\(\int \frac {(a+b x+c x^2)^2}{(d+e x)^{5/2}} \, dx\) [524]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 \sqrt {d+e x}}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt {d+e x}}{e^5}-\frac {4 c (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \left (c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-5 e^2 \left (a^2 e^2+2 a b e (2 d+3 e x)-b^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+10 c e \left (a e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )\right )\right )}{15 e^5 (d+e x)^{3/2}} \] Input:

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

Output:

(2*(c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) 
 - 5*e^2*(a^2*e^2 + 2*a*b*e*(2*d + 3*e*x) - b^2*(8*d^2 + 12*d*e*x + 3*e^2* 
x^2)) + 10*c*e*(a*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b*(-16*d^3 - 24*d^2*e 
*x - 6*d*e^2*x^2 + e^3*x^3))))/(15*e^5*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 2009}

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.

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

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 \sqrt {d+e x}}+\frac {2 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)^{3/2}}+\frac {\left (a e^2-b d e+c d^2\right )^2}{e^4 (d+e x)^{5/2}}-\frac {2 c \sqrt {d+e x} (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^{3/2}}{e^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {d+e x} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 \sqrt {d+e x}}-\frac {2 \left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^{3/2}}-\frac {4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5}\)

Input:

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

Output:

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(3*e^5*(d + e*x)^(3/2)) + (4*(2*c*d - b*e)* 
(c*d^2 - b*d*e + a*e^2))/(e^5*Sqrt[d + e*x]) + (2*(6*c^2*d^2 + b^2*e^2 - 2 
*c*e*(3*b*d - a*e))*Sqrt[d + e*x])/e^5 - (4*c*(2*c*d - b*e)*(d + e*x)^(3/2 
))/(3*e^5) + (2*c^2*(d + e*x)^(5/2))/(5*e^5)
 

Defintions of rubi rules used

rule 1140

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {2 \left (3 c^{2} e^{2} x^{2}+10 e^{2} x b c -14 c^{2} d e x +30 a c \,e^{2}+15 b^{2} e^{2}-80 b c d e +73 c^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (6 x b \,e^{2}-12 c d x e +a \,e^{2}+5 b d e -11 c \,d^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(129\)
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {3 c^{2} x^{4}}{5}+\left (-2 b \,x^{3}-6 a \,x^{2}\right ) c -3 b^{2} x^{2}+6 a b x +a^{2}\right ) e^{4}+4 d \left (\frac {2 x^{3} c^{2}}{5}+\left (3 b \,x^{2}-6 a x \right ) c +b \left (-3 b x +a \right )\right ) e^{3}-16 d^{2} \left (\frac {3 c^{2} x^{2}}{5}+\left (-3 b x +a \right ) c +\frac {b^{2}}{2}\right ) e^{2}+32 d^{3} \left (-\frac {6 c x}{5}+b \right ) c e -\frac {128 c^{2} d^{4}}{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(145\)
gosper	\(-\frac {2 \left (-3 c^{2} x^{4} e^{4}-10 x^{3} b c \,e^{4}+8 d \,c^{2} x^{3} e^{3}-30 x^{2} a c \,e^{4}-15 x^{2} b^{2} e^{4}+60 x^{2} b c d \,e^{3}-48 x^{2} c^{2} d^{2} e^{2}+30 x a b \,e^{4}-120 x a c d \,e^{3}-60 x \,b^{2} d \,e^{3}+240 x b c \,d^{2} e^{2}-192 x \,c^{2} d^{3} e +5 a^{2} e^{4}+20 d \,e^{3} a b -80 a c \,d^{2} e^{2}-40 d^{2} e^{2} b^{2}+160 b c \,d^{3} e -128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(194\)
trager	\(-\frac {2 \left (-3 c^{2} x^{4} e^{4}-10 x^{3} b c \,e^{4}+8 d \,c^{2} x^{3} e^{3}-30 x^{2} a c \,e^{4}-15 x^{2} b^{2} e^{4}+60 x^{2} b c d \,e^{3}-48 x^{2} c^{2} d^{2} e^{2}+30 x a b \,e^{4}-120 x a c d \,e^{3}-60 x \,b^{2} d \,e^{3}+240 x b c \,d^{2} e^{2}-192 x \,c^{2} d^{3} e +5 a^{2} e^{4}+20 d \,e^{3} a b -80 a c \,d^{2} e^{2}-40 d^{2} e^{2} b^{2}+160 b c \,d^{3} e -128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(194\)
orering	\(-\frac {2 \left (-3 c^{2} x^{4} e^{4}-10 x^{3} b c \,e^{4}+8 d \,c^{2} x^{3} e^{3}-30 x^{2} a c \,e^{4}-15 x^{2} b^{2} e^{4}+60 x^{2} b c d \,e^{3}-48 x^{2} c^{2} d^{2} e^{2}+30 x a b \,e^{4}-120 x a c d \,e^{3}-60 x \,b^{2} d \,e^{3}+240 x b c \,d^{2} e^{2}-192 x \,c^{2} d^{3} e +5 a^{2} e^{4}+20 d \,e^{3} a b -80 a c \,d^{2} e^{2}-40 d^{2} e^{2} b^{2}+160 b c \,d^{3} e -128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(194\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c \,e^{2} \sqrt {e x +d}+2 b^{2} e^{2} \sqrt {e x +d}-12 b c d e \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (2 a \,e^{3} b -4 a d \,e^{2} c -2 d \,e^{2} b^{2}+6 d^{2} e b c -4 c^{2} d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\)	\(210\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c \,e^{2} \sqrt {e x +d}+2 b^{2} e^{2} \sqrt {e x +d}-12 b c d e \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (2 a \,e^{3} b -4 a d \,e^{2} c -2 d \,e^{2} b^{2}+6 d^{2} e b c -4 c^{2} d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\)	\(210\)

