left (c d^2-a e^2\right )}\right )}{(p+4) \left (c d^2-a e^2\right )}\)

Input:

Int[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
 

Output:

((d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c 
*d^2 - a*e^2)*(4 + p)) + (3*c*d*((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^( 
1 + p)/((c*d^2 - a*e^2)*(3 + p)*(d + e*x)^(2*(2 + p))) + (2*c*d*(((d + e*x 
)^(-3 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c*d^2 - a* 
e^2)*(2 + p)) + (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c* 
d^2 - a*e^2)^2*(1 + p)*(2 + p)*(d + e*x)^(2*(1 + p)))))/((c*d^2 - a*e^2)*( 
3 + p))))/((c*d^2 - a*e^2)*(4 + p))
 

Defintions of rubi rules used

rule 1123

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

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(744\) vs. \(2(292)=584\).

Time = 1.83 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.59

method	result	size
gosper	\(-\frac {\left (e x +d \right )^{-4-2 p} \left (c d x +a e \right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} \left (a^{3} e^{6} p^{3}-3 a^{2} c \,d^{2} e^{4} p^{3}-3 a^{2} c d \,e^{5} p^{2} x +3 a \,c^{2} d^{4} e^{2} p^{3}+6 a \,c^{2} d^{3} e^{3} p^{2} x +6 a \,c^{2} d^{2} e^{4} p \,x^{2}-c^{3} d^{6} p^{3}-3 c^{3} d^{5} e \,p^{2} x -6 c^{3} d^{4} e^{2} p \,x^{2}-6 c^{3} d^{3} e^{3} x^{3}+6 a^{3} e^{6} p^{2}-21 a^{2} c \,d^{2} e^{4} p^{2}-9 a^{2} c d \,e^{5} p x +24 a \,c^{2} d^{4} e^{2} p^{2}+30 a \,c^{2} d^{3} e^{3} p x +6 x^{2} a \,c^{2} d^{2} e^{4}-9 c^{3} d^{6} p^{2}-21 c^{3} d^{5} e p x -24 c^{3} d^{4} e^{2} x^{2}+11 a^{3} e^{6} p -42 a^{2} c \,d^{2} e^{4} p -6 x \,a^{2} c d \,e^{5}+57 a \,c^{2} d^{4} e^{2} p +24 x a \,c^{2} d^{3} e^{3}-26 c^{3} d^{6} p -36 c^{3} d^{5} e x +6 e^{6} a^{3}-24 d^{2} e^{4} a^{2} c +36 d^{4} e^{2} a \,c^{2}-24 d^{6} c^{3}\right )}{a^{4} e^{8} p^{4}-4 a^{3} c \,d^{2} e^{6} p^{4}+6 a^{2} c^{2} d^{4} e^{4} p^{4}-4 a \,c^{3} d^{6} e^{2} p^{4}+c^{4} d^{8} p^{4}+10 a^{4} e^{8} p^{3}-40 a^{3} c \,d^{2} e^{6} p^{3}+60 a^{2} c^{2} d^{4} e^{4} p^{3}-40 