Integrand size = 27, antiderivative size = 186 \[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {2 x \left (d-e x^2\right )}{3 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x \left (d-3 e x^2\right )}{6 d^2 \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/3*x*(-e*x^2+d)/(-e^2*x^4+d^2)^(3/2)+1/6*x*(-3*e*x^2+d)/d^2/(-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^(1/2)/e ^(1/2)/(-e^2*x^4+d^2)^(1/2)-1/3*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/ d^(1/2),I)/d^(1/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {d x \left (5 d^2-2 d e x^2-e^2 x^4\right )+d x \left (d^2-e^2 x^4\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )-4 e x^3 \left (d^2-e^2 x^4\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )}{6 d^2 \left (d^2-e^2 x^4\right )^{3/2}} \] Input:
Integrate[(d - e*x^2)^3/(d^2 - e^2*x^4)^(5/2),x]
Output:
(d*x*(5*d^2 - 2*d*e*x^2 - e^2*x^4) + d*x*(d^2 - e^2*x^4)*Sqrt[1 - (e^2*x^4 )/d^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (e^2*x^4)/d^2] - 4*e*x^3*(d^2 - e^ 2*x^4)*Sqrt[1 - (e^2*x^4)/d^2]*Hypergeometric2F1[3/4, 5/2, 7/4, (e^2*x^4)/ d^2])/(6*d^2*(d^2 - e^2*x^4)^(3/2))
Time = 0.70 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1396, 314, 25, 402, 25, 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.
| \(\displaystyle \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {\sqrt {d-e x^2}}{\left (e x^2+d\right )^{5/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {2 d-e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{3 d}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {2 d-e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {d e \left (3 e x^2+d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {d e \left (3 e x^2+d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {3 e x^2+d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-2 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[(d - e*x^2)^3/(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])/(3*d*(d + e*x^2)^(3/ 2)) + ((3*x*Sqrt[d - e*x^2])/(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]) - (2*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] ))/(2*d))/(3*d)))/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[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* (p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d *(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, 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 = 6.14 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.19
| method | result | size |
| elliptic | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3 d \,e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(222\) |
| default | \(d^{3} \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{6 d^{2} e^{4} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}+\frac {5 x}{12 d^{4} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {5 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{12 d^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-e^{3} \left (\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{6 e^{6} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}-\frac {x^{3}}{4 e^{2} d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{4 e^{3} d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d \,e^{2} \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{6 e^{6} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}-\frac {x}{12 e^{2} d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{12 e^{2} d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-3 d^{2} e \left (\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{6 d^{2} e^{4} \left (x^{4}-\frac {d^{2}}{e^{2}}\right )^{2}}+\frac {x^{3}}{4 d^{4} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{4 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(574\) |
Input:
int((-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3/d*x/e^2*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+1/2*(-e^2*x^2+d*e)/d^2*x/e/(( x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/6/d/(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/(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 = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {3 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (d e^{2} - 3 \, e^{3}\right )} x^{4} + d^{3} - 3 \, d^{2} e + 2 \, {\left (d^{2} e - 3 \, d e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (3 \, e^{2} x^{3} + 5 \, d e x\right )}}{6 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}} \] Input:
integrate((-e*x^2+d)^3/(-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) + ((d*e^2 - 3*e^3)*x^4 + d^3 - 3*d^2*e + 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 + 5*d*e*x))/(d^2*e^3*x^4 + 2*d^3*e^2*x^2 + d^4*e)
\[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=- \int \left (- \frac {d^{3}}{d^{4} \sqrt {d^{2} - e^{2} x^{4}} - 2 d^{2} e^{2} x^{4} \sqrt {d^{2} - e^{2} x^{4}} + e^{4} x^{8} \sqrt {d^{2} - e^{2} x^{4}}}\right )\, dx - \int \frac {e^{3} x^{6}}{d^{4} \sqrt {d^{2} - e^{2} x^{4}} - 2 d^{2} e^{2} x^{4} \sqrt {d^{2} - e^{2} x^{4}} + e^{4} x^{8} \sqrt {d^{2} - e^{2} x^{4}}}\, dx - \int \left (- \frac {3 d e^{2} x^{4}}{d^{4} \sqrt {d^{2} - e^{2} x^{4}} - 2 d^{2} e^{2} x^{4} \sqrt {d^{2} - e^{2} x^{4}} + e^{4} x^{8} \sqrt {d^{2} - e^{2} x^{4}}}\right )\, dx - \int \frac {3 d^{2} e x^{2}}{d^{4} \sqrt {d^{2} - e^{2} x^{4}} - 2 d^{2} e^{2} x^{4} \sqrt {d^{2} - e^{2} x^{4}} + e^{4} x^{8} \sqrt {d^{2} - e^{2} x^{4}}}\, dx \] Input:
integrate((-e*x**2+d)**3/(-e**2*x**4+d**2)**(5/2),x)
Output:
-Integral(-d**3/(d**4*sqrt(d**2 - e**2*x**4) - 2*d**2*e**2*x**4*sqrt(d**2 - e**2*x**4) + e**4*x**8*sqrt(d**2 - e**2*x**4)), x) - Integral(e**3*x**6/ (d**4*sqrt(d**2 - e**2*x**4) - 2*d**2*e**2*x**4*sqrt(d**2 - e**2*x**4) + e **4*x**8*sqrt(d**2 - e**2*x**4)), x) - Integral(-3*d*e**2*x**4/(d**4*sqrt( d**2 - e**2*x**4) - 2*d**2*e**2*x**4*sqrt(d**2 - e**2*x**4) + e**4*x**8*sq rt(d**2 - e**2*x**4)), x) - Integral(3*d**2*e*x**2/(d**4*sqrt(d**2 - e**2* x**4) - 2*d**2*e**2*x**4*sqrt(d**2 - e**2*x**4) + e**4*x**8*sqrt(d**2 - e* *2*x**4)), x)
\[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { -\frac {{\left (e x^{2} - d\right )}^{3}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
Output:
-integrate((e*x^2 - d)^3/(-e^2*x^4 + d^2)^(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { -\frac {{\left (e x^{2} - d\right )}^{3}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
Output:
integrate(-(e*x^2 - d)^3/(-e^2*x^4 + d^2)^(5/2), x)
Timed out. \[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^3}{{\left (d^2-e^2\,x^4\right )}^{5/2}} \,d x \] Input:
int((d - e*x^2)^3/(d^2 - e^2*x^4)^(5/2),x)
Output:
int((d - e*x^2)^3/(d^2 - e^2*x^4)^(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \] Input:
int((-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x* *6),x)