\(\int \frac {1}{x^7 (a+b x^4) \sqrt {c+d x^4}} \, dx\) [244]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4}}{6 a c x^6}+\frac {(3 b c+2 a d) \sqrt {c+d x^4}}{6 a^2 c^2 x^2}+\frac {b^2 \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2} \sqrt {b c-a d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\sqrt {c+d x^4} \left (-a c+3 b c x^4+2 a d x^4\right )}{6 a^2 c^2 x^6}+\frac {b^2 \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2} \sqrt {b c-a d}} \] Input:

Integrate[1/(x^7*(a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {965, 382, 25, 445, 27, 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.

\(\displaystyle \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\)

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 382

\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {2 b d x^4+3 b c+2 a d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{3 a c}-\frac {\sqrt {c+d x^4}}{3 a c x^6}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {2 b d x^4+3 b c+2 a d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{3 a c}-\frac {\sqrt {c+d x^4}}{3 a c x^6}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\int \frac {3 b^2 c^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a c}-\frac {\sqrt {c+d x^4} (2 a d+3 b c)}{a c x^2}}{3 a c}-\frac {\sqrt {c+d x^4}}{3 a c x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 b^2 c \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a}-\frac {\sqrt {c+d x^4} (2 a d+3 b c)}{a c x^2}}{3 a c}-\frac {\sqrt {c+d x^4}}{3 a c x^6}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 b^2 c \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{a}-\frac {\sqrt {c+d x^4} (2 a d+3 b c)}{a c x^2}}{3 a c}-\frac {\sqrt {c+d x^4}}{3 a c x^6}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 b^2 c \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^4} (2 a d+3 b c)}{a c x^2}}{3 a c}-\frac {\sqrt {c+d x^4}}{3 a c x^6}\right )\)

Input:

Int[1/(x^7*(a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

(-1/3*Sqrt[c + d*x^4]/(a*c*x^6) - (-(((3*b*c + 2*a*d)*Sqrt[c + d*x^4])/(a* 
c*x^2)) - (3*b^2*c*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])] 
)/(a^(3/2)*Sqrt[b*c - a*d]))/(3*a*c))/2
 

Defintions of rubi rules used

rule 25

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

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 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 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 382

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ 
(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b* 
x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m 
+ 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ 
b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 445

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

rule 965

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

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 a d \,x^{4}-3 b c \,x^{4}+a c \right )}{3 x^{6}}+\frac {b^{2} c^{2} \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}}{2 a^{2} c^{2}}\)	\(88\)
risch	\(-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 a d \,x^{4}-3 b c \,x^{4}+a c \right )}{6 c^{2} a^{2} x^{6}}+\frac {b^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{a^{2}}\)	\(368\)
default	\(-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 d \,x^{4}+c \right )}{6 a \,x^{6} c^{2}}+\frac {b^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{a^{2}}+\frac {b \sqrt {d \,x^{4}+c}}{2 a^{2} x^{2} c}\)	\(379\)
elliptic	\(-\frac {\sqrt {d \,x^{4}+c}}{6 a c \,x^{6}}+\frac {d \sqrt {d \,x^{4}+c}}{3 a \,c^{2} x^{2}}+\frac {b \sqrt {d \,x^{4}+c}}{2 a^{2} x^{2} c}-\frac {b^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 a^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {b^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 a^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\)	\(396\)

Input:

int(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.63 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [-\frac {3 \, \sqrt {-a b c + a^{2} d} b^{2} c^{2} x^{6} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, {\left (a^{2} b c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{24 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{6}}, \frac {3 \, \sqrt {a b c - a^{2} d} b^{2} c^{2} x^{6} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) - 2 \, {\left (a^{2} b c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{6}}\right ] \] Input:

integrate(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

[-1/24*(3*sqrt(-a*b*c + a^2*d)*b^2*c^2*x^6*log(((b^2*c^2 - 8*a*b*c*d + 8*a 
^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 - 4*((b*c - 2*a*d)*x 
^6 - a*c*x^2)*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + 
 a^2)) + 4*(a^2*b*c^2 - a^3*c*d - (3*a*b^2*c^2 - a^2*b*c*d - 2*a^3*d^2)*x^ 
4)*sqrt(d*x^4 + c))/((a^3*b*c^3 - a^4*c^2*d)*x^6), 1/12*(3*sqrt(a*b*c - a^ 
2*d)*b^2*c^2*x^6*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt 
(a*b*c - a^2*d)/((a*b*c*d - a^2*d^2)*x^6 + (a*b*c^2 - a^2*c*d)*x^2)) - 2*( 
a^2*b*c^2 - a^3*c*d - (3*a*b^2*c^2 - a^2*b*c*d - 2*a^3*d^2)*x^4)*sqrt(d*x^ 
4 + c))/((a^3*b*c^3 - a^4*c^2*d)*x^6)]
 

Sympy [F]

\[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^{7} \left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:

integrate(1/x**7/(b*x**4+a)/(d*x**4+c)**(1/2),x)
 

Output:

Integral(1/(x**7*(a + b*x**4)*sqrt(c + d*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c} x^{7}} \,d x } \] Input:

integrate(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^7), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (95) = 190\).

Time = 0.26 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.78 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {1}{6} \, d^{\frac {5}{2}} {\left (\frac {3 \, b^{2} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} a^{2} d^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + 3 \, b c^{2} + 2 \, a c d\right )}}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )}^{3} a^{2} d^{2}}\right )} \] Input:

integrate(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")
 

Output:

-1/6*d^(5/2)*(3*b^2*arctan(1/2*((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b - b*c 
+ 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a^2*d^2) + 2*(3 
*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^4*b - 6*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2 
*b*c - 6*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*a*d + 3*b*c^2 + 2*a*c*d)/(((sqr 
t(d)*x^2 - sqrt(d*x^4 + c))^2 - c)^3*a^2*d^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^7\,\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \] Input:

int(1/(x^7*(a + b*x^4)*(c + d*x^4)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x^7*(a + b*x^4)*(c + d*x^4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 1240, normalized size of antiderivative = 10.78 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:

int(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x)
 

Output:

(3*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*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)*b**2*c**2*x**6 + 12*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*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)*b**2*c*d*x**10 + 3*sqrt(d 
)*sqrt(a)*sqrt(c + d*x**4)*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)*b**2*c**2*x**6 + 12*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*l 
og(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**2*c*d*x**10 - 3*sqrt(d)*sqrt(a)*sqrt( 
c + d*x**4)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt 
(d)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x**4)*b**2*c**2*x**6 - 12*sqrt 
(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a* 
d - b*c) + 2*sqrt(d)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x**4)*b**2*c* 
d*x**10 + 9*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**4) + sqrt(d)*sqrt(b)*x**2)* 
b**2*c**2*d*x**8 + 12*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**4) + sqrt(d)*sqrt 
(b)*x**2)*b**2*c*d**2*x**12 + 9*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**4) + sq...