\(\int \frac {x^4}{(a+b x^2)^2 \sqrt {c+d x^2}} \, dx\) [1014]

Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {a x \sqrt {c+d x^2}}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^2 (b c-a d)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b^2 \sqrt {d}} \] Output:

1/2*a*x*(d*x^2+c)^(1/2)/b/(-a*d+b*c)/(b*x^2+a)-1/2*a^(1/2)*(-2*a*d+3*b*c)* 
arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2))/b^2/(-a*d+b*c)^(3/2)+ar 
ctanh(d^(1/2)*x/(d*x^2+c)^(1/2))/b^2/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.12 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\frac {a b x \sqrt {c+d x^2}}{(b c-a d) \left (a+b x^2\right )}+\frac {\sqrt {a} (3 b c-2 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {2 \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{\sqrt {d}}}{2 b^2} \] Input:

Integrate[x^4/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
 

Output:

((a*b*x*Sqrt[c + d*x^2])/((b*c - a*d)*(a + b*x^2)) + (Sqrt[a]*(3*b*c - 2*a 
*d)*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b 
*c - a*d])])/(b*c - a*d)^(3/2) - (2*Log[-(Sqrt[d]*x) + Sqrt[c + d*x^2]])/S 
qrt[d])/(2*b^2)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {372, 398, 224, 219, 291, 218}

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.

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

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[x^4/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
 

Output:

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

Defintions of rubi rules used

rule 218

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

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

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

rule 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

rule 398

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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {-\frac {a \left (-\frac {\sqrt {x^{2} d +c}\, b x}{b \,x^{2}+a}-\frac {\left (2 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{a d -b c}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )}{\sqrt {d}}}{2 b^{2}}\)	\(115\)
default	\(\frac {\ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b^{2} \sqrt {d}}-\frac {a \left (\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b^{3}}-\frac {a \left (\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b^{3}}+\frac {3 a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {3 a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\)	\(843\)

Input:

int(x^4/(b*x^2+a)^2/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1053, normalized size of antiderivative = 7.98 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^{4}}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
 

Output:

Integral(x**4/((a + b*x**2)**2*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^4/((b*x^2 + a)^2*sqrt(d*x^2 + c)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (110) = 220\).

Time = 0.15 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.15 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {{\left (3 \, a b c \sqrt {d} - 2 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{3} c - a b^{2} d\right )}} - \frac {\log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, b^{2} \sqrt {d}} \] Input:

integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

-1/2*(3*a*b*c*sqrt(d) - 2*a^2*d^(3/2))*arctan(-1/2*((sqrt(d)*x - sqrt(d*x^ 
2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((b^3*c - a*b^2*d)*sqr 
t(a*b*c*d - a^2*d^2)) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c*sqrt(d) - 2 
*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(3/2) - a*b*c^2*sqrt(d))/(((sqrt(d) 
*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqr 
t(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*(b^3*c - a*b^2*d)) - 1/2*log((sqr 
t(d)*x - sqrt(d*x^2 + c))^2)/(b^2*sqrt(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^4}{{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}} \,d x \] Input:

int(x^4/((a + b*x^2)^2*(c + d*x^2)^(1/2)),x)
 

Output:

int(x^4/((a + b*x^2)^2*(c + d*x^2)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1041, normalized size of antiderivative = 7.89 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

int(x^4/(b*x^2+a)^2/(d*x^2+c)^(1/2),x)
 

Output:

( - 2*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c 
) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*d**2 
 + 3*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
 - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*c*d - 
2*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 
2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*d**2*x**2 
 + 3*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
 - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b**2*c*d*x 
**2 - 2*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
 - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*d**2 
+ 3*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2 
*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*c*d - 2*sq 
rt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + 
 b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*d**2*x**2 + 3*sq 
rt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + 
 b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b**2*c*d*x**2 + 2*sq 
rt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sq 
rt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*a**2*d**2 - 3*sqrt(a)*sqrt(a*d - 
b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b* 
x + 2*a*d + 2*b*d*x**2)*a*b*c*d + 2*sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(...