Integrand size = 27, antiderivative size = 191 \[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x \left (d-e x^2\right )}{3 d \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x \left (2 d-3 e x^2\right )}{6 d^3 \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{6 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
1/3*x*(-e*x^2+d)/d/(-e^2*x^4+d^2)^(3/2)+1/6*x*(-3*e*x^2+2*d)/d^3/(-e^2*x^4 +d^2)^(1/2)+1/2*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(3/ 2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)-1/6*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2 )*x/d^(1/2),I)/d^(3/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (4 d^2-d e x^2-3 e^2 x^4\right )-3 i d \left (d+e x^2\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )+i d \left (d+e x^2\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{6 d^3 \sqrt {-\frac {e}{d}} \left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[(d - e*x^2)^2/(d^2 - e^2*x^4)^(5/2),x]
Output:
(Sqrt[-(e/d)]*x*(4*d^2 - d*e*x^2 - 3*e^2*x^4) - (3*I)*d*(d + e*x^2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] + I*d*(d + e*x^ 2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1])/(6*d^ 3*Sqrt[-(e/d)]*(d + e*x^2)*Sqrt[d^2 - e^2*x^4])
Time = 0.69 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.30, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1396, 316, 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.
\(\displaystyle \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 316 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {e \left (5 d-e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (5 d-e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {5 d-e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\int -\frac {2 d e \left (3 e x^2+2 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {3 e x^2+2 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 289 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 329 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {3 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d^2}+\frac {x \sqrt {d-e x^2}}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[(d - e*x^2)^2/(d^2 - e^2*x^4)^(5/2),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((x*Sqrt[d - e*x^2])/(6*d^2*(d + e*x^2)^( 3/2)) + ((3*x*Sqrt[d - e*x^2])/(d*Sqrt[d + e*x^2]) + ((3*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]) - (d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[Ar cSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])) /d)/(6*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 = 3.98 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.16
method | result | size |
elliptic | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{6 e^{2} d^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 e \,d^{3} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(222\) |
default | \(d^{2} \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{6 d^{2} e^{4} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}+\frac {5 x}{12 d^{4} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {5 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{12 d^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{2} \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{6 e^{6} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}-\frac {x}{12 e^{2} d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{12 e^{2} d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-2 d e \left (\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{6 d^{2} e^{4} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}+\frac {x^{3}}{4 d^{4} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{4 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(417\) |
Input:
int((-e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/6/e^2/d^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+1/2*(-e^2*x^2+d*e)/e/d^3*x/ ((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/3/d^2/(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)-1/2/d^2/(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)*(Ellipti cF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {3 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (2 \, d e^{2} - 3 \, e^{3}\right )} x^{4} + 2 \, d^{3} - 3 \, d^{2} e + 2 \, {\left (2 \, d^{2} e - 3 \, d e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (3 \, e^{2} x^{3} + 4 \, d e x\right )}}{6 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}} \] Input:
integrate((-e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
Output:
1/6*(3*(e^3*x^4 + 2*d*e^2*x^2 + d^2*e)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt( e/d)), -1) + ((2*d*e^2 - 3*e^3)*x^4 + 2*d^3 - 3*d^2*e + 2*(2*d^2*e - 3*d*e ^2)*x^2)*sqrt(e/d)*elliptic_f(arcsin(x*sqrt(e/d)), -1) + sqrt(-e^2*x^4 + d ^2)*(3*e^2*x^3 + 4*d*e*x))/(d^3*e^3*x^4 + 2*d^4*e^2*x^2 + d^5*e)
\[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (- d + e x^{2}\right )^{2}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((-e*x**2+d)**2/(-e**2*x**4+d**2)**(5/2),x)
Output:
Integral((-d + e*x**2)**2/(-(-d + e*x**2)*(d + e*x**2))**(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} - d\right )}^{2}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
Output:
integrate((e*x^2 - d)^2/(-e^2*x^4 + d^2)^(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} - d\right )}^{2}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
Output:
integrate((e*x^2 - d)^2/(-e^2*x^4 + d^2)^(5/2), x)
Timed out. \[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^2}{{\left (d^2-e^2\,x^4\right )}^{5/2}} \,d x \] Input:
int((d - e*x^2)^2/(d^2 - e^2*x^4)^(5/2),x)
Output:
int((d - e*x^2)^2/(d^2 - e^2*x^4)^(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^2}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \] Input:
int((-e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x* *8),x)