a \,c^{3} d^{6} e^{2} p^{3}+10 c^{4} d^{8} p^{3}+35 a^{4} e^{8} p^{2}-140 a^{3} c \,d^{2} e^{6} p^{2}+210 a^{2} c^{2} d^{4} e^{4} p^{2}-140 a \,c^{3} d^{6} e^{2} p^{2}+35 c^{4} d^{8} p^{2}+50 a^{4} e^{8} p -200 a^{3} c \,d^{2} e^{6} p +300 a^{2} c^{2} d^{4} e^{4} p -200 a \,c^{3} d^{6} e^{2} p +50 c^{4} d^{8} p +24 a^{4} e^{8}-96 a^{3} c \,d^{2} e^{6}+144 a^{2} c^{2} d^{4} e^{4}-96 a \,c^{3} d^{6} e^{2}+24 c^{4} d^{8}}\)	\(745\)
orering	\(-\frac {\left (a^{3} e^{6} p^{3}-3 a^{2} c \,d^{2} e^{4} p^{3}-3 a^{2} c d \,e^{5} p^{2} x +3 a \,c^{2} d^{4} e^{2} p^{3}+6 a \,c^{2} d^{3} e^{3} p^{2} x +6 a \,c^{2} d^{2} e^{4} p \,x^{2}-c^{3} d^{6} p^{3}-3 c^{3} d^{5} e \,p^{2} x -6 c^{3} d^{4} e^{2} p \,x^{2}-6 c^{3} d^{3} e^{3} x^{3}+6 a^{3} e^{6} p^{2}-21 a^{2} c \,d^{2} e^{4} p^{2}-9 a^{2} c d \,e^{5} p x +24 a \,c^{2} d^{4} e^{2} p^{2}+30 a \,c^{2} d^{3} e^{3} p x +6 x^{2} a \,c^{2} d^{2} e^{4}-9 c^{3} d^{6} p^{2}-21 c^{3} d^{5} e p x -24 c^{3} d^{4} e^{2} x^{2}+11 a^{3} e^{6} p -42 a^{2} c \,d^{2} e^{4} p -6 x \,a^{2} c d \,e^{5}+57 a \,c^{2} d^{4} e^{2} p +24 x a \,c^{2} d^{3} e^{3}-26 c^{3} d^{6} p -36 c^{3} d^{5} e x +6 e^{6} a^{3}-24 d^{2} e^{4} a^{2} c +36 d^{4} e^{2} a \,c^{2}-24 d^{6} c^{3}\right ) \left (e x +d \right ) \left (c d x +a e \right ) \left (e x +d \right )^{-5-2 p} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{p}}{a^{4} e^{8} p^{4}-4 a^{3} c \,d^{2} e^{6} p^{4}+6 a^{2} c^{2} d^{4} e^{4} p^{4}-4 a \,c^{3} d^{6} e^{2} p^{4}+c^{4} d^{8} p^{4}+10 a^{4} e^{8} p^{3}-40 a^{3} c \,d^{2} e^{6} p^{3}+60 a^{2} c^{2} d^{4} e^{4} p^{3}-40 a \,c^{3} d^{6} e^{2} p^{3}+10 c^{4} d^{8} p^{3}+35 a^{4} e^{8} p^{2}-140 a^{3} c \,d^{2} e^{6} p^{2}+210 a^{2} c^{2} d^{4} e^{4} p^{2}-140 a \,c^{3} d^{6} e^{2} p^{2}+35 c^{4} d^{8} p^{2}+50 a^{4} e^{8} p -200 a^{3} c \,d^{2} e^{6} p +300 a^{2} c^{2} d^{4} e^{4} p -200 a \,c^{3} d^{6} e^{2} p +50 c^{4} d^{8} p +24 a^{4} e^{8}-96 a^{3} c \,d^{2} e^{6}+144 a^{2} c^{2} d^{4} e^{4}-96 a \,c^{3} d^{6} e^{2}+24 c^{4} d^{8}}\)	\(751\)

