\(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\) [631]

Optimal result

Integrand size = 22, antiderivative size = 497 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}+\frac {7 b^2 c d e-20 a c^2 d e-5 b^3 e^2-2 b c \left (c d^2-9 a e^2\right )-c \left (4 c^2 d^2+5 b^2 e^2-4 c e (b d+4 a e)\right ) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {25 b^4 c d e^3+240 a^2 c^3 d e^3-15 b^5 e^4-32 b^2 c^2 d e \left (c d^2+6 a e^2\right )+2 b^3 c e^2 \left (3 c d^2+55 a e^2\right )+8 b c^2 \left (2 c^2 d^4+9 a c d^2 e^2-23 a^2 e^4\right )+c \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) x}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}+\frac {5 e^4 (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:

-e/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^(3/2)+1/3*(7*b^2*c*d*e-20*a*c 
^2*d*e-5*b^3*e^2-2*b*c*(-9*a*e^2+c*d^2)-c*(4*c^2*d^2+5*b^2*e^2-4*c*e*(4*a* 
e+b*d))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(3/2)+1/3*(25* 
b^4*c*d*e^3+240*a^2*c^3*d*e^3-15*b^5*e^4-32*b^2*c^2*d*e*(6*a*e^2+c*d^2)+2* 
b^3*c*e^2*(55*a*e^2+3*c*d^2)+8*b*c^2*(-23*a^2*e^4+9*a*c*d^2*e^2+2*c^2*d^4) 
+c*(32*c^4*d^4-15*b^4*e^4-16*c^3*d^2*e*(-9*a*e+4*b*d)+20*b^2*c*e^3*(5*a*e+ 
b*d)+4*c^2*e^2*(-32*a^2*e^2-36*a*b*d*e+3*b^2*d^2))*x)/(-4*a*c+b^2)^2/(a*e^ 
2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)+5/2*e^4*(-b*e+2*c*d)*arctanh(1/2*(b*d 
-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e 
^2-b*d*e+c*d^2)^(7/2)
 

Mathematica [A] (verified)

Time = 11.32 (sec) , antiderivative size = 482, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)-4 c^2 e^2 \left (-3 b^2 d^2+36 a b d e+32 a^2 e^2\right )\right ) \sqrt {a+x (b+c x)}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x) (a+x (b+c x))^{3/2}}+\frac {-5 b^4 e^3+b^3 c e^2 (3 d-5 e x)-8 c^2 \left (4 a^2 e^3+2 c^2 d^3 x-a c d e (d-7 e x)\right )-4 b c^2 \left (a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right )+2 b^2 c e \left (16 a e^2+c d (5 d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (d+e x) \sqrt {a+x (b+c x)}}+\frac {15 \left (b^2-4 a c\right ) e^4 (-2 c d+b e) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 \left (c d^2+e (-b d+a e)\right )^{5/2}}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

(2*((e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c* 
e^3*(b*d + 5*a*e) - 4*c^2*e^2*(-3*b^2*d^2 + 36*a*b*d*e + 32*a^2*e^2))*Sqrt 
[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x) 
) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)*(a + x*(b + c* 
x))^(3/2)) + (-5*b^4*e^3 + b^3*c*e^2*(3*d - 5*e*x) - 8*c^2*(4*a^2*e^3 + 2* 
c^2*d^3*x - a*c*d*e*(d - 7*e*x)) - 4*b*c^2*(a*e^2*(9*d - 7*e*x) + 2*c*d^2* 
(d - 3*e*x)) + 2*b^2*c*e*(16*a*e^2 + c*d*(5*d + e*x)))/((b^2 - 4*a*c)*(-(c 
*d^2) + e*(b*d - a*e))*(d + e*x)*Sqrt[a + x*(b + c*x)]) + (15*(b^2 - 4*a*c 
)*e^4*(-2*c*d + b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c* 
d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a 
*e))^(5/2))))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1165, 27, 1235, 27, 1228, 1154, 219}

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.

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {2 \int \frac {8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)+6 c e (2 c d-b e) x}{2 (d+e x)^2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)+6 c e (2 c d-b e) x}{(d+e x)^2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {e \left (6 c e (2 c d-b e) \left (d b^2+2 a e b-8 a c d\right )+\left (-3 e b^2+2 c d b+8 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )+2 c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \int \frac {6 c e (2 c d-b e) \left (d b^2+2 a e b-8 a c d\right )+\left (-3 e b^2+2 c d b+8 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )+2 c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {15 e^3 \left (b^2-4 a c\right )^2 (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {15 e^3 \left (b^2-4 a c\right )^2 (2 c d-b e) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac {15 e^3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 
 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^(3/2)) - ((2*(6*a*c*e*(2*c*d 
 - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d 
- 8*a*e)) - c*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e) 
)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b*x + c*x^ 
2]) - (e*(((32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^ 
2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*S 
qrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) + (15*(b^2 - 4*a 
*c)^2*e^3*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c* 
d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^( 
3/2))))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(3*(b^2 - 4*a*c)*(c*d^2 - 
 b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1142\) vs. \(2(477)=954\).

