Integrand size = 27, antiderivative size = 258 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{7 \left (d-e x^2\right )^4}+\frac {4 x \sqrt {d^2-e^2 x^4}}{35 d \left (d-e x^2\right )^3}+\frac {11 x \sqrt {d^2-e^2 x^4}}{70 d^2 \left (d-e x^2\right )^2}+\frac {3 x \sqrt {d^2-e^2 x^4}}{10 d^3 \left (d-e x^2\right )}-\frac {3 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{10 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {31 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{70 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/7*x*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^4+4/35*x*(-e^2*x^4+d^2)^(1/2)/d/(-e* x^2+d)^3+11/70*x*(-e^2*x^4+d^2)^(1/2)/d^2/(-e*x^2+d)^2+3/10*x*(-e^2*x^4+d^ 2)^(1/2)/d^3/(-e*x^2+d)-3/10*(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)+31/70*(1-e^2*x^4/d^2)^(1/2)*E llipticF(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 = 11.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.59 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\frac {-\frac {x \left (d+e x^2\right ) \left (-60 d^3+93 d^2 e x^2-74 d e^2 x^4+21 e^3 x^6\right )}{\left (d-e x^2\right )^3}+\frac {i d \sqrt {1-\frac {e^2 x^4}{d^2}} \left (21 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-31 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\sqrt {-\frac {e}{d}}}}{70 d^3 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[(d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^6,x]
Output:
(-((x*(d + e*x^2)*(-60*d^3 + 93*d^2*e*x^2 - 74*d*e^2*x^4 + 21*e^3*x^6))/(d - e*x^2)^3) + (I*d*Sqrt[1 - (e^2*x^4)/d^2]*(21*EllipticE[I*ArcSinh[Sqrt[- (e/d)]*x], -1] - 31*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/Sqrt[-(e/d) ])/(70*d^3*Sqrt[d^2 - e^2*x^4])
Time = 0.84 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.24, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {1396, 315, 25, 27, 402, 27, 402, 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.
\(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\left (e x^2+d\right )^{3/2}}{\left (d-e x^2\right )^{9/2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}-\frac {\int -\frac {d e \left (3 e x^2+5 d\right )}{\left (d-e x^2\right )^{7/2} \sqrt {e x^2+d}}dx}{7 d e}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int \frac {d e \left (3 e x^2+5 d\right )}{\left (d-e x^2\right )^{7/2} \sqrt {e x^2+d}}dx}{7 d e}+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \int \frac {3 e x^2+5 d}{\left (d-e x^2\right )^{7/2} \sqrt {e x^2+d}}dx+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {\int \frac {6 d e \left (4 e x^2+7 d\right )}{\left (d-e x^2\right )^{5/2} \sqrt {e x^2+d}}dx}{10 d^2 e}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \int \frac {4 e x^2+7 d}{\left (d-e x^2\right )^{5/2} \sqrt {e x^2+d}}dx}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\int \frac {d e \left (11 e x^2+31 d\right )}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{6 d^2 e}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\int \frac {11 e x^2+31 d}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\int \frac {2 d e \left (10 d-21 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\int \frac {10 d-21 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {31 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 289 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {31 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 329 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {31 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {31 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {31 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {31 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {11 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{7 \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^6,x]
Output:
(Sqrt[d^2 - e^2*x^4]*((2*x*Sqrt[d + e*x^2])/(7*(d - e*x^2)^(7/2)) + ((4*x* Sqrt[d + e*x^2])/(5*d*(d - e*x^2)^(5/2)) + (3*((11*x*Sqrt[d + e*x^2])/(6*d *(d - e*x^2)^(3/2)) + ((21*x*Sqrt[d + e*x^2])/(d*Sqrt[d - e*x^2]) + ((-21* 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]) + (31*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]))/d)/(6*d)))/(5*d))/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[(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[(-(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 = 5.11 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{7 e^{4} \left (x^{2}-\frac {d}{e}\right )^{4}}-\frac {4 x \sqrt {-e^{2} x^{4}+d^{2}}}{35 d \,e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 d^{2} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {3 \left (-e^{2} x^{2}-d e \right ) x}{10 e \,d^{3} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(293\) |
elliptic | \(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{7 e^{4} \left (x^{2}-\frac {d}{e}\right )^{4}}-\frac {4 x \sqrt {-e^{2} x^{4}+d^{2}}}{35 d \,e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 d^{2} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {3 \left (-e^{2} x^{2}-d e \right ) x}{10 e \,d^{3} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(293\) |
Input:
int((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^6,x,method=_RETURNVERBOSE)
Output:
2/7*x/e^4*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^4-4/35/d*x/e^3*(-e^2*x^4+d^2)^(1/ 2)/(x^2-d/e)^3+11/70/d^2*x/e^2*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^2-3/10*(-e^2 *x^2-d*e)/e/d^3*x/((x^2-d/e)*(-e^2*x^2-d*e))^(1/2)+1/7/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)+3/10/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)-EllipticE(x*(e/d)^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=-\frac {21 \, {\left (e^{5} x^{8} - 4 \, d e^{4} x^{6} + 6 \, d^{2} e^{3} x^{4} - 4 \, d^{3} e^{2} x^{2} + d^{4} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (10 \, d e^{4} + 21 \, e^{5}\right )} x^{8} - 4 \, {\left (10 \, d^{2} e^{3} + 21 \, d e^{4}\right )} x^{6} + 10 \, d^{5} + 21 \, d^{4} e + 6 \, {\left (10 \, d^{3} e^{2} + 21 \, d^{2} e^{3}\right )} x^{4} - 4 \, {\left (10 \, d^{4} e + 21 \, d^{3} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (21 \, e^{4} x^{7} - 74 \, d e^{3} x^{5} + 93 \, d^{2} e^{2} x^{3} - 60 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{70 \, {\left (d^{3} e^{5} x^{8} - 4 \, d^{4} e^{4} x^{6} + 6 \, d^{5} e^{3} x^{4} - 4 \, d^{6} e^{2} x^{2} + d^{7} e\right )}} \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^6,x, algorithm="fricas")
Output:
-1/70*(21*(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) - ((10*d*e^4 + 21*e^5)*x^8 - 4*(10*d^2*e^3 + 21*d*e^4)*x^6 + 10*d^5 + 21*d^4*e + 6*(10*d^3*e^2 + 21*d^ 2*e^3)*x^4 - 4*(10*d^4*e + 21*d^3*e^2)*x^2)*sqrt(e/d)*elliptic_f(arcsin(x* sqrt(e/d)), -1) + (21*e^4*x^7 - 74*d*e^3*x^5 + 93*d^2*e^2*x^3 - 60*d^3*e*x )*sqrt(-e^2*x^4 + d^2))/(d^3*e^5*x^8 - 4*d^4*e^4*x^6 + 6*d^5*e^3*x^4 - 4*d ^6*e^2*x^2 + d^7*e)
Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\text {Timed out} \] Input:
integrate((-e**2*x**4+d**2)**(3/2)/(-e*x**2+d)**6,x)
Output:
Timed out
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