Integrand size = 24, antiderivative size = 1146 \[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\sqrt {d} x \sqrt {c+d x^4}}{4 b (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right )}-\frac {x^3 \sqrt {c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 \sqrt [4]{-a} b^{5/4} (b c-a d)^{3/2}}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 \sqrt [4]{-a} b^{5/4} (-b c+a d)^{3/2}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b (b c-a d) \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b (b c-a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 b \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 b \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{32 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{32 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^4}} \]
1/16*(-a*d+3*b*c)*arctan(x*(-a*d+b*c)^(1/2)/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^( 1/2))/(-a)^(1/4)/b^(5/4)/(-a*d+b*c)^(3/2)+1/16*(-a*d+3*b*c)*arctan(x*(a*d- b*c)^(1/2)/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(1/4)/b^(5/4)/(a*d-b*c )^(3/2)-1/4*x^3*(d*x^4+c)^(1/2)/(-a*d+b*c)/(b*x^4+a)+1/4*x*d^(1/2)*(d*x^4+ c)^(1/2)/b/(-a*d+b*c)/(c^(1/2)+x^2*d^(1/2))-1/4*c^(1/4)*d^(1/4)*(cos(2*arc tan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*Elliptic E(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*((d* x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b/(-a*d+b*c)/(d*x^4+c)^(1/2)+1/8*c^( 1/4)*d^(1/4)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/ 4)*x/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^ (1/2)+x^2*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b/(-a*d+b*c)/ (d*x^4+c)^(1/2)-1/16*d^(1/4)*(-a*d+3*b*c)*(cos(2*arctan(d^(1/4)*x/c^(1/4)) )^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4) *x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)-(-a)^(1/2 )*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^(3/2)/c^(1/4)/(-a^2 *d^2+b^2*c^2)/(d*x^4+c)^(1/2)-1/32*(-a*d+3*b*c)*(cos(2*arctan(d^(1/4)*x/c^ (1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan( d^(1/4)*x/c^(1/4))),1/4*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/ b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2 )-(-a)^(1/2)*d^(1/2))^2*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^(3/...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.39 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.14 \[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {-7 a x^3 \left (c+d x^4\right )+7 c x^3 \left (a+b x^4\right ) \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+d x^7 \left (a+b x^4\right ) \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{28 a (b c-a d) \left (a+b x^4\right ) \sqrt {c+d x^4}} \]
(-7*a*x^3*(c + d*x^4) + 7*c*x^3*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[3 /4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + d*x^7*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(28*a* (b*c - a*d)*(a + b*x^4)*Sqrt[c + d*x^4])
Time = 1.55 (sec) , antiderivative size = 1085, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {971, 1054, 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.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle \frac {\int \frac {x^2 \left (d x^4+3 c\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{4 (b c-a d)}-\frac {x^3 \sqrt {c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {\int \left (\frac {d x^2}{b \sqrt {d x^4+c}}+\frac {(3 b c-a d) x^2}{b \left (b x^4+a\right ) \sqrt {d x^4+c}}\right )dx}{4 (b c-a d)}-\frac {x^3 \sqrt {c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {(3 b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 \sqrt [4]{-a} b^{5/4} \sqrt {b c-a d}}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 \sqrt [4]{-a} b^{5/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{b \sqrt {d x^4+c}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt {d x^4+c}}-\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}-\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}+\frac {\sqrt {d} x \sqrt {d x^4+c}}{b \left (\sqrt {d} x^2+\sqrt {c}\right )}}{4 (b c-a d)}-\frac {x^3 \sqrt {d x^4+c}}{4 (b c-a d) \left (b x^4+a\right )}\) |
-1/4*(x^3*Sqrt[c + d*x^4])/((b*c - a*d)*(a + b*x^4)) + ((Sqrt[d]*x*Sqrt[c + d*x^4])/(b*(Sqrt[c] + Sqrt[d]*x^2)) + ((3*b*c - a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(1/4)*b^(5/4)*Sqrt[ b*c - a*d]) - ((3*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/ 4)*Sqrt[c + d*x^4])])/(4*(-a)^(1/4)*b^(5/4)*Sqrt[b*c - a*d]) - (c^(1/4)*d^ (1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]* EllipticE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(b*Sqrt[c + d*x^4]) + (c^(1 /4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^ 2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*b*Sqrt[c + d*x^4]) - ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(3*b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*Arc Tan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(3*b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcT an[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(3*b*c - a*d)*(Sqrt[c] + Sqrt[d]* x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[ c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d ^(1/4)*x)/c^(1/4)], 1/2])/(8*Sqrt[-a]*b^(3/2)*c^(1/4)*d^(1/4)*(b*c + a*d)* Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(3*b*c - a*d...
3.9.38.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.02 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.31
| method | result | size |
| elliptic | \(\frac {x^{3} \sqrt {d \,x^{4}+c}}{4 \left (a d -b c \right ) \left (b \,x^{4}+a \right )}-\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{4 \left (a d -b c \right ) b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-a d +3 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha }}{32 b^{2}}\) | \(353\) |
| default | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha }}{8 b^{2}}-\frac {a \left (-\frac {b \,x^{3} \sqrt {d \,x^{4}+c}}{4 \left (a d -b c \right ) a \left (b \,x^{4}+a \right )}+\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{4 \left (a d -b c \right ) a \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-3 a d +b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha }}{32 b a}\right )}{b}\) | \(556\) |
1/4/(a*d-b*c)*x^3*(d*x^4+c)^(1/2)/(b*x^4+a)-1/4*I*d^(1/2)/(a*d-b*c)/b*c^(1 /2)/(I/c^(1/2)*d^(1/2))^(1/2)*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2) *d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*(EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2) ,I)-EllipticE(x*(I/c^(1/2)*d^(1/2))^(1/2),I))-1/32/b^2*sum((-a*d+3*b*c)/(a *d-b*c)/_alpha*(-1/((-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c) /((-a*d+b*c)/b)^(1/2)/(d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^ 3*b/a*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x ^4+c)^(1/2)*EllipticPi(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_alph a^2/a*b,(-I/c^(1/2)*d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=Root Of(_Z^4*b+a))
Timed out. \[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{6}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \]
\[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {x^{6}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}} \,d x } \]
\[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {x^{6}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}} \,d x } \]
Timed out. \[ \int \frac {x^6}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^6}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \]