Integrand size = 26, antiderivative size = 257 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{7 d \left (d+e x^2\right )^4}+\frac {11 x \sqrt {d^2-e^2 x^4}}{70 d^2 \left (d+e x^2\right )^3}+\frac {41 x \sqrt {d^2-e^2 x^4}}{210 d^3 \left (d+e x^2\right )^2}+\frac {7 x \sqrt {d^2-e^2 x^4}}{20 d^4 \left (d+e x^2\right )}+\frac {7 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{20 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {41 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{210 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
1/7*x*(-e^2*x^4+d^2)^(1/2)/d/(e*x^2+d)^4+11/70*x*(-e^2*x^4+d^2)^(1/2)/d^2/ (e*x^2+d)^3+41/210*x*(-e^2*x^4+d^2)^(1/2)/d^3/(e*x^2+d)^2+7/20*x*(-e^2*x^4 +d^2)^(1/2)/d^4/(e*x^2+d)+7/20*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d ^(1/2),I)/d^(5/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)-41/210*(1-e^2*x^4/d^2)^(1/2 )*EllipticF(e^(1/2)*x/d^(1/2),I)/d^(5/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 11.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\frac {-\frac {x \left (-d+e x^2\right ) \left (355 d^3+671 d^2 e x^2+523 d e^2 x^4+147 e^3 x^6\right )}{\left (d+e x^2\right )^3}+\frac {i e \sqrt {1-\frac {e^2 x^4}{d^2}} \left (147 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-82 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\left (-\frac {e}{d}\right )^{3/2}}}{420 d^4 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[Sqrt[d^2 - e^2*x^4]/(d + e*x^2)^5,x]
Output:
(-((x*(-d + e*x^2)*(355*d^3 + 671*d^2*e*x^2 + 523*d*e^2*x^4 + 147*e^3*x^6) )/(d + e*x^2)^3) + (I*e*Sqrt[1 - (e^2*x^4)/d^2]*(147*EllipticE[I*ArcSinh[S qrt[-(e/d)]*x], -1] - 82*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/(-(e/d ))^(3/2))/(420*d^4*Sqrt[d^2 - e^2*x^4])
Time = 0.86 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.29, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {1396, 314, 25, 402, 25, 27, 402, 27, 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.
| \(\displaystyle \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\sqrt {d-e x^2}}{\left (e x^2+d\right )^{9/2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}-\frac {\int -\frac {6 d-5 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{7 d}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int \frac {6 d-5 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}-\frac {\int -\frac {d e \left (49 d-33 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\int \frac {d e \left (49 d-33 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\int \frac {49 d-33 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {2 d e \left (106 d-41 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\int \frac {106 d-41 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {d e \left (147 e x^2+65 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\int \frac {d e \left (147 e x^2+65 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\int \frac {147 e x^2+65 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {147 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-82 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {147 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {82 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {147 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {82 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {147 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {82 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {147 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {82 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {147 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {82 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}+\frac {41 x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[Sqrt[d^2 - e^2*x^4]/(d + e*x^2)^5,x]
Output:
(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d - e*x^2])/(7*d*(d + e*x^2)^(7/2)) + ((11*x *Sqrt[d - e*x^2])/(10*d*(d + e*x^2)^(5/2)) + ((41*x*Sqrt[d - e*x^2])/(3*d* (d + e*x^2)^(3/2)) + ((147*x*Sqrt[d - e*x^2])/(2*d*Sqrt[d + e*x^2]) + ((14 7*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]) - (82*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))/(10*d))/(7*d)))/(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[(-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 = 4.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{7 d \,e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 d^{2} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {41 x \sqrt {-e^{2} x^{4}+d^{2}}}{210 d^{3} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {7 \left (-e^{2} x^{2}+d e \right ) x}{20 e \,d^{4} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {13 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{84 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {7 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{20 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(290\) |
