Integrand size = 22, antiderivative size = 204 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=-\frac {c}{3 a x^3 \left (a+b x^4\right )^{5/2}}-\frac {(13 b c-3 a d) x}{30 a^2 \left (a+b x^4\right )^{5/2}}-\frac {(13 b c-3 a d) x}{20 a^3 \left (a+b x^4\right )^{3/2}}-\frac {(13 b c-3 a d) x}{8 a^4 \sqrt {a+b x^4}}-\frac {(13 b c-3 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{16 a^{17/4} \sqrt [4]{b} \sqrt {a+b x^4}} \] Output:
-1/3*c/a/x^3/(b*x^4+a)^(5/2)-1/30*(-3*a*d+13*b*c)*x/a^2/(b*x^4+a)^(5/2)-1/ 20*(-3*a*d+13*b*c)*x/a^3/(b*x^4+a)^(3/2)-1/8*(-3*a*d+13*b*c)*x/a^4/(b*x^4+ a)^(1/2)-1/16*(-3*a*d+13*b*c)*(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^(17/4)/b^(1/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.55 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\frac {-40 a^3 c-(13 b c-3 a d) x^4 \left (4 a^2+6 a \left (a+b x^4\right )+15 \left (a+b x^4\right )^2+15 \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{120 a^4 x^3 \left (a+b x^4\right )^{5/2}} \] Input:
Integrate[(c + d*x^4)/(x^4*(a + b*x^4)^(7/2)),x]
Output:
(-40*a^3*c - (13*b*c - 3*a*d)*x^4*(4*a^2 + 6*a*(a + b*x^4) + 15*(a + b*x^4 )^2 + 15*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4 , -((b*x^4)/a)]))/(120*a^4*x^3*(a + b*x^4)^(5/2))
Time = 0.47 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {955, 749, 749, 749, 761}
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.
| \(\displaystyle \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(13 b c-3 a d) \int \frac {1}{\left (b x^4+a\right )^{7/2}}dx}{3 a}-\frac {c}{3 a x^3 \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {(13 b c-3 a d) \left (\frac {9 \int \frac {1}{\left (b x^4+a\right )^{5/2}}dx}{10 a}+\frac {x}{10 a \left (a+b x^4\right )^{5/2}}\right )}{3 a}-\frac {c}{3 a x^3 \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {(13 b c-3 a d) \left (\frac {9 \left (\frac {5 \int \frac {1}{\left (b x^4+a\right )^{3/2}}dx}{6 a}+\frac {x}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x}{10 a \left (a+b x^4\right )^{5/2}}\right )}{3 a}-\frac {c}{3 a x^3 \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {(13 b c-3 a d) \left (\frac {9 \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {b x^4+a}}dx}{2 a}+\frac {x}{2 a \sqrt {a+b x^4}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x}{10 a \left (a+b x^4\right )^{5/2}}\right )}{3 a}-\frac {c}{3 a x^3 \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle -\frac {(13 b c-3 a d) \left (\frac {9 \left (\frac {5 \left (\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {x}{2 a \sqrt {a+b x^4}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x}{10 a \left (a+b x^4\right )^{5/2}}\right )}{3 a}-\frac {c}{3 a x^3 \left (a+b x^4\right )^{5/2}}\) |
Input:
Int[(c + d*x^4)/(x^4*(a + b*x^4)^(7/2)),x]
Output:
-1/3*c/(a*x^3*(a + b*x^4)^(5/2)) - ((13*b*c - 3*a*d)*(x/(10*a*(a + b*x^4)^ (5/2)) + (9*(x/(6*a*(a + b*x^4)^(3/2)) + (5*(x/(2*a*Sqrt[a + b*x^4]) + ((S qrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elliptic F[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*b^(1/4)*Sqrt[a + b*x^4]) ))/(6*a)))/(10*a)))/(3*a)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Result contains complex when optimal does not.
