\(\int \frac {1}{\sqrt {a+b x+c x^2} (d+b x+c x^2)^3} \, dx\) [38]

Optimal result

Integrand size = 27, antiderivative size = 224 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac {\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}} \] Output:

-1/2*(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)^2+3/4*( 
b^2+4*c*(a-2*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)^2/(b^2-4*c*d)^2/(c*x^ 
2+b*x+d)-1/4*(3*b^4+8*b^2*c*(a-4*d)+16*c^2*(3*a^2-8*a*d+8*d^2))*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)^(5/2)/(b^ 
2-4*c*d)^(5/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1746\) vs. \(2(224)=448\).

Time = 16.59 (sec) , antiderivative size = 1746, normalized size of antiderivative = 7.79 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)^3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1305, 27, 2135, 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.

\(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} \left (b x+c x^2+d\right )^3} \, dx\)

\(\Big \downarrow \) 1305

\(\displaystyle \frac {\int -\frac {8 (a-d) x^2 c^4+8 b (a-d) x c^3+(a-d) \left (3 b^2+12 a c-16 c d\right ) c^2}{2 \sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )^2}dx}{2 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {8 (a-d) x^2 c^4+8 b (a-d) x c^3+(a-d) \left (3 b^2+12 a c-16 c d\right ) c^2}{\sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )^2}dx}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}\)

\(\Big \downarrow \) 2135

\(\displaystyle -\frac {-\frac {\int \frac {c^4 (a-d)^2 \left (3 b^4+8 c (a-4 d) b^2+16 c^2 \left (3 a^2-8 d a+8 d^2\right )\right )}{2 \sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )}dx}{c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {3 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {c^2 \left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a} \left (c x^2+b x+d\right )}dx}{2 \left (b^2-4 c d\right )}-\frac {3 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}\)

\(\Big \downarrow \) 1313

\(\displaystyle -\frac {\frac {b c^2 \left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \int \frac {1}{b \left (b^2-4 c d\right )-\frac {b (a-d) (b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{b^2-4 c d}-\frac {3 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {c^2 \left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a-d} \left (b^2-4 c d\right )^{3/2}}-\frac {3 c^2 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 c d\right ) \left (b x+c x^2+d\right )}}{4 c^2 (a-d)^2 \left (b^2-4 c d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2}\)

Input:

Int[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)^3),x]
 

Output:

-1/2*((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/((a - d)*(b^2 - 4*c*d)*(d + b*x + 
 c*x^2)^2) - ((-3*c^2*(b^2 + 4*c*(a - 2*d))*(b + 2*c*x)*Sqrt[a + b*x + c*x 
^2])/((b^2 - 4*c*d)*(d + b*x + c*x^2)) + (c^2*(3*b^4 + 8*b^2*c*(a - 4*d) + 
 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*ArcTanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b 
^2 - 4*c*d]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a - d]*(b^2 - 4*c*d)^(3/2)))/(4 
*c^2*(a - d)^2*(b^2 - 4*c*d))
 

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]
 

rule 2135

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. 
)*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] 
, C = Coeff[Px, x, 2]}, Simp[(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)))*( 
(A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b 
*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* 
e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), 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*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 
 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + 
 a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p 
+ 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B 
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a 
*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p 
 + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( 
c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) 
*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d 
 - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F 
reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1867\) vs. \(2(204)=408\).

Time = 3.13 (sec) , antiderivative size = 1868, normalized size of antiderivative = 8.34

method	result	size
default	\(\text {Expression too large to display}\)	\(1868\)

Input:

int(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (204) = 408\).

Time = 2.53 (sec) , antiderivative size = 3818, normalized size of antiderivative = 17.04 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(c*x**2+b*x+a)**(1/2)/(c*x**2+b*x+d)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (c x^{2} + b x + d\right )}^{3}} \,d x } \] Input:

integrate(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d)^3,x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(c*x^2 + b*x + a)*(c*x^2 + b*x + d)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2986 vs. \(2 (204) = 408\).

Time = 0.93 (sec) , antiderivative size = 2986, normalized size of antiderivative = 13.33 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d)^3,x, algorithm="giac")
 

Output:

-1/8*((3*b^4 + 8*a*b^2*c + 48*a^2*c^2 - 32*b^2*c*d - 128*a*c^2*d + 128*c^2 
*d^2)*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 - sqrt(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 - 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) - (3*b^4 + 8*a*b^2*c + 48*a^2*c^2 - 32*b^2*c*d - 12 
8*a*c^2*d + 128*c^2*d^2)*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 - sqrt( 
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 - sqr 
t(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^2*b^4 - 2*a*b^4*d - 8*a^2*b^ 
2*c*d + b^4*d^2 + 16*a*b^2*c*d^2 + 16*a^2*c^2*d^2 - 8*b^2*c*d^3 - 32*a*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+d\right )}^3} \,d x \] Input:

int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)^3),x)
 

Output:

int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 18107, normalized size of antiderivative = 80.83 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d)^3,x)
 

Output:

(192*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**3*b**2*c**3*x**2 + 384*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**3*b*c**4*x**3 + 384*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**3*b*c**3*d*x + 192*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**3*c**5*x**4 + 384*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**3*c**4*d 
*x**2 + 192*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**3*c**3*d**2 + 80*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**2*b**4*c**2*x**2 + 160* 
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**2*b**3*c**3*x**3 + 160*sqrt(a - d)*sqrt(b**2 - 4*c*d)*log( -...