Integrand size = 27, antiderivative size = 227 \[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x}{10 d^2 \left (d-e x^2\right )^2 \sqrt {d^2-e^2 x^4}}+\frac {7 x}{30 d^3 \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}}+\frac {x \left (20 d+21 e x^2\right )}{60 d^5 \sqrt {d^2-e^2 x^4}}-\frac {7 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{20 d^{7/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {41 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{60 d^{7/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
1/10*x/d^2/(-e*x^2+d)^2/(-e^2*x^4+d^2)^(1/2)+7/30*x/d^3/(-e*x^2+d)/(-e^2*x ^4+d^2)^(1/2)+1/60*x*(21*e*x^2+20*d)/d^5/(-e^2*x^4+d^2)^(1/2)-7/20*(1-e^2* x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(7/2)/e^(1/2)/(-e^2*x^4+d^ 2)^(1/2)+41/60*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/d^(7/2 )/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.66 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (40 d^3-33 d^2 e x^2-22 d e^2 x^4+21 e^3 x^6\right )+21 i d \left (d-e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-41 i d \left (d-e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{60 d^5 \sqrt {-\frac {e}{d}} \left (d-e x^2\right )^2 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[1/((d - e*x^2)^2*(d^2 - e^2*x^4)^(3/2)),x]
Output:
(Sqrt[-(e/d)]*x*(40*d^3 - 33*d^2*e*x^2 - 22*d*e^2*x^4 + 21*e^3*x^6) + (21* I)*d*(d - e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d) ]*x], -1] - (41*I)*d*(d - e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*Arc Sinh[Sqrt[-(e/d)]*x], -1])/(60*d^5*Sqrt[-(e/d)]*(d - e*x^2)^2*Sqrt[d^2 - e ^2*x^4])
Time = 0.84 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.44, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {1396, 316, 27, 402, 27, 402, 25, 27, 402, 27, 399, 289, 329, 327, 765, 762}
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.
| \(\displaystyle \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (d-e x^2\right )^{7/2} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (5 e x^2+9 d\right )}{\left (d-e x^2\right )^{5/2} \left (e x^2+d\right )^{3/2}}dx}{10 d^2 e}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {5 e x^2+9 d}{\left (d-e x^2\right )^{5/2} \left (e x^2+d\right )^{3/2}}dx}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {2 d e \left (21 e x^2+20 d\right )}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {21 e x^2+20 d}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int -\frac {d e \left (d-41 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{2 d^2 e}+\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\int \frac {d e \left (d-41 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{2 d^2 e}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\int \frac {d-41 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\int \frac {2 d e \left (20 d-21 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\int \frac {20 d-21 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {41 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\frac {41 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\frac {41 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\frac {41 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\frac {41 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {41 x}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\frac {41 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}}{2 d}}{3 d}+\frac {7 x}{3 d \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}}}{10 d^2}+\frac {x}{10 d^2 \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[1/((d - e*x^2)^2*(d^2 - e^2*x^4)^(3/2)),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(10*d^2*(d - e*x^2)^(5/2)*Sqrt[d + e*x ^2]) + ((7*x)/(3*d*(d - e*x^2)^(3/2)*Sqrt[d + e*x^2]) + ((41*x)/(2*d*Sqrt[ d - e*x^2]*Sqrt[d + e*x^2]) - ((21*x*Sqrt[d - e*x^2])/(d*Sqrt[d + e*x^2]) - ((-21*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[ d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) + (41*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[ d - e*x^2]*Sqrt[d + e*x^2]))/d)/(2*d))/(3*d))/(10*d^2)))/Sqrt[d^2 - e^2*x^ 4]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart [p]) Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* c + a*d, 0] && !IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, 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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) Int[Sqrt[1 + b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c + a*d, 0] && !(LtQ[a*c, 0] && GtQ[a*b, 0])