Input:

int((c*x^2+b*x+a)^2/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/15*(3*c^2*e^2*x^2+10*b*c*e^2*x-14*c^2*d*e*x+30*a*c*e^2+15*b^2*e^2-80*b*c 
*d*e+73*c^2*d^2)*(e*x+d)^(1/2)/e^5-2/3*(6*b*e^2*x-12*c*d*e*x+a*e^2+5*b*d*e 
-11*c*d^2)*(a*e^2-b*d*e+c*d^2)/e^5/(e*x+d)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 160 \, b c d^{3} e - 20 \, a b d e^{3} - 5 \, a^{2} e^{4} + 40 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 2 \, {\left (4 \, c^{2} d e^{3} - 5 \, b c e^{4}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{2} e^{2} - 20 \, b c d e^{3} + 5 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \, {\left (32 \, c^{2} d^{3} e - 40 \, b c d^{2} e^{2} - 5 \, a b e^{4} + 10 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \] Input:

integrate((c*x^2+b*x+a)^2/(e*x+d)^(5/2),x, algorithm="fricas")
 

Output:

2/15*(3*c^2*e^4*x^4 + 128*c^2*d^4 - 160*b*c*d^3*e - 20*a*b*d*e^3 - 5*a^2*e 
^4 + 40*(b^2 + 2*a*c)*d^2*e^2 - 2*(4*c^2*d*e^3 - 5*b*c*e^4)*x^3 + 3*(16*c^ 
2*d^2*e^2 - 20*b*c*d*e^3 + 5*(b^2 + 2*a*c)*e^4)*x^2 + 6*(32*c^2*d^3*e - 40 
*b*c*d^2*e^2 - 5*a*b*e^4 + 10*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/(e^7*x 
^2 + 2*d*e^6*x + d^2*e^5)
 

Sympy [A] (verification not implemented)

Time = 5.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{e^{4}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \sqrt {d + e x}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{4} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:

integrate((c*x**2+b*x+a)**2/(e*x+d)**(5/2),x)
 

Output:

Piecewise((2*(c**2*(d + e*x)**(5/2)/(5*e**4) + (d + e*x)**(3/2)*(2*b*c*e - 
 4*c**2*d)/(3*e**4) + sqrt(d + e*x)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 
6*c**2*d**2)/e**4 - 2*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)/(e**4*sqrt(d 
 + e*x)) - (a*e**2 - b*d*e + c*d**2)**2/(3*e**4*(d + e*x)**(3/2)))/e, Ne(e 
, 0)), ((a**2*x + a*b*x**2 + b*c*x**4/2 + c**2*x**5/5 + x**3*(2*a*c + b**2 
)/3)/d**(5/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} - 10 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {5 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{4}}\right )}}{15 \, e} \] Input:

integrate((c*x^2+b*x+a)^2/(e*x+d)^(5/2),x, algorithm="maxima")
 

Output:

2/15*((3*(e*x + d)^(5/2)*c^2 - 10*(2*c^2*d - b*c*e)*(e*x + d)^(3/2) + 15*( 
6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*sqrt(e*x + d))/e^4 - 5*(c^2*d^4 
 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 - 6*(2*c^2* 
d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d))/((e*x + d)^( 
3/2)*e^4))/e
 