Input:

int((e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^p,x,method=_RETURNV 
ERBOSE)
 

Output:

-(e*x+d)^(-4-2*p)*(c*d*x+a*e)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^p/(a^4*e^8 
*p^4-4*a^3*c*d^2*e^6*p^4+6*a^2*c^2*d^4*e^4*p^4-4*a*c^3*d^6*e^2*p^4+c^4*d^8 
*p^4+10*a^4*e^8*p^3-40*a^3*c*d^2*e^6*p^3+60*a^2*c^2*d^4*e^4*p^3-40*a*c^3*d 
^6*e^2*p^3+10*c^4*d^8*p^3+35*a^4*e^8*p^2-140*a^3*c*d^2*e^6*p^2+210*a^2*c^2 
*d^4*e^4*p^2-140*a*c^3*d^6*e^2*p^2+35*c^4*d^8*p^2+50*a^4*e^8*p-200*a^3*c*d 
^2*e^6*p+300*a^2*c^2*d^4*e^4*p-200*a*c^3*d^6*e^2*p+50*c^4*d^8*p+24*a^4*e^8 
-96*a^3*c*d^2*e^6+144*a^2*c^2*d^4*e^4-96*a*c^3*d^6*e^2+24*c^4*d^8)*(a^3*e^ 
6*p^3-3*a^2*c*d^2*e^4*p^3-3*a^2*c*d*e^5*p^2*x+3*a*c^2*d^4*e^2*p^3+6*a*c^2* 
d^3*e^3*p^2*x+6*a*c^2*d^2*e^4*p*x^2-c^3*d^6*p^3-3*c^3*d^5*e*p^2*x-6*c^3*d^ 
4*e^2*p*x^2-6*c^3*d^3*e^3*x^3+6*a^3*e^6*p^2-21*a^2*c*d^2*e^4*p^2-9*a^2*c*d 
*e^5*p*x+24*a*c^2*d^4*e^2*p^2+30*a*c^2*d^3*e^3*p*x+6*a*c^2*d^2*e^4*x^2-9*c 
^3*d^6*p^2-21*c^3*d^5*e*p*x-24*c^3*d^4*e^2*x^2+11*a^3*e^6*p-42*a^2*c*d^2*e 
^4*p-6*a^2*c*d*e^5*x+57*a*c^2*d^4*e^2*p+24*a*c^2*d^3*e^3*x-26*c^3*d^6*p-36 
*c^3*d^5*e*x+6*a^3*e^6-24*a^2*c*d^2*e^4+36*a*c^2*d^4*e^2-24*c^3*d^6)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (292) = 584\).

Time = 0.17 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.65 \[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorith 
m="fricas")
 

Output:

(6*c^4*d^4*e^4*x^5 + 24*a*c^3*d^7*e - 36*a^2*c^2*d^5*e^3 + 24*a^3*c*d^3*e^ 
5 - 6*a^4*d*e^7 + 6*(5*c^4*d^5*e^3 + (c^4*d^5*e^3 - a*c^3*d^3*e^5)*p)*x^4 
+ (a*c^3*d^7*e - 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 - a^4*d*e^7)*p^3 + 3* 
(20*c^4*d^6*e^2 + (c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*p^2 + 
(9*c^4*d^6*e^2 - 10*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*p)*x^3 + 3*(3*a*c^3*d 
^7*e - 8*a^2*c^2*d^5*e^3 + 7*a^3*c*d^3*e^5 - 2*a^4*d*e^7)*p^2 + (60*c^4*d^ 
7*e + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*p^3 
+ 3*(4*c^4*d^7*e - 9*a*c^3*d^5*e^3 + 6*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*p^2 
+ (47*c^4*d^7*e - 60*a*c^3*d^5*e^3 + 15*a^2*c^2*d^3*e^5 - 2*a^3*c*d*e^7)*p 
)*x^2 + (26*a*c^3*d^7*e - 57*a^2*c^2*d^5*e^3 + 42*a^3*c*d^3*e^5 - 11*a^4*d 
*e^7)*p + (24*c^4*d^8 + 24*a*c^3*d^6*e^2 - 36*a^2*c^2*d^4*e^4 + 24*a^3*c*d 
^2*e^6 - 6*a^4*e^8 + (c^4*d^8 - 2*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 - a^4*e^ 
8)*p^3 + 3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 6*a^3*c*d^2* 
e^6 - 2*a^4*e^8)*p^2 + (26*c^4*d^8 - 10*a*c^3*d^6*e^2 - 45*a^2*c^2*d^4*e^4 
 + 40*a^3*c*d^2*e^6 - 11*a^4*e^8)*p)*x)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^ 
2)*x)^p*(e*x + d)^(-2*p - 5)/(24*c^4*d^8 - 96*a*c^3*d^6*e^2 + 144*a^2*c^2* 
d^4*e^4 - 96*a^3*c*d^2*e^6 + 24*a^4*e^8 + (c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a 
^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^4 + 10*(c^4*d^8 - 4*a*c^3*d^ 
6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^3 + 35*(c^4*d^8 - 
 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^2 +...
 

