Integrand size = 30, antiderivative size = 354 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^{3/2} x \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {3 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{7/4} \sqrt {a+b x^4}} \] Output:
-1/5*c*(b*x^4+a)^(1/2)/a/x^5-1/4*d*(b*x^4+a)^(1/2)/a/x^4-1/3*e*(b*x^4+a)^( 1/2)/a/x^3-1/2*f*(b*x^4+a)^(1/2)/a/x^2+3/5*b*c*(b*x^4+a)^(1/2)/a^(3/2)/x/( a^(1/2)+b^(1/2)*x^2)+1/4*b*d*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)+3/5* b^(5/4)*c*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)* EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(7/4)/(b*x^4+a)^ (1/2)-1/30*b^(3/4)*(9*b^(1/2)*c+5*a^(1/2)*e)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4 +a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1 /4)),1/2*2^(1/2))/a^(7/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.38 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=-\frac {\sqrt {a+b x^4} \left (12 a c \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )+5 x \left (3 a \left (d+2 f x^2\right ) \sqrt {1+\frac {b x^4}{a}}-3 b d x^4 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+4 a e x \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )\right )\right )}{60 a^2 x^5 \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]
Output:
-1/60*(Sqrt[a + b*x^4]*(12*a*c*Hypergeometric2F1[-5/4, 1/2, -1/4, -((b*x^4 )/a)] + 5*x*(3*a*(d + 2*f*x^2)*Sqrt[1 + (b*x^4)/a] - 3*b*d*x^4*ArcTanh[Sqr t[1 + (b*x^4)/a]] + 4*a*e*x*Hypergeometric2F1[-3/4, 1/2, 1/4, -((b*x^4)/a) ])))/(a^2*x^5*Sqrt[1 + (b*x^4)/a])
Time = 0.99 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2372, 2009}
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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (\frac {c+e x^2}{x^6 \sqrt {a+b x^4}}+\frac {d+f x^2}{x^5 \sqrt {a+b x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+9 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{7/4} \sqrt {a+b x^4}}+\frac {3 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]
Output:
-1/5*(c*Sqrt[a + b*x^4])/(a*x^5) - (d*Sqrt[a + b*x^4])/(4*a*x^4) - (e*Sqrt [a + b*x^4])/(3*a*x^3) - (f*Sqrt[a + b*x^4])/(2*a*x^2) + (3*b*c*Sqrt[a + b *x^4])/(5*a^2*x) - (3*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*a^2*(Sqrt[a] + Sqrt[ b]*x^2)) + (b*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) + (3*b^(5/4) *c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ell ipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*a^(7/4)*Sqrt[a + b*x^4]) - (b^(3/4)*(9*Sqrt[b]*c + 5*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x ^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/ 2])/(30*a^(7/4)*Sqrt[a + b*x^4])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 2.00 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-36 b c \,x^{4}+30 a f \,x^{3}+20 a e \,x^{2}+15 a d x +12 a c \right )}{60 a^{2} x^{5}}-\frac {b \left (\frac {10 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {15 \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {18 i c \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )}{30 a^{2}}\) | \(255\) |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{5 a \,x^{5}}-\frac {d \sqrt {b \,x^{4}+a}}{4 a \,x^{4}}-\frac {e \sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {f \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+\frac {3 b c \sqrt {b \,x^{4}+a}}{5 a^{2} x}-\frac {e b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 i b^{\frac {3}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {d b \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4 a^{\frac {3}{2}}}\) | \(286\) |
| default | \(c \left (-\frac {\sqrt {b \,x^{4}+a}}{5 a \,x^{5}}+\frac {3 b \sqrt {b \,x^{4}+a}}{5 a^{2} x}-\frac {3 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {\sqrt {b \,x^{4}+a}}{4 a \,x^{4}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )+e \left (-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {f \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}\) | \(297\) |
Input:
