Integrand size = 26, antiderivative size = 178 \[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\frac {2}{105} d x \left (20 d+21 e x^2\right ) \sqrt {d^2-e^2 x^4}-\frac {1}{7} x \left (d^2-e^2 x^4\right )^{3/2}+\frac {4 d^{9/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{5 \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {4 d^{9/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{105 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/105*d*x*(21*e*x^2+20*d)*(-e^2*x^4+d^2)^(1/2)-1/7*x*(-e^2*x^4+d^2)^(3/2)+ 4/5*d^(9/2)*(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)-4/105*d^(9/2)*(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 = 7.57 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\frac {x \sqrt {d^2-e^2 x^4} \left (-3 \left (d^2-e^2 x^4\right ) \sqrt {1-\frac {e^2 x^4}{d^2}}+24 d^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+14 d e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{21 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:
Integrate[(d + e*x^2)^2*Sqrt[d^2 - e^2*x^4],x]
Output:
(x*Sqrt[d^2 - e^2*x^4]*(-3*(d^2 - e^2*x^4)*Sqrt[1 - (e^2*x^4)/d^2] + 24*d^ 2*Hypergeometric2F1[-1/2, 1/4, 5/4, (e^2*x^4)/d^2] + 14*d*e*x^2*Hypergeome tric2F1[-1/2, 3/4, 7/4, (e^2*x^4)/d^2]))/(21*Sqrt[1 - (e^2*x^4)/d^2])
Time = 0.75 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.56, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1396, 318, 27, 403, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -2 d e \sqrt {d-e x^2} \sqrt {e x^2+d} \left (7 e x^2+4 d\right )dx}{7 e}-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \int \sqrt {d-e x^2} \sqrt {e x^2+d} \left (7 e x^2+4 d\right )dx-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (-\frac {\int -\frac {d e \sqrt {d-e x^2} \left (41 e x^2+27 d\right )}{\sqrt {e x^2+d}}dx}{5 e}-\frac {7}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {\int \frac {d e \sqrt {d-e x^2} \left (41 e x^2+27 d\right )}{\sqrt {e x^2+d}}dx}{5 e}-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \int \frac {\sqrt {d-e x^2} \left (41 e x^2+27 d\right )}{\sqrt {e x^2+d}}dx-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {\int \frac {2 d e \left (21 e x^2+20 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \int \frac {21 e x^2+20 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \left (21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx\right )+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \left (21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \left (\frac {21 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \left (\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \left (\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{7} d \left (\frac {1}{5} d \left (\frac {2}{3} d \left (\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {41}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {1}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d + e*x^2)^2*Sqrt[d^2 - e^2*x^4],x]
Output:
(Sqrt[d^2 - e^2*x^4]*(-1/7*(x*(d - e*x^2)^(3/2)*(d + e*x^2)^(3/2)) + (2*d* ((-7*x*(d - e*x^2)^(3/2)*Sqrt[d + e*x^2])/5 + (d*((41*x*Sqrt[d - e*x^2]*Sq rt[d + e*x^2])/3 + (2*d*((21*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[Arc Sin[(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[ArcSin[(Sqrt[e]*x)/Sqrt[d]], - 1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])))/3))/5))/7))/(Sqrt[d - e*x^ 2]*Sqrt[d + e*x^2])
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[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 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 = 5.60 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {x \left (15 e^{2} x^{4}+42 d e \,x^{2}+25 d^{2}\right ) \sqrt {-e^{2} x^{4}+d^{2}}}{105}+\frac {4 d^{3} \left (\frac {20 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {21 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{105}\) | \(183\) |
| elliptic | \(\frac {e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {2 d e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}+\frac {5 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{21}+\frac {16 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{21 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(206\) |
| default | \(d^{2} \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{2} \left (\frac {x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}-\frac {2 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{21 e^{2}}+\frac {2 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{21 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+2 d e \left (\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}-\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(305\) |
Input:
int((e*x^2+d)^2*(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/105*x*(15*e^2*x^4+42*d*e*x^2+25*d^2)*(-e^2*x^4+d^2)^(1/2)+4/105*d^3*(20* 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)*Ell ipticF(x*(e/d)^(1/2),I)-21*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 = 137, normalized size of antiderivative = 0.77 \[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=-\frac {84 \, \sqrt {-e^{2}} d^{4} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 4 \, {\left (21 \, d^{4} + 20 \, d^{3} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left (15 \, e^{4} x^{6} + 42 \, d e^{3} x^{4} + 25 \, d^{2} e^{2} x^{2} - 84 \, d^{3} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{105 \, e^{2} x} \] Input:
integrate((e*x^2+d)^2*(-e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
Output:
-1/105*(84*sqrt(-e^2)*d^4*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - 4*(21*d^4 + 20*d^3*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e) /x), -1) - (15*e^4*x^6 + 42*d*e^3*x^4 + 25*d^2*e^2*x^2 - 84*d^3*e)*sqrt(-e ^2*x^4 + d^2))/(e^2*x)
Time = 1.38 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.75 \[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\frac {d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {d^{2} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {d e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((e*x**2+d)**2*(-e**2*x**4+d**2)**(1/2),x)
Output:
d**3*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d **2)/(4*gamma(5/4)) + d**2*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), e* *2*x**4*exp_polar(2*I*pi)/d**2)/(2*gamma(7/4)) + d*e**2*x**5*gamma(5/4)*hy per((-1/2, 5/4), (9/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(9/4))
\[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\int { \sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-e^2*x^4+d^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-e^2*x^4 + d^2)*(e*x^2 + d)^2, x)
\[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\int { \sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-e^2*x^4+d^2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-e^2*x^4 + d^2)*(e*x^2 + d)^2, x)
Timed out. \[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\int \sqrt {d^2-e^2\,x^4}\,{\left (e\,x^2+d\right )}^2 \,d x \] Input:
int((d^2 - e^2*x^4)^(1/2)*(d + e*x^2)^2,x)
Output:
int((d^2 - e^2*x^4)^(1/2)*(d + e*x^2)^2, x)
\[ \int \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4} \, dx=\frac {5 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} x}{21}+\frac {2 \sqrt {-e^{2} x^{4}+d^{2}}\, d e \,x^{3}}{5}+\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{2} x^{5}}{7}+\frac {16 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{4}}{21}+\frac {4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{3} e}{5} \] Input:
int((e*x^2+d)^2*(-e^2*x^4+d^2)^(1/2),x)
Output:
(25*sqrt(d**2 - e**2*x**4)*d**2*x + 42*sqrt(d**2 - e**2*x**4)*d*e*x**3 + 1 5*sqrt(d**2 - e**2*x**4)*e**2*x**5 + 80*int(sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**4 + 84*int((sqrt(d**2 - e**2*x**4)*x**2)/(d**2 - e**2*x* *4),x)*d**3*e)/105