Giac [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, {\left (e x + d\right )} c^{2} d^{3} - c^{2} d^{4} - 18 \, {\left (e x + d\right )} b c d^{2} e + 2 \, b c d^{3} e + 6 \, {\left (e x + d\right )} b^{2} d e^{2} + 12 \, {\left (e x + d\right )} a c d e^{2} - b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 6 \, {\left (e x + d\right )} a b e^{3} + 2 \, a b d e^{3} - a^{2} e^{4}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{5}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} e^{20} - 20 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d e^{20} + 90 \, \sqrt {e x + d} c^{2} d^{2} e^{20} + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} b c e^{21} - 90 \, \sqrt {e x + d} b c d e^{21} + 15 \, \sqrt {e x + d} b^{2} e^{22} + 30 \, \sqrt {e x + d} a c e^{22}\right )}}{15 \, e^{25}} \] Input:

integrate((c*x^2+b*x+a)^2/(e*x+d)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2/3*(12*(e*x + d)*c^2*d^3 - c^2*d^4 - 18*(e*x + d)*b*c*d^2*e + 2*b*c*d^3*e 
 + 6*(e*x + d)*b^2*d*e^2 + 12*(e*x + d)*a*c*d*e^2 - b^2*d^2*e^2 - 2*a*c*d^ 
2*e^2 - 6*(e*x + d)*a*b*e^3 + 2*a*b*d*e^3 - a^2*e^4)/((e*x + d)^(3/2)*e^5) 
 + 2/15*(3*(e*x + d)^(5/2)*c^2*e^20 - 20*(e*x + d)^(3/2)*c^2*d*e^20 + 90*s 
qrt(e*x + d)*c^2*d^2*e^20 + 10*(e*x + d)^(3/2)*b*c*e^21 - 90*sqrt(e*x + d) 
*b*c*d*e^21 + 15*sqrt(e*x + d)*b^2*e^22 + 30*sqrt(e*x + d)*a*c*e^22)/e^25
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {\frac {2\,a^2\,e^4}{3}-\left (d+e\,x\right )\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e-4\,a\,b\,e^3+8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )+\frac {2\,c^2\,d^4}{3}+\frac {2\,b^2\,d^2\,e^2}{3}-\frac {4\,a\,b\,d\,e^3}{3}-\frac {4\,b\,c\,d^3\,e}{3}+\frac {4\,a\,c\,d^2\,e^2}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {\sqrt {d+e\,x}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \] Input:

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

Output:

(2*c^2*(d + e*x)^(5/2))/(5*e^5) - ((2*a^2*e^4)/3 - (d + e*x)*(8*c^2*d^3 + 
4*b^2*d*e^2 - 4*a*b*e^3 + 8*a*c*d*e^2 - 12*b*c*d^2*e) + (2*c^2*d^4)/3 + (2 
*b^2*d^2*e^2)/3 - (4*a*b*d*e^3)/3 - (4*b*c*d^3*e)/3 + (4*a*c*d^2*e^2)/3)/( 
e^5*(d + e*x)^(3/2)) + ((d + e*x)^(1/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^ 
2 - 12*b*c*d*e))/e^5 - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(3/2))/(3*e^5)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {\frac {2}{5} c^{2} e^{4} x^{4}+\frac {4}{3} b c \,e^{4} x^{3}-\frac {16}{15} c^{2} d \,e^{3} x^{3}+4 a c \,e^{4} x^{2}+2 b^{2} e^{4} x^{2}-8 b c d \,e^{3} x^{2}+\frac {32}{5} c^{2} d^{2} e^{2} x^{2}-4 a b \,e^{4} x +16 a c d \,e^{3} x +8 b^{2} d \,e^{3} x -32 b c \,d^{2} e^{2} x +\frac {128}{5} c^{2} d^{3} e x -\frac {2}{3} a^{2} e^{4}-\frac {8}{3} a b d \,e^{3}+\frac {32}{3} a c \,d^{2} e^{2}+\frac {16}{3} b^{2} d^{2} e^{2}-\frac {64}{3} b c \,d^{3} e +\frac {256}{15} c^{2} d^{4}}{\sqrt {e x +d}\, e^{5} \left (e x +d \right )} \] Input:

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

Output:

(2*( - 5*a**2*e**4 - 20*a*b*d*e**3 - 30*a*b*e**4*x + 80*a*c*d**2*e**2 + 12 
0*a*c*d*e**3*x + 30*a*c*e**4*x**2 + 40*b**2*d**2*e**2 + 60*b**2*d*e**3*x + 
 15*b**2*e**4*x**2 - 160*b*c*d**3*e - 240*b*c*d**2*e**2*x - 60*b*c*d*e**3* 
x**2 + 10*b*c*e**4*x**3 + 128*c**2*d**4 + 192*c**2*d**3*e*x + 48*c**2*d**2 
*e**2*x**2 - 8*c**2*d*e**3*x**3 + 3*c**2*e**4*x**4))/(15*sqrt(d + e*x)*e** 
5*(d + e*x))