\(\int \frac {1}{\sqrt {a+b x+c x^2} (d+b x+c x^2)^2} \, dx\) [37]

Optimal result

Integrand size = 27, antiderivative size = 129 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )}+\frac {\left (b^2+4 c (a-2 d)\right ) \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{(a-d)^{3/2} \left (b^2-4 c d\right )^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 4.60 (sec) , antiderivative size = 1333, normalized size of antiderivative = 10.33 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-(RootSum[c^2*d - 2*Sqrt[a]*b*c*#1 + b^2*#1^2 + 4*a*c*#1^2 - 2*c*d*#1^2 - 
2*Sqrt[a]*b*#1^3 + d*#1^4 & , (-4*a*b^2*Log[x] + b^2*d*Log[x] + 4*a*c*d*Lo 
g[x] + c*d^2*Log[x] + 4*a*b^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] 
 - b^2*d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 4*a*c*d*Log[-Sqrt[ 
a] + Sqrt[a + b*x + c*x^2] - x*#1] - c*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c 
*x^2] - x*#1] - 2*Sqrt[a]*b*d*Log[x]*#1 + 2*Sqrt[a]*b*d*Log[-Sqrt[a] + Sqr 
t[a + b*x + c*x^2] - x*#1]*#1 - d^2*Log[x]*#1^2 + d^2*Log[-Sqrt[a] + Sqrt[ 
a + b*x + c*x^2] - x*#1]*#1^2)/(-(Sqrt[a]*b*c) + b^2*#1 + 4*a*c*#1 - 2*c*d 
*#1 - 3*Sqrt[a]*b*#1^2 + 2*d*#1^3) & ]/d^3) + ((2*(b + 2*c*x)*Sqrt[a + x*( 
b + c*x)])/(d + x*(b + c*x)) - RootSum[c^2*d - 2*Sqrt[a]*b*c*#1 + b^2*#1^2 
 + 4*a*c*#1^2 - 2*c*d*#1^2 - 2*Sqrt[a]*b*#1^3 + d*#1^4 & , (-8*a^2*b^4*Log 
[x] + 10*a*b^4*d*Log[x] + 40*a^2*b^2*c*d*Log[x] - 2*b^4*d^2*Log[x] - 46*a* 
b^2*c*d^2*Log[x] - 32*a^2*c^2*d^2*Log[x] + 7*b^2*c*d^3*Log[x] + 28*a*c^2*d 
^3*Log[x] + 8*a^2*b^4*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 10*a* 
b^4*d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 40*a^2*b^2*c*d*Log[-S 
qrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + 2*b^4*d^2*Log[-Sqrt[a] + Sqrt[a + 
 b*x + c*x^2] - x*#1] + 46*a*b^2*c*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2 
] - x*#1] + 32*a^2*c^2*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 
7*b^2*c*d^3*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 28*a*c^2*d^3*Lo 
g[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 4*a^(3/2)*b^3*d*Log[x]*#1 ...
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1305, 27, 1313, 221}

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.

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

\(\Big \downarrow \) 1305

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (4 c (a-2 d)+b^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )}dx}{2 (a-d) \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}\)

\(\Big \downarrow \) 1313

\(\displaystyle \frac {b \left (4 c (a-2 d)+b^2\right ) \int \frac {1}{b \left (b^2-4 c d\right )-\frac {b (a-d) (b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{(a-d) \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 221

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

rule 1305

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x 
_)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a 
*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( 
d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - 
 b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( 
c*e - b*f))*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si 
mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f 
 - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - 
b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f 
+ b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f 
*(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* 
(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b 
^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - 
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q 
, 0]
 

rule 1313

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( 
x_)^2]), x_Symbol] :> Simp[-2*e   Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e 
)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 
0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs. \(2(117)=234\).

Time = 2.62 (sec) , antiderivative size = 827, normalized size of antiderivative = 6.41

method	result	size
default	\(\frac {-\frac {\sqrt {c {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}+\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{\left (a -d \right ) \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}+\frac {\sqrt {b^{2}-4 c d}\, \ln \left (\frac {2 a -2 d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {c {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}+\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{2 \left (a -d \right )^{\frac {3}{2}}}}{b^{2}-4 c d}+\frac {2 c \ln \left (\frac {2 a -2 d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {c {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}+\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\left (b^{2}-4 c d \right )^{\frac {3}{2}} \sqrt {a -d}}+\frac {-\frac {\sqrt {c {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}-\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{\left (a -d \right ) \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}-\frac {\sqrt {b^{2}-4 c d}\, \ln \left (\frac {2 a -2 d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {c {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}-\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{2 \left (a -d \right )^{\frac {3}{2}}}}{b^{2}-4 c d}-\frac {2 c \ln \left (\frac {2 a -2 d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {c {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}-\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\left (b^{2}-4 c d \right )^{\frac {3}{2}} \sqrt {a -d}}\)	\(827\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (117) = 234\).

