\(\int \frac {(c+d x)^2}{x^3 (a+b x^2)^{3/2}} \, dx\) [1225]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int \frac {(c+d x)^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {b c^2-a d^2+2 b c d x}{a^2 \sqrt {a+b x^2}}-\frac {c^2 \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 c d \sqrt {a+b x^2}}{a^2 x}+\frac {\left (3 b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-b c x^2 (3 c+8 d x)-a \left (c^2+4 c d x-2 d^2 x^2\right )}{2 a^2 x^2 \sqrt {a+b x^2}}+\frac {\left (-3 b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:

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

Output:

(-(b*c*x^2*(3*c + 8*d*x)) - a*(c^2 + 4*c*d*x - 2*d^2*x^2))/(2*a^2*x^2*Sqrt 
[a + b*x^2]) + ((-3*b*c^2 + 2*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2]) 
/Sqrt[a]])/a^(5/2)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {532, 25, 2338, 25, 534, 243, 73, 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.

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

\(\Big \downarrow \) 532

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, c \left (4 d x +c \right )}{2 a^{2} x^{2}}+\frac {\frac {c^{2} b}{\sqrt {b \,x^{2}+a}}+a \left (2 a \,d^{2}-3 b \,c^{2}\right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {4 b c d x}{\sqrt {b \,x^{2}+a}}}{2 a^{2}}\)	\(118\)
default	\(c^{2} \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )+d^{2} \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )+2 c d \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )\)	\(156\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.45 \[ \int \frac {(c+d x)^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (3 \, b^{2} c^{2} - 2 \, a b d^{2}\right )} x^{4} + {\left (3 \, a b c^{2} - 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, a b c d x^{3} + 4 \, a^{2} c d x + a^{2} c^{2} + {\left (3 \, a b c^{2} - 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {{\left ({\left (3 \, b^{2} c^{2} - 2 \, a b d^{2}\right )} x^{4} + {\left (3 \, a b c^{2} - 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (8 \, a b c d x^{3} + 4 \, a^{2} c d x + a^{2} c^{2} + {\left (3 \, a b c^{2} - 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(((3*b^2*c^2 - 2*a*b*d^2)*x^4 + (3*a*b*c^2 - 2*a^2*d^2)*x^2)*sqrt(a) 
*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(8*a*b*c*d*x^3 + 
4*a^2*c*d*x + a^2*c^2 + (3*a*b*c^2 - 2*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3 
*b*x^4 + a^4*x^2), -1/2*(((3*b^2*c^2 - 2*a*b*d^2)*x^4 + (3*a*b*c^2 - 2*a^2 
*d^2)*x^2)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (8*a*b*c*d*x^3 + 
4*a^2*c*d*x + a^2*c^2 + (3*a*b*c^2 - 2*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3 
*b*x^4 + a^4*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x)^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {4 \, b c d x}{\sqrt {b x^{2} + a} a^{2}} + \frac {3 \, b c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} - \frac {3 \, b c^{2}}{2 \, \sqrt {b x^{2} + a} a^{2}} + \frac {d^{2}}{\sqrt {b x^{2} + a} a} - \frac {2 \, c d}{\sqrt {b x^{2} + a} a x} - \frac {c^{2}}{2 \, \sqrt {b x^{2} + a} a x^{2}} \] Input:

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

Output:

-4*b*c*d*x/(sqrt(b*x^2 + a)*a^2) + 3/2*b*c^2*arcsinh(a/(sqrt(a*b)*abs(x))) 
/a^(5/2) - d^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/2*b*c^2/(sqrt(b*x 
^2 + a)*a^2) + d^2/(sqrt(b*x^2 + a)*a) - 2*c*d/(sqrt(b*x^2 + a)*a*x) - 1/2 
*c^2/(sqrt(b*x^2 + a)*a*x^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {2 \, b c d x}{a^{2}} + \frac {a^{2} b c^{2} - a^{3} d^{2}}{a^{4}}}{\sqrt {b x^{2} + a}} - \frac {{\left (3 \, b c^{2} - 2 \, a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b c^{2} + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b c^{2} - 4 \, a^{2} \sqrt {b} c d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \] Input:

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

Output:

-(2*b*c*d*x/a^2 + (a^2*b*c^2 - a^3*d^2)/a^4)/sqrt(b*x^2 + a) - (3*b*c^2 - 
2*a*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) + 
((sqrt(b)*x - sqrt(b*x^2 + a))^3*b*c^2 + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2 
*a*sqrt(b)*c*d + (sqrt(b)*x - sqrt(b*x^2 + a))*a*b*c^2 - 4*a^2*sqrt(b)*c*d 
)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^2*a^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.36 \[ \int \frac {(c+d x)^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a^{2} c^{2}-4 \sqrt {b \,x^{2}+a}\, a^{2} c d x +2 \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x^{2}-3 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{2}-8 \sqrt {b \,x^{2}+a}\, a b c d \,x^{3}+2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d^{2} x^{2}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} x^{2}+2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{4}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{4}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d^{2} x^{2}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} x^{2}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{4}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{4}+8 \sqrt {b}\, a^{2} c d \,x^{2}+8 \sqrt {b}\, a b c d \,x^{4}}{2 a^{3} x^{2} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

( - sqrt(a + b*x**2)*a**2*c**2 - 4*sqrt(a + b*x**2)*a**2*c*d*x + 2*sqrt(a 
+ b*x**2)*a**2*d**2*x**2 - 3*sqrt(a + b*x**2)*a*b*c**2*x**2 - 8*sqrt(a + b 
*x**2)*a*b*c*d*x**3 + 2*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)* 
x)/sqrt(a))*a**2*d**2*x**2 - 3*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + s 
qrt(b)*x)/sqrt(a))*a*b*c**2*x**2 + 2*sqrt(a)*log((sqrt(a + b*x**2) - sqrt( 
a) + sqrt(b)*x)/sqrt(a))*a*b*d**2*x**4 - 3*sqrt(a)*log((sqrt(a + b*x**2) - 
 sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c**2*x**4 - 2*sqrt(a)*log((sqrt(a + b* 
x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*d**2*x**2 + 3*sqrt(a)*log((sqrt 
(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c**2*x**2 - 2*sqrt(a)*log 
((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*d**2*x**4 + 3*sqrt( 
a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c**2*x**4 + 
8*sqrt(b)*a**2*c*d*x**2 + 8*sqrt(b)*a*b*c*d*x**4)/(2*a**3*x**2*(a + b*x**2 
))