int((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/60*(b*x^4+a)^(1/2)*(-36*b*c*x^4+30*a*f*x^3+20*a*e*x^2+15*a*d*x+12*a*c)/ a^2/x^5-1/30*b/a^2*(10*a*e/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)* x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/ a^(1/2)*b^(1/2))^(1/2),I)-15/2*a^(1/2)*d*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2) )/x^2)+18*I*c*b^(1/2)*a^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/ 2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*(EllipticF(x *(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)))
Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.45 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\frac {72 \, \sqrt {a} b c x^{5} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 15 \, \sqrt {a} b d x^{5} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 8 \, {\left (9 \, b c - 5 \, a e\right )} \sqrt {a} x^{5} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (36 \, b c x^{4} - 30 \, a f x^{3} - 20 \, a e x^{2} - 15 \, a d x - 12 \, a c\right )} \sqrt {b x^{4} + a}}{120 \, a^{2} x^{5}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/120*(72*sqrt(a)*b*c*x^5*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) + 15*sqrt(a)*b*d*x^5*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^ 4) - 8*(9*b*c - 5*a*e)*sqrt(a)*x^5*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a) ^(1/4)), -1) + 2*(36*b*c*x^4 - 30*a*f*x^3 - 20*a*e*x^2 - 15*a*d*x - 12*a*c )*sqrt(b*x^4 + a))/(a^2*x^5)
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.46 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=- \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 a x^{2}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} + \frac {c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {e \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {3}{2}}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/x**6/(b*x**4+a)**(1/2),x)
Output:
-sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(4*a*x**2) - sqrt(b)*f*sqrt(a/(b*x**4) + 1 )/(2*a) + c*gamma(-5/4)*hyper((-5/4, 1/2), (-1/4,), b*x**4*exp_polar(I*pi) /a)/(4*sqrt(a)*x**5*gamma(-1/4)) + e*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,) , b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*x**3*gamma(1/4)) + b*d*asinh(sqrt(a )/(sqrt(b)*x**2))/(4*a**(3/2))
\[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{6}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6), x)
\[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{6}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6), x)
Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\int \frac {f\,x^3+e\,x^2+d\,x+c}{x^6\,\sqrt {b\,x^4+a}} \,d x \] Input:
int((c + d*x + e*x^2 + f*x^3)/(x^6*(a + b*x^4)^(1/2)),x)
Output:
int((c + d*x + e*x^2 + f*x^3)/(x^6*(a + b*x^4)^(1/2)), x)
\[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx =\text {Too large to display} \] Input:
int((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x)
Output:
( - 6*sqrt(b)*sqrt(a)*sqrt(a + b*x**4)*a*d*x**2 - 16*sqrt(b)*sqrt(a)*sqrt( a + b*x**4)*a*f*x**4 - 8*sqrt(b)*sqrt(a)*sqrt(a + b*x**4)*b*d*x**6 - 32*sq rt(b)*sqrt(a)*sqrt(a + b*x**4)*b*f*x**8 + 8*sqrt(a)*sqrt(a + b*x**4)*int(s qrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a**2*c*x**4 + 32*sqrt(a)*sqrt(a + b* x**4)*int(sqrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a*b*c*x**8 + 8*sqrt(a)*sq rt(a + b*x**4)*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x)*a**2*e*x**4 + 32* sqrt(a)*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x)*a*b*e*x **8 - sqrt(a + b*x**4)*log(sqrt(a + b*x**4) - sqrt(a))*a*b*d*x**4 - 4*sqrt (a + b*x**4)*log(sqrt(a + b*x**4) - sqrt(a))*b**2*d*x**8 + sqrt(a + b*x**4 )*log(sqrt(a + b*x**4) + sqrt(a))*a*b*d*x**4 + 4*sqrt(a + b*x**4)*log(sqrt (a + b*x**4) + sqrt(a))*b**2*d*x**8 + 24*sqrt(b)*sqrt(a)*int(sqrt(a + b*x* *4)/(a*x**6 + b*x**10),x)*a**2*c*x**6 + 32*sqrt(b)*sqrt(a)*int(sqrt(a + b* x**4)/(a*x**6 + b*x**10),x)*a*b*c*x**10 + 24*sqrt(b)*sqrt(a)*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x)*a**2*e*x**6 + 32*sqrt(b)*sqrt(a)*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x)*a*b*e*x**10 - 2*sqrt(a)*a**2*d - 4*sqrt(a)*a **2*f*x**2 - 10*sqrt(a)*a*b*d*x**4 - 32*sqrt(a)*a*b*f*x**6 - 8*sqrt(a)*b** 2*d*x**8 - 32*sqrt(a)*b**2*f*x**10 - 3*sqrt(b)*log(sqrt(a + b*x**4) - sqrt (a))*a*b*d*x**6 - 4*sqrt(b)*log(sqrt(a + b*x**4) - sqrt(a))*b**2*d*x**10 + 3*sqrt(b)*log(sqrt(a + b*x**4) + sqrt(a))*a*b*d*x**6 + 4*sqrt(b)*log(sqrt (a + b*x**4) + sqrt(a))*b**2*d*x**10)/(8*sqrt(a)*a*x**4*(sqrt(a + b*x**...