Time = 0.32 (sec) , antiderivative size = 1544, normalized size of antiderivative = 11.97 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*(8*c*d^2 - (b^2*c + 4*a*c^2 
- 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d - (b^3 + 4*a*b*c - 8*b*c*d)*x)*log((8*a^2 
*b^4 + (b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 
4*a*c^4)*d)*x^4 + 2*(b^5*c + 24*a*b^3*c^2 + 16*a^2*b*c^3 + 128*b*c^3*d^2 - 
 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (b^4 + 24*a*b^2*c + 16*a^2*c^2)*d^2 + ( 
b^6 + 32*a*b^4*c + 48*a^2*b^2*c^2 + 32*(5*b^2*c^2 + 4*a*c^3)*d^2 - 2*(19*b 
^4*c + 104*a*b^2*c^2 + 48*a^2*c^3)*d)*x^2 - 4*(2*a*b^3 + 2*(b^2*c^2 + 4*a* 
c^3 - 8*c^3*d)*x^3 + 3*(b^3*c + 4*a*b*c^2 - 8*b*c^2*d)*x^2 - (b^3 + 4*a*b* 
c)*d + (b^4 + 8*a*b^2*c - 2*(5*b^2*c + 4*a*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 
 - (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a) - 8*(a*b^4 + 4*a^2*b^2*c)*d + 2* 
(4*a*b^5 + 16*a^2*b^3*c + 16*(b^3*c + 4*a*b*c^2)*d^2 - (3*b^5 + 40*a*b^3*c 
 + 48*a^2*b*c^2)*d)*x)/(c^2*x^4 + 2*b*c*x^3 + 2*b*d*x + (b^2 + 2*c*d)*x^2 
+ d^2)) - 4*(a*b^3 + 4*b*c*d^2 - (b^3 + 4*a*b*c)*d + 2*(a*b^2*c + 4*c^2*d^ 
2 - (b^2*c + 4*a*c^2)*d)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*d + 16*c^2*d^5 
 - 8*(b^2*c + 4*a*c^2)*d^4 + (b^4 + 16*a*b^2*c + 16*a^2*c^2)*d^3 - 2*(a*b^ 
4 + 4*a^2*b^2*c)*d^2 + (a^2*b^4*c + 16*c^3*d^4 - 8*(b^2*c^2 + 4*a*c^3)*d^3 
 + (b^4*c + 16*a*b^2*c^2 + 16*a^2*c^3)*d^2 - 2*(a*b^4*c + 4*a^2*b^2*c^2)*d 
)*x^2 + (a^2*b^5 + 16*b*c^2*d^4 - 8*(b^3*c + 4*a*b*c^2)*d^3 + (b^5 + 16*a* 
b^3*c + 16*a^2*b*c^2)*d^2 - 2*(a*b^5 + 4*a^2*b^3*c)*d)*x), -1/2*(sqrt(-a*b 
^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*(8*c*d^2 - (b^2*c + 4*a*c^2 - 8*c^2*d)*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(1/(sqrt(c*x^2 + b*x + a)*(c*x^2 + b*x + d)^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (117) = 234\).

Time = 0.30 (sec) , antiderivative size = 1170, normalized size of antiderivative = 9.07 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*((b^2 + 4*a*c - 8*c*d)*log(abs(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2* 
b^2*c - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^2 + 8*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^2*c^2*d - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqrt 
(c) - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) + 8*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))*b*c^(3/2)*d - 3*a*b^2*c + 4*sqrt(a*b^2 - b^2*d - 4*a 
*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) + 4*a^2*c^2 
+ 2*b^2*c*d + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))*b*c + sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c 
)))/sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2) - (b^2 + 4*a*c - 8*c*d)*log(ab 
s(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c - 4*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^2*a*c^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^2*d - (sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqrt(c) - 4*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))*a*b*c^(3/2) + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^(3/2)*d 
 - 3*a*b^2*c - 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^2*c^(3/2) + 4*a^2*c^2 + 2*b^2*c*d - 4*sqrt(a*b^2 - b^2* 
d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c - sqrt(a*b^ 
2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^2*d - 4*a*c*d 
+ 4*c*d^2))/(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2) + ((sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^2*b^2*sqrt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^( 
3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2)*d + (sqrt(c)*x -...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 2661, normalized size of antiderivative = 20.63 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(a - d)*sqrt(b**2 - 4*c*d)*log( - sqrt(4*sqrt(c)*sqrt(a - d)*sqr 
t(b**2 - 4*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) 
 + b + 2*c*x)*a*b*c*x - 4*sqrt(a - d)*sqrt(b**2 - 4*c*d)*log( - sqrt(4*sqr 
t(c)*sqrt(a - d)*sqrt(b**2 - 4*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sq 
rt(a + b*x + c*x**2) + b + 2*c*x)*a*c**2*x**2 - 4*sqrt(a - d)*sqrt(b**2 - 
4*c*d)*log( - sqrt(4*sqrt(c)*sqrt(a - d)*sqrt(b**2 - 4*c*d) + 4*a*c + b**2 
 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*a*c*d - sqrt(a - 
 d)*sqrt(b**2 - 4*c*d)*log( - sqrt(4*sqrt(c)*sqrt(a - d)*sqrt(b**2 - 4*c*d 
) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)* 
b**3*x - sqrt(a - d)*sqrt(b**2 - 4*c*d)*log( - sqrt(4*sqrt(c)*sqrt(a - d)* 
sqrt(b**2 - 4*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x* 
*2) + b + 2*c*x)*b**2*c*x**2 - sqrt(a - d)*sqrt(b**2 - 4*c*d)*log( - sqrt( 
4*sqrt(c)*sqrt(a - d)*sqrt(b**2 - 4*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt( 
c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*b**2*d + 8*sqrt(a - d)*sqrt(b**2 - 
4*c*d)*log( - sqrt(4*sqrt(c)*sqrt(a - d)*sqrt(b**2 - 4*c*d) + 4*a*c + b**2 
 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*b*c*d*x + 8*sqrt 
(a - d)*sqrt(b**2 - 4*c*d)*log( - sqrt(4*sqrt(c)*sqrt(a - d)*sqrt(b**2 - 4 
*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c 
*x)*c**2*d*x**2 + 8*sqrt(a - d)*sqrt(b**2 - 4*c*d)*log( - sqrt(4*sqrt(c)*s 
qrt(a - d)*sqrt(b**2 - 4*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sqrt(...