| elliptic | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{7 d \,e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 d^{2} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {41 x \sqrt {-e^{2} x^{4}+d^{2}}}{210 d^{3} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {7 \left (-e^{2} x^{2}+d e \right ) x}{20 e \,d^{4} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {13 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{84 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {7 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{20 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(290\) |
Input:
int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^5,x,method=_RETURNVERBOSE)
Output:
1/7/d*x/e^4*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^4+11/70/d^2/e^3*x*(-e^2*x^4+d^2 )^(1/2)/(x^2+d/e)^3+41/210/d^3/e^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+7/20 *(-e^2*x^2+d*e)/e/d^4*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+13/84/d^3/(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^3/(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 = 280, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\frac {147 \, {\left (e^{5} x^{8} + 4 \, d e^{4} x^{6} + 6 \, d^{2} e^{3} x^{4} + 4 \, d^{3} e^{2} x^{2} + d^{4} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (65 \, d e^{4} - 147 \, e^{5}\right )} x^{8} + 4 \, {\left (65 \, d^{2} e^{3} - 147 \, d e^{4}\right )} x^{6} + 65 \, d^{5} - 147 \, d^{4} e + 6 \, {\left (65 \, d^{3} e^{2} - 147 \, d^{2} e^{3}\right )} x^{4} + 4 \, {\left (65 \, d^{4} e - 147 \, d^{3} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (147 \, e^{4} x^{7} + 523 \, d e^{3} x^{5} + 671 \, d^{2} e^{2} x^{3} + 355 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{420 \, {\left (d^{4} e^{5} x^{8} + 4 \, d^{5} e^{4} x^{6} + 6 \, d^{6} e^{3} x^{4} + 4 \, d^{7} e^{2} x^{2} + d^{8} e\right )}} \] Input:
integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^5,x, algorithm="fricas")
Output:
1/420*(147*(e^5*x^8 + 4*d*e^4*x^6 + 6*d^2*e^3*x^4 + 4*d^3*e^2*x^2 + d^4*e) *sqrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), -1) + ((65*d*e^4 - 147*e^5)*x^8 + 4*(65*d^2*e^3 - 147*d*e^4)*x^6 + 65*d^5 - 147*d^4*e + 6*(65*d^3*e^2 - 1 47*d^2*e^3)*x^4 + 4*(65*d^4*e - 147*d^3*e^2)*x^2)*sqrt(e/d)*elliptic_f(arc sin(x*sqrt(e/d)), -1) + (147*e^4*x^7 + 523*d*e^3*x^5 + 671*d^2*e^2*x^3 + 3 55*d^3*e*x)*sqrt(-e^2*x^4 + d^2))/(d^4*e^5*x^8 + 4*d^5*e^4*x^6 + 6*d^6*e^3 *x^4 + 4*d^7*e^2*x^2 + d^8*e)
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\int \frac {\sqrt {- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )}}{\left (d + e x^{2}\right )^{5}}\, dx \] Input:
integrate((-e**2*x**4+d**2)**(1/2)/(e*x**2+d)**5,x)
Output:
Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/(d + e*x**2)**5, x)
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\int { \frac {\sqrt {-e^{2} x^{4} + d^{2}}}{{\left (e x^{2} + d\right )}^{5}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^5,x, algorithm="maxima")
Output:
integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d)^5, x)
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\int { \frac {\sqrt {-e^{2} x^{4} + d^{2}}}{{\left (e x^{2} + d\right )}^{5}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^5,x, algorithm="giac")
Output:
integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d)^5, x)
Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\int \frac {\sqrt {d^2-e^2\,x^4}}{{\left (e\,x^2+d\right )}^5} \,d x \] Input:
int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^5,x)
Output:
int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^5, x)
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^5} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{5} x^{10}+5 d \,e^{4} x^{8}+10 d^{2} e^{3} x^{6}+10 d^{3} e^{2} x^{4}+5 d^{4} e \,x^{2}+d^{5}}d x \] Input:
int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^5,x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**5 + 5*d**4*e*x**2 + 10*d**3*e**2*x**4 + 10* d**2*e**3*x**6 + 5*d*e**4*x**8 + e**5*x**10),x)