Time = 3.25 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.05
| method | result | size |
| elliptic | \(\frac {x \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{10 a^{2} b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {x \left (9 a d -19 c b \right ) \sqrt {b \,x^{4}+a}}{60 a^{3} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x \left (9 a d -31 c b \right )}{24 a^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {c \sqrt {b \,x^{4}+a}}{3 a^{4} x^{3}}+\frac {\left (\frac {9 a d -31 c b}{24 a^{4}}-\frac {b c}{3 a^{4}}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(214\) |
| default | \(d \left (\frac {x \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {3 x \sqrt {b \,x^{4}+a}}{20 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {3 x}{8 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 a^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+c \left (-\frac {x \sqrt {b \,x^{4}+a}}{10 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {19 x \sqrt {b \,x^{4}+a}}{60 a^{3} b \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {31 b x}{24 a^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{3 a^{4} x^{3}}-\frac {13 b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 a^{4} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(327\) |
| risch | \(-\frac {c \sqrt {b \,x^{4}+a}}{3 a^{4} x^{3}}-\frac {\frac {c b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-3 a^{3} \left (a d -c b \right ) \left (\frac {x \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {3 x \sqrt {b \,x^{4}+a}}{20 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {3 x}{8 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 a^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+3 a b c \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+3 a^{2} b c \left (\frac {x \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {5 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )}{3 a^{4}}\) | \(488\) |
Input:
int((d*x^4+c)/x^4/(b*x^4+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/10/a^2*x/b^3*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^3+1/60/a^3*x*(9*a*d-19* b*c)/b^2*(b*x^4+a)^(1/2)/(x^4+a/b)^2+1/24/a^4*x*(9*a*d-31*b*c)/((x^4+a/b)* b)^(1/2)-1/3/a^4*c*(b*x^4+a)^(1/2)/x^3+(1/24/a^4*(9*a*d-31*b*c)-1/3*b/a^4* c)/(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 )
Time = 0.11 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\frac {15 \, {\left ({\left (13 \, b^{4} c - 3 \, a b^{3} d\right )} x^{15} + 3 \, {\left (13 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} x^{11} + 3 \, {\left (13 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{7} + {\left (13 \, a^{3} b c - 3 \, a^{4} d\right )} x^{3}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (15 \, {\left (13 \, b^{4} c - 3 \, a b^{3} d\right )} x^{12} + 36 \, {\left (13 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} x^{8} + 40 \, a^{3} b c + 25 \, {\left (13 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{4}\right )} \sqrt {b x^{4} + a}}{120 \, {\left (a^{4} b^{4} x^{15} + 3 \, a^{5} b^{3} x^{11} + 3 \, a^{6} b^{2} x^{7} + a^{7} b x^{3}\right )}} \] Input:
integrate((d*x^4+c)/x^4/(b*x^4+a)^(7/2),x, algorithm="fricas")
Output:
1/120*(15*((13*b^4*c - 3*a*b^3*d)*x^15 + 3*(13*a*b^3*c - 3*a^2*b^2*d)*x^11 + 3*(13*a^2*b^2*c - 3*a^3*b*d)*x^7 + (13*a^3*b*c - 3*a^4*d)*x^3)*sqrt(a)* (-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) - (15*(13*b^4*c - 3*a* b^3*d)*x^12 + 36*(13*a*b^3*c - 3*a^2*b^2*d)*x^8 + 40*a^3*b*c + 25*(13*a^2* b^2*c - 3*a^3*b*d)*x^4)*sqrt(b*x^4 + a))/(a^4*b^4*x^15 + 3*a^5*b^3*x^11 + 3*a^6*b^2*x^7 + a^7*b*x^3)
Result contains complex when optimal does not.
Time = 164.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.40 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {7}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((d*x**4+c)/x**4/(b*x**4+a)**(7/2),x)
Output:
c*gamma(-3/4)*hyper((-3/4, 7/2), (1/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**( 7/2)*x**3*gamma(1/4)) + d*x*gamma(1/4)*hyper((1/4, 7/2), (5/4,), b*x**4*ex p_polar(I*pi)/a)/(4*a**(7/2)*gamma(5/4))
\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{2}} x^{4}} \,d x } \] Input:
integrate((d*x^4+c)/x^4/(b*x^4+a)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(7/2)*x^4), x)
\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{2}} x^{4}} \,d x } \] Input:
integrate((d*x^4+c)/x^4/(b*x^4+a)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(7/2)*x^4), x)
Timed out. \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\int \frac {d\,x^4+c}{x^4\,{\left (b\,x^4+a\right )}^{7/2}} \,d x \] Input:
int((c + d*x^4)/(x^4*(a + b*x^4)^(7/2)),x)
Output:
int((c + d*x^4)/(x^4*(a + b*x^4)^(7/2)), x)
\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/2}} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, d -3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a^{4} d \,x^{3}+13 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a^{3} b c \,x^{3}-9 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a^{3} b d \,x^{7}+39 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a^{2} b^{2} c \,x^{7}-9 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a^{2} b^{2} d \,x^{11}+39 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a \,b^{3} c \,x^{11}-3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) a \,b^{3} d \,x^{15}+13 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{20}+4 a \,b^{3} x^{16}+6 a^{2} b^{2} x^{12}+4 a^{3} b \,x^{8}+a^{4} x^{4}}d x \right ) b^{4} c \,x^{15}}{13 b \,x^{3} \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )} \] Input:
int((d*x^4+c)/x^4/(b*x^4+a)^(7/2),x)
Output:
( - sqrt(a + b*x**4)*d - 3*int(sqrt(a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2*b**2*x**12 + 4*a*b**3*x**16 + b**4*x**20),x)*a**4*d*x**3 + 13*in t(sqrt(a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2*b**2*x**12 + 4*a*b* *3*x**16 + b**4*x**20),x)*a**3*b*c*x**3 - 9*int(sqrt(a + b*x**4)/(a**4*x** 4 + 4*a**3*b*x**8 + 6*a**2*b**2*x**12 + 4*a*b**3*x**16 + b**4*x**20),x)*a* *3*b*d*x**7 + 39*int(sqrt(a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2* b**2*x**12 + 4*a*b**3*x**16 + b**4*x**20),x)*a**2*b**2*c*x**7 - 9*int(sqrt (a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2*b**2*x**12 + 4*a*b**3*x** 16 + b**4*x**20),x)*a**2*b**2*d*x**11 + 39*int(sqrt(a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2*b**2*x**12 + 4*a*b**3*x**16 + b**4*x**20),x)*a*b **3*c*x**11 - 3*int(sqrt(a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2*b **2*x**12 + 4*a*b**3*x**16 + b**4*x**20),x)*a*b**3*d*x**15 + 13*int(sqrt(a + b*x**4)/(a**4*x**4 + 4*a**3*b*x**8 + 6*a**2*b**2*x**12 + 4*a*b**3*x**16 + b**4*x**20),x)*b**4*c*x**15)/(13*b*x**3*(a**3 + 3*a**2*b*x**4 + 3*a*b** 2*x**8 + b**3*x**12))