\(\int \frac {\sqrt {c+d x^4}}{x (a+b x^4)} \, dx\) [222]

Optimal result

Integrand size = 24, antiderivative size = 85 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{2 a}+\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a \sqrt {b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\frac {\frac {\sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{\sqrt {b}}-\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{2 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {948, 94, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 94

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 94

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(a + b*x), x], 
x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(c + d*x), x], x] 
 /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
 

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 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {-\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{\sqrt {c}}\right )+\frac {\left (a d -c b \right ) \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{\sqrt {\left (a d -c b \right ) b}}}{2 a}\)	\(71\)
elliptic	\(\frac {\sqrt {d \,x^{4}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{2 a}-\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 a}-\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 a}\)	\(726\)
default	\(\frac {\frac {\sqrt {d \,x^{4}+c}}{2}-\frac {\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{2}}{a}-\frac {b \left (\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}+\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}\right )}{a}\)	\(734\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 377, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\left [\frac {\sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right ) + \sqrt {c} \log \left (\frac {d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right )}{4 \, a}, \frac {2 \, \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + \sqrt {c} \log \left (\frac {d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right )}{4 \, a}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{4} + c}}\right ) + \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right )}{4 \, a}, \frac {\sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{4} + c}}\right )}{2 \, a}\right ] \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (70) = 140\).

Time = 7.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\begin {cases} \frac {2 \left (\frac {c d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {- c}} \right )}}{4 a \sqrt {- c}} + \frac {d \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 a b \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (- \frac {b \left (\begin {cases} \frac {\frac {a}{2 b} + x^{4}}{a} & \text {for}\: b = 0 \\- \frac {\log {\left (a - 2 b \left (\frac {a}{2 b} + x^{4}\right ) \right )}}{2 b} & \text {otherwise} \end {cases}\right )}{2 a} - \frac {b \left (\begin {cases} \frac {\frac {a}{2 b} + x^{4}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + 2 b \left (\frac {a}{2 b} + x^{4}\right ) \right )}}{2 b} & \text {otherwise} \end {cases}\right )}{2 a}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(c*d*atan(sqrt(c + d*x**4)/sqrt(-c))/(4*a*sqrt(-c)) + d*(a*d 
- b*c)*atan(sqrt(c + d*x**4)/sqrt((a*d - b*c)/b))/(4*a*b*sqrt((a*d - b*c)/ 
b)))/d, Ne(d, 0)), (sqrt(c)*(-b*Piecewise(((a/(2*b) + x**4)/a, Eq(b, 0)), 
(-log(a - 2*b*(a/(2*b) + x**4))/(2*b), True))/(2*a) - b*Piecewise(((a/(2*b 
) + x**4)/a, Eq(b, 0)), (log(a + 2*b*(a/(2*b) + x**4))/(2*b), True))/(2*a) 
), True))
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=-\frac {{\left (b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a} + \frac {c \arctan \left (\frac {\sqrt {d x^{4} + c}}{\sqrt {-c}}\right )}{2 \, a \sqrt {-c}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.69 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c}\,\left (\sqrt {d\,x^4+c}\,\left (\frac {a^2\,b\,d^4}{2}-a\,b^2\,c\,d^3+b^3\,c^2\,d^2\right )+\frac {c\,\left (8\,a^3\,b^2\,d^3-16\,a^2\,b^3\,c\,d^2\right )\,\sqrt {d\,x^4+c}}{16\,a^2}\right )}{2\,a\,\left (\frac {b^2\,c^2\,d^3}{4}-\frac {a\,b\,c\,d^4}{4}\right )}\right )}{2\,a}+\frac {\mathrm {atanh}\left (\frac {a\,b^2\,c\,d^3\,\sqrt {d\,x^4+c}\,\sqrt {b^2\,c-a\,b\,d}}{4\,\left (\frac {a\,b^3\,c^2\,d^3}{4}-\frac {a^2\,b^2\,c\,d^4}{4}\right )}\right )\,\sqrt {b^2\,c-a\,b\,d}}{2\,a\,b} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 802, normalized size of antiderivative = 9.44 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d 
- b*c)*sqrt(a*d - b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x* 
*2)/sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)) - 2*sqrt(b)*sqr 
t(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(b)*sqrt(c + 
d*x**4) + sqrt(d)*sqrt(b)*x**2)/sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2 
*a*d - b*c))*a*d + 2*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a* 
d - b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)/sqrt(2*sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c))*b*c + sqrt(d)*sqrt(b)*sqrt(a) 
*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c)*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**4) + sqrt(d)*sqrt(b)*x**2) - sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d 
)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c)*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**4) + sqrt 
(d)*sqrt(b)*x**2) + sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d 
 + b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqr 
t(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)*a*d - sqrt(b)*sqrt(2*sqrt(d) 
*sqrt(a)*sqrt(a*d - b*c) - 2*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**4) + sqrt(d)*sqrt(b)*x* 
*2)*b*c - sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c)*lo 
g(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(...