Sympy [F]

\[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p} \left (d + e x\right )^{- 2 p - 5}\, dx \] Input:

integrate((e*x+d)**(-5-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
                                                                                    
                                                                                    
 

Output:

Integral(((d + e*x)*(a*e + c*d*x))**p*(d + e*x)**(-2*p - 5), x)
 

Maxima [F]

\[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \] Input:

integrate((e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorith 
m="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 5), 
x)
 

Giac [F]

\[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \] Input:

integrate((e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorith 
m="giac")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 5), 
x)
 

Mupad [B] (verification not implemented)

Time = 6.42 (sec) , antiderivative size = 1036, normalized size of antiderivative = 3.60 \[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx =\text {Too large to display} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^(2*p + 5),x)
 

Output:

(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p*((6*c^4*d^4*e^4*x^5)/((a*e^2 - c 
*d^2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) - (x*(6*a 
^4*e^8 - 24*c^4*d^8 + 11*a^4*e^8*p - 26*c^4*d^8*p + 6*a^4*e^8*p^2 + a^4*e^ 
8*p^3 - 9*c^4*d^8*p^2 - c^4*d^8*p^3 - 24*a*c^3*d^6*e^2 - 24*a^3*c*d^2*e^6 
+ 36*a^2*c^2*d^4*e^4 + 9*a^2*c^2*d^4*e^4*p^2 + 10*a*c^3*d^6*e^2*p - 40*a^3 
*c*d^2*e^6*p + 45*a^2*c^2*d^4*e^4*p + 12*a*c^3*d^6*e^2*p^2 - 18*a^3*c*d^2* 
e^6*p^2 + 2*a*c^3*d^6*e^2*p^3 - 2*a^3*c*d^2*e^6*p^3))/((a*e^2 - c*d^2)^4*( 
d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) - (a*d*e*(6*a^3*e^ 
6 - 24*c^3*d^6 + 11*a^3*e^6*p - 26*c^3*d^6*p + 6*a^3*e^6*p^2 + a^3*e^6*p^3 
 - 9*c^3*d^6*p^2 - c^3*d^6*p^3 + 36*a*c^2*d^4*e^2 - 24*a^2*c*d^2*e^4 + 57* 
a*c^2*d^4*e^2*p - 42*a^2*c*d^2*e^4*p + 24*a*c^2*d^4*e^2*p^2 - 21*a^2*c*d^2 
*e^4*p^2 + 3*a*c^2*d^4*e^2*p^3 - 3*a^2*c*d^2*e^4*p^3))/((a*e^2 - c*d^2)^4* 
(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (6*c^3*d^3*e^3* 
x^4*(5*c*d^2 - a*e^2*p + c*d^2*p))/((a*e^2 - c*d^2)^4*(d + e*x)^(2*p + 5)* 
(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (3*c^2*d^2*e^2*x^3*(20*c^2*d^4 + a^ 
2*e^4*p + 9*c^2*d^4*p + a^2*e^4*p^2 + c^2*d^4*p^2 - 10*a*c*d^2*e^2*p - 2*a 
*c*d^2*e^2*p^2))/((a*e^2 - c*d^2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 1 
0*p^3 + p^4 + 24)) + (c*d*e*x^2*(60*c^3*d^6 - 2*a^3*e^6*p + 47*c^3*d^6*p - 
 3*a^3*e^6*p^2 - a^3*e^6*p^3 + 12*c^3*d^6*p^2 + c^3*d^6*p^3 - 60*a*c^2*d^4 
*e^2*p + 15*a^2*c*d^2*e^4*p - 27*a*c^2*d^4*e^2*p^2 + 18*a^2*c*d^2*e^4*p...
 

Reduce [F]

\[ \int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p}}{\left (e x +d \right )^{2 p} d^{5}+5 \left (e x +d \right )^{2 p} d^{4} e x +10 \left (e x +d \right )^{2 p} d^{3} e^{2} x^{2}+10 \left (e x +d \right )^{2 p} d^{2} e^{3} x^{3}+5 \left (e x +d \right )^{2 p} d \,e^{4} x^{4}+\left (e x +d \right )^{2 p} e^{5} x^{5}}d x \] Input:

int((e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^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)*d**5 + 
 5*(d + e*x)**(2*p)*d**4*e*x + 10*(d + e*x)**(2*p)*d**3*e**2*x**2 + 10*(d 
+ e*x)**(2*p)*d**2*e**3*x**3 + 5*(d + e*x)**(2*p)*d*e**4*x**4 + (d + e*x)* 
*(2*p)*e**5*x**5),x)