Integrand size = 22, antiderivative size = 299 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 d \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}} \]
-1/2*e^2*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)-1/2*a^(3/4)*c^(1/4) *e*EllipticE(c^(1/4)*x/a^(1/4),I)*(1-c*x^4/a)^(1/2)/d/(-a*e^2+c*d^2)/(-c*x ^4+a)^(1/2)+1/2*a^(1/4)*(-a*e^2+3*c*d^2)*EllipticPi(c^(1/4)*x/a^(1/4),-e*a ^(1/2)/d/c^(1/2),I)*(1-c*x^4/a)^(1/2)/c^(1/4)/d^2/(-a*e^2+c*d^2)/(-c*x^4+a )^(1/2)-1/2*a^(1/4)*c^(1/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(1-c*x^4/a)^(1/ 2)/d/(e*a^(1/2)+d*c^(1/2))/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.81 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\frac {-a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d e^2 x+\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d e^2 x^5+i \sqrt {a} \sqrt {c} d e \left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {c} d \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i c d^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a d e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i c d^2 e x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a e^3 x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 \left (c d^2-a e^2\right ) \left (d+e x^2\right ) \sqrt {a-c x^4}} \]
(-(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^2*x) + Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*e^2* x^5 + I*Sqrt[a]*Sqrt[c]*d*e*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*Ar cSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*Sqrt[c]*d*(-(Sqrt[c]*d) + Sqrt[ a]*e)*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/S qrt[a])]*x], -1] - (3*I)*c*d^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e )/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*d*e^2*Sqr t[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(S qrt[c]/Sqrt[a])]*x], -1] - (3*I)*c*d^2*e*x^2*Sqrt[1 - (c*x^4)/a]*EllipticP i[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*e^3*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I* ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2* (c*d^2 - a*e^2)*(d + e*x^2)*Sqrt[a - c*x^4])
Time = 0.72 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1552, 2235, 27, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
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.
\(\displaystyle \int \frac {1}{\sqrt {a-c x^4} \left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1552 |
\(\displaystyle \frac {\int \frac {-c e^2 x^4-2 c d e x^2+2 c d^2-a e^2}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 2235 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-\frac {\int \frac {c e^2 \left (e x^2+d\right )}{\sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \int \frac {e x^2+d}{\sqrt {a-c x^4}}dx}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1513 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {\sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1390 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1543 |
\(\displaystyle \frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-c \left (\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1542 |
\(\displaystyle \frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}}-c \left (\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\) |
-1/2*(e^2*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e*x^2)) + (-(c*((a^(3 /4)*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^( 3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(d - (Sqrt[a]*e)/Sqrt[c])*Sqrt[1 - (c*x^4 )/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]) )) + (a^(1/4)*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]* e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*Sqrt[a - c*x ^4]))/(2*d*(c*d^2 - a*e^2))
3.2.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp [(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2) )), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sq rt[a + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, -1]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si mp[-(e^2)^(-1) Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( C*d^2 - B*d*e + A*e^2)/e^2 Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] / ; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (245 ) = 490\).
Time = 0.77 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{2 \left (a \,e^{2}-c \,d^{2}\right ) d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(523\) |
elliptic | \(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{2 \left (a \,e^{2}-c \,d^{2}\right ) d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(523\) |
1/2*e^2/(a*e^2-c*d^2)/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)+1/2*c/(a*e^2-c*d^2)/( 1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1 /2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-1 /2*c^(1/2)*e/(a*e^2-c*d^2)/d*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2 )*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*Elli pticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+1/2*c^(1/2)*e/(a*e^2-c*d^2)/d*a^(1/2) /(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^ (1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I) +1/2/(a*e^2-c*d^2)/d^2*e^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)* x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*( 1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c^(1/2))^(1/2)/( 1/a^(1/2)*c^(1/2))^(1/2))*a-3/2/(a*e^2-c*d^2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1 -1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^( 1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/ 2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*c
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^2} \,d x \]