Time = 1.16 (sec) , antiderivative size = 1143, normalized size of antiderivative = 2.30

method	result	size
default	\(\text {Expression too large to display}\)	\(1143\)

Input:

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/e^2*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/ 
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1 
/3/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e 
+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e) 
+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e) 
^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2 
-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(c*(x+d 
/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e 
+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+ 
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d 
/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d 
/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+ 
c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+( 
b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2 
*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))-4*c/(a*e^2-b*d* 
e+c*d^2)*e^2*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2 
-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2) 
/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c* 
(x+d/e)+(b*e-2*c*d)/e)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d 
^2)/e^2)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4416 vs. \(2 (477) = 954\).

Time = 5.27 (sec) , antiderivative size = 8874, normalized size of antiderivative = 17.86 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Integral(1/((d + e*x)**2*(a + b*x + c*x**2)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5572 vs. \(2 (477) = 954\).

Time = 0.66 (sec) , antiderivative size = 5572, normalized size of antiderivative = 11.21 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/6*((30*b^4*c^(3/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e 
+ a*e^2)*sqrt(c)*abs(e))) - 240*a*b^2*c^(5/2)*d*e^7*log(abs(2*c*d*e - b*e^ 
2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 480*a^2*c^(7/2)*d*e^7 
*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) 
- 15*b^5*sqrt(c)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^ 
2)*sqrt(c)*abs(e))) + 120*a*b^3*c^(3/2)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sq 
rt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 240*a^2*b*c^(5/2)*e^8*log(abs 
(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 64*sqr 
t(c*d^2 - b*d*e + a*e^2)*c^5*d^4*e^2*abs(e) + 128*sqrt(c*d^2 - b*d*e + a*e 
^2)*b*c^4*d^3*e^3*abs(e) - 24*sqrt(c*d^2 - b*d*e + a*e^2)*b^2*c^3*d^2*e^4* 
abs(e) - 288*sqrt(c*d^2 - b*d*e + a*e^2)*a*c^4*d^2*e^4*abs(e) - 40*sqrt(c* 
d^2 - b*d*e + a*e^2)*b^3*c^2*d*e^5*abs(e) + 288*sqrt(c*d^2 - b*d*e + a*e^2 
)*a*b*c^3*d*e^5*abs(e) + 30*sqrt(c*d^2 - b*d*e + a*e^2)*b^4*c*e^6*abs(e) - 
 200*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c^2*e^6*abs(e) + 256*sqrt(c*d^2 - b 
*d*e + a*e^2)*a^2*c^3*e^6*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d^2 - b* 
d*e + a*e^2)*b^4*c^(7/2)*d^6*abs(e) - 8*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2* 
c^(9/2)*d^6*abs(e) + 16*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*c^(11/2)*d^6*abs(e 
) - 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^5*c^(5/2)*d^5*e*abs(e) + 24*sqrt(c*d^2 
 - b*d*e + a*e^2)*a*b^3*c^(7/2)*d^5*e*abs(e) - 48*sqrt(c*d^2 - b*d*e + a*e 
^2)*a^2*b*c^(9/2)*d^5*e*abs(e) + 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^6*c^(3...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int(1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x)
 

Output:

int(1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.99 (sec) , antiderivative size = 12526, normalized size of antiderivative = 25.20 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x)
 

Output:

(240*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a* 
e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**4*b*c**2*d*e**5 
 + 240*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt( 
a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**4*b*c**2*e**6 
*x - 480*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr 
t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**4*c**3*d**2 
*e**4 - 480*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)* 
sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**4*c**3*d 
*e**5*x - 120*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2 
)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*b**3 
*c*d*e**5 - 120*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x* 
*2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*b* 
*3*c*e**6*x + 240*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c* 
x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3* 
b**2*c**2*d**2*e**4 + 720*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + 
b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d* 
x)*a**3*b**2*c**2*d*e**5*x + 480*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sq 
rt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 
 2*c*d*x)*a**3*b**2*c**2*e**6*x**2 - 960*sqrt(a*e**2 - b*d*e + c*d**2)*log 
( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*...