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 1.74 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(\frac {\left (-e^{2} x^{2}+d e \right ) x}{16 e \,d^{5} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{20 d^{3} e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {17 x \sqrt {-e^{2} x^{4}+d^{2}}}{120 d^{4} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {33 \left (-e^{2} x^{2}-d e \right ) x}{80 e \,d^{5} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 d^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {7 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{20 d^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(306\) |
| elliptic | \(\frac {\left (-e^{2} x^{2}+d e \right ) x}{16 e \,d^{5} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{20 d^{3} e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {17 x \sqrt {-e^{2} x^{4}+d^{2}}}{120 d^{4} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {33 \left (-e^{2} x^{2}-d e \right ) x}{80 e \,d^{5} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 d^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {7 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{20 d^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(306\) |
Input:
int(1/(-e*x^2+d)^2/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/16*(-e^2*x^2+d*e)/e/d^5*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)-1/20/d^3*x/e^ 3*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^3+17/120/d^4/e^2*x*(-e^2*x^4+d^2)^(1/2)/( x^2-d/e)^2-33/80*(-e^2*x^2-d*e)/e/d^5*x/((x^2-d/e)*(-e^2*x^2-d*e))^(1/2)+1 /3/d^4/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2 )*EllipticF(x*(e/d)^(1/2),I)+7/20/d^4/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x ^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*( e/d)^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=-\frac {21 \, {\left (e^{5} x^{8} - 2 \, d e^{4} x^{6} + 2 \, d^{3} e^{2} x^{2} - d^{4} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (20 \, d e^{4} + 21 \, e^{5}\right )} x^{8} - 2 \, {\left (20 \, d^{2} e^{3} + 21 \, d e^{4}\right )} x^{6} - 20 \, d^{5} - 21 \, d^{4} e + 2 \, {\left (20 \, d^{4} e + 21 \, d^{3} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (21 \, e^{4} x^{7} - 22 \, d e^{3} x^{5} - 33 \, d^{2} e^{2} x^{3} + 40 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{60 \, {\left (d^{5} e^{5} x^{8} - 2 \, d^{6} e^{4} x^{6} + 2 \, d^{8} e^{2} x^{2} - d^{9} e\right )}} \] Input:
integrate(1/(-e*x^2+d)^2/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
-1/60*(21*(e^5*x^8 - 2*d*e^4*x^6 + 2*d^3*e^2*x^2 - d^4*e)*sqrt(e/d)*ellipt ic_e(arcsin(x*sqrt(e/d)), -1) - ((20*d*e^4 + 21*e^5)*x^8 - 2*(20*d^2*e^3 + 21*d*e^4)*x^6 - 20*d^5 - 21*d^4*e + 2*(20*d^4*e + 21*d^3*e^2)*x^2)*sqrt(e /d)*elliptic_f(arcsin(x*sqrt(e/d)), -1) + (21*e^4*x^7 - 22*d*e^3*x^5 - 33* d^2*e^2*x^3 + 40*d^3*e*x)*sqrt(-e^2*x^4 + d^2))/(d^5*e^5*x^8 - 2*d^6*e^4*x ^6 + 2*d^8*e^2*x^2 - d^9*e)
\[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}} \left (- d + e x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(-e*x**2+d)**2/(-e**2*x**4+d**2)**(3/2),x)
Output:
Integral(1/((-(-d + e*x**2)*(d + e*x**2))**(3/2)*(-d + e*x**2)**2), x)
\[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} - d\right )}^{2}} \,d x } \] Input:
integrate(1/(-e*x^2+d)^2/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-e^2*x^4 + d^2)^(3/2)*(e*x^2 - d)^2), x)
\[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} - d\right )}^{2}} \,d x } \] Input:
integrate(1/(-e*x^2+d)^2/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-e^2*x^4 + d^2)^(3/2)*(e*x^2 - d)^2), x)
Timed out. \[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{3/2}\,{\left (d-e\,x^2\right )}^2} \,d x \] Input:
int(1/((d^2 - e^2*x^4)^(3/2)*(d - e*x^2)^2),x)
Output:
int(1/((d^2 - e^2*x^4)^(3/2)*(d - e*x^2)^2), x)
\[ \int \frac {1}{\left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{6} x^{12}-2 d \,e^{5} x^{10}-d^{2} e^{4} x^{8}+4 d^{3} e^{3} x^{6}-d^{4} e^{2} x^{4}-2 d^{5} e \,x^{2}+d^{6}}d x \] Input:
int(1/(-e*x^2+d)^2/(-e^2*x^4+d^2)^(3/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**6 - 2*d**5*e*x**2 - d**4*e**2*x**4 + 4*d**3 *e**3*x**6 - d**2*e**4*x**8 - 2*d*e**5*x**10 + e**6*x**12),x)