Integrand size = 24, antiderivative size = 809 \[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4}}{a x}+\frac {\sqrt {d} x \sqrt {c+d x^4}}{a \left (\sqrt {c}+\sqrt {d} x^2\right )}-\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \arctan \left (\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 a}-\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \arctan \left (\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 a}-\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 )}{a \sqrt {c+d x^4}}+\frac {b c^{5/4} \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 )}{a (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (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 )}{8 \sqrt {b} \sqrt [4]{c} \left ((-a)^{3/2} \sqrt {b} \sqrt {c}+a^2 \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (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 {\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 a \sqrt {b} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \] Output:
-(d*x^4+c)^(1/2)/a/x+d^(1/2)*x*(d*x^4+c)^(1/2)/a/(c^(1/2)+d^(1/2)*x^2)-1/4 *(-(-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)*arctan((-(-a*d+b*c)/(-a)^(1/2)/b^( 1/2))^(1/2)*x/(d*x^4+c)^(1/2))/a-1/4*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2) *arctan(((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)*x/(d*x^4+c)^(1/2))/a-c^(1/4) *d^(1/4)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*E llipticE(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))/a/(d*x^4+c)^(1/2)+b *c^(5/4)*d^(1/4)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2) ^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*x/c^(1/4)),1/2*2^(1/2))/a/(a*d+b*c )/(d*x^4+c)^(1/2)-1/8*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*(-a*d+b*c)*(c^( 1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*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))/b^(1/2)/c^(1/4)/((-a)^(3/2) *b^(1/2)*c^(1/2)+a^2*d^(1/2))/d^(1/4)/(d*x^4+c)^(1/2)-1/8*(b^(1/2)*c^(1/2) +(-a)^(1/2)*d^(1/2))*(-a*d+b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+ d^(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(1/4))),-1/4*c^( 1/2)*(b^(1/2)-(-a)^(1/2)*d^(1/2)/c^(1/2))^2/(-a)^(1/2)/b^(1/2)/d^(1/2),1/2 *2^(1/2))/a/b^(1/2)/c^(1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^(1/2))/d^(1/4) /(d*x^4+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\frac {-21 a \left (c+d x^4\right )-7 (b c-2 a d) x^4 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+3 b d x^8 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{21 a^2 x \sqrt {c+d x^4}} \] Input:
Integrate[Sqrt[c + d*x^4]/(x^2*(a + b*x^4)),x]
Output:
(-21*a*(c + d*x^4) - 7*(b*c - 2*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + 3*b*d*x^8*Sqrt[1 + (d*x^4)/c]* AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(21*a^2*x*Sqrt[c + d*x^4])
Time = 2.28 (sec) , antiderivative size = 1021, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {975, 25, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx\) |
| \(\Big \downarrow \) 975 |
| \(\displaystyle \frac {\int -\frac {x^2 \left (-b d x^4+b c-2 a d\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{a}-\frac {\sqrt {c+d x^4}}{a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {x^2 \left (-b d x^4+b c-2 a d\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{a}-\frac {\sqrt {c+d x^4}}{a x}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle -\frac {\int \left (\frac {(b c-a d) x^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}-\frac {d x^2}{\sqrt {d x^4+c}}\right )dx}{a}-\frac {\sqrt {c+d x^4}}{a x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {(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} \sqrt {b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}+\frac {\sqrt {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} \sqrt [4]{b}}-\frac {\sqrt {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} \sqrt [4]{b}}+\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 )}{\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 \sqrt {d x^4+c}}-\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (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 \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} (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 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (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} \sqrt {b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt {d} x \sqrt {d x^4+c}}{\sqrt {d} x^2+\sqrt {c}}}{a}-\frac {\sqrt {d x^4+c}}{a x}\) |
Input:
Int[Sqrt[c + d*x^4]/(x^2*(a + b*x^4)),x]
Output:
-(Sqrt[c + d*x^4]/(a*x)) - (-((Sqrt[d]*x*Sqrt[c + d*x^4])/(Sqrt[c] + Sqrt[ d]*x^2)) + (Sqrt[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^(1/4)) - (Sqrt[b*c - a*d]*ArcTanh[(Sqr t[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(1/4)*b^(1/ 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]*EllipticE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/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*Sqrt[ c + d*x^4]) - ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - a*d)* (Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellipt icF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*c^(1/4)*(b*c + a*d)*Sqrt[c + d *x^4]) - ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - a*d)*(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])/(4*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4] ) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(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]*Sqrt[b]*c^(1/4)*d^(1/4)*(b*c + a*d) *Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(b*c - a*d)*(S qrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellip...
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ (a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n) ^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi alQ[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 complex when optimal does not.
Time = 4.72 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(-\frac {\sqrt {d \,x^{4}+c}}{a x}+\frac {\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}+\frac {\left (a d -c b \right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 b}}{a}\) | \(318\) |
| elliptic | \(-\frac {\sqrt {d \,x^{4}+c}}{a x}+\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{a \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-a d +c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 b a}\) | \(319\) |
| default | \(\frac {-\frac {\sqrt {d \,x^{4}+c}}{x}+\frac {2 i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}}{a}-\frac {b \left (\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 b^{2}}\right )}{a}\) | \(421\) |
Input:
int((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
-(d*x^4+c)^(1/2)/a/x+1/a*(I*d^(1/2)*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))+1/8*(a*d-b*c)/b*sum(1/_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)*_alpha^2/a*b,(-I/c^(1/2)*d^(1/2))^(1/2)/(I/c^(1/2)*d^ (1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a)))
Timed out. \[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {c + d x^{4}}}{x^{2} \left (a + b x^{4}\right )}\, dx \] Input:
integrate((d*x**4+c)**(1/2)/x**2/(b*x**4+a),x)
Output:
Integral(sqrt(c + d*x**4)/(x**2*(a + b*x**4)), x)
\[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{2}} \,d x } \] Input:
integrate((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2), x)
\[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{2}} \,d x } \] Input:
integrate((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2), x)
Timed out. \[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {d\,x^4+c}}{x^2\,\left (b\,x^4+a\right )} \,d x \] Input:
int((c + d*x^4)^(1/2)/(x^2*(a + b*x^4)),x)
Output:
int((c + d*x^4)^(1/2)/(x^2*(a + b*x^4)), x)
\[ \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {d \,x^{4}+c}}{b \,x^{6}+a \,x^{2}}d x \] Input:
int((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x)
Output:
int(sqrt(c + d*x**4)/(a*x**2 + b*x**6),x)