Integrand size = 27, antiderivative size = 172 \[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {4 d x \left (d-e x^2\right )}{\sqrt {d^2-e^2 x^4}}+\frac {1}{3} x \sqrt {d^2-e^2 x^4}+\frac {8 d^{5/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {34 d^{5/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
4*d*x*(-e*x^2+d)/(-e^2*x^4+d^2)^(1/2)+1/3*x*(-e^2*x^4+d^2)^(1/2)+8*d^(5/2) *(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^ 2)^(1/2)-34/3*d^(5/2)*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I) /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 = 10.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {13 d^2 x+12 d e x^3-e^2 x^5-10 d^2 x \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )-16 d e x^3 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )}{3 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[(d - e*x^2)^4/(d^2 - e^2*x^4)^(3/2),x]
Output:
(13*d^2*x + 12*d*e*x^3 - e^2*x^5 - 10*d^2*x*Sqrt[1 - (e^2*x^4)/d^2]*Hyperg eometric2F1[1/4, 1/2, 5/4, (e^2*x^4)/d^2] - 16*d*e*x^3*Sqrt[1 - (e^2*x^4)/ d^2]*Hypergeometric2F1[3/4, 3/2, 7/4, (e^2*x^4)/d^2])/(3*Sqrt[d^2 - e^2*x^ 4])
Time = 0.66 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.37, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1396, 315, 25, 27, 403, 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.
| \(\displaystyle \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {\left (d-e x^2\right )^{5/2}}{\left (e x^2+d\right )^{3/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int -\frac {d e \left (d-7 e x^2\right ) \sqrt {d-e x^2}}{\sqrt {e x^2+d}}dx}{d e}+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}-\frac {\int \frac {d e \left (d-7 e x^2\right ) \sqrt {d-e x^2}}{\sqrt {e x^2+d}}dx}{d e}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}-\int \frac {\left (d-7 e x^2\right ) \sqrt {d-e x^2}}{\sqrt {e x^2+d}}dx\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {\int \frac {2 d e \left (5 d-12 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \int \frac {5 d-12 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \left (17 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-12 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx\right )+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \left (\frac {17 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-12 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx\right )+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \left (\frac {17 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \left (\frac {17 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \left (\frac {17 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {2}{3} d \left (\frac {17 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^{3/2}}{\sqrt {d+e x^2}}+\frac {7}{3} x \sqrt {d+e x^2} \sqrt {d-e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[(d - e*x^2)^4/(d^2 - e^2*x^4)^(3/2),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((2*x*(d - e*x^2)^(3/2))/Sqrt[d + e*x^2] + (7*x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])/3 - (2*d*((-12*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]) + (17*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[A rcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) ))/3))/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[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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[(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 = 17.72 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.18
| method | result | size |
| elliptic | \(\frac {4 \left (-e^{2} x^{2}+d e \right ) d x}{e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}-\frac {10 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {8 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(203\) |
| risch | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d \left (-\frac {11 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {6 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+12 d^{2} \left (\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )\right )}{3}\) | \(352\) |
| default | \(d^{4} \left (\frac {x}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{4} \left (\frac {d^{2} x}{2 e^{4} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3 e^{4}}-\frac {5 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 e^{4} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-4 e^{3} d \left (\frac {x^{3}}{2 e^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {3 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 e^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{2 e^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-4 d^{3} e \left (\frac {x^{3}}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(539\) |
Input:
int((-e*x^2+d)^4/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
4*(-e^2*x^2+d*e)*d/e*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/3*x*(-e^2*x^4+d^ 2)^(1/2)-10/3*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)-8*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)-Ellip ticE(x*(e/d)^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=-\frac {24 \, {\left (d^{2} e x^{3} + d^{3} x\right )} \sqrt {-e^{2}} \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 2 \, {\left ({\left (12 \, d^{2} e - 5 \, d e^{2}\right )} x^{3} + {\left (12 \, d^{3} - 5 \, d^{2} e\right )} x\right )} \sqrt {-e^{2}} \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left (e^{3} x^{4} - 11 \, d e^{2} x^{2} - 24 \, d^{2} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{3 \, {\left (e^{3} x^{3} + d e^{2} x\right )}} \] Input:
integrate((-e*x^2+d)^4/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
-1/3*(24*(d^2*e*x^3 + d^3*x)*sqrt(-e^2)*sqrt(d/e)*elliptic_e(arcsin(sqrt(d /e)/x), -1) - 2*((12*d^2*e - 5*d*e^2)*x^3 + (12*d^3 - 5*d^2*e)*x)*sqrt(-e^ 2)*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), -1) - (e^3*x^4 - 11*d*e^2*x^2 - 24*d^2*e)*sqrt(-e^2*x^4 + d^2))/(e^3*x^3 + d*e^2*x)
\[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\left (- d + e x^{2}\right )^{4}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-e*x**2+d)**4/(-e**2*x**4+d**2)**(3/2),x)
Output:
Integral((-d + e*x**2)**4/(-(-d + e*x**2)*(d + e*x**2))**(3/2), x)
\[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} - d\right )}^{4}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^4/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x^2 - d)^4/(-e^2*x^4 + d^2)^(3/2), x)
\[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} - d\right )}^{4}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^4/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
integrate((e*x^2 - d)^4/(-e^2*x^4 + d^2)^(3/2), x)
Timed out. \[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^4}{{\left (d^2-e^2\,x^4\right )}^{3/2}} \,d x \] Input:
int((d - e*x^2)^4/(d^2 - e^2*x^4)^(3/2),x)
Output:
int((d - e*x^2)^4/(d^2 - e^2*x^4)^(3/2), x)
\[ \int \frac {\left (d-e x^2\right )^4}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {6 \sqrt {-e^{2} x^{4}+d^{2}}\, d x +\sqrt {-e^{2} x^{4}+d^{2}}\, e \,x^{3}-3 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \right ) d^{4}-3 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \right ) d^{3} e \,x^{2}+17 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \right ) d^{2} e^{2}+17 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \right ) d \,e^{3} x^{2}}{3 e \,x^{2}+3 d} \] Input:
int((-e*x^2+d)^4/(-e^2*x^4+d^2)^(3/2),x)
Output:
(6*sqrt(d**2 - e**2*x**4)*d*x + sqrt(d**2 - e**2*x**4)*e*x**3 - 3*int(sqrt (d**2 - e**2*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6),x)*d**4 - 3*int(sqrt(d**2 - e**2*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x* *6),x)*d**3*e*x**2 + 17*int((sqrt(d**2 - e**2*x**4)*x**4)/(d**3 + d**2*e*x **2 - d*e**2*x**4 - e**3*x**6),x)*d**2*e**2 + 17*int((sqrt(d**2 - e**2*x** 4)*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6),x)*d*e**3*x**2)/(3 *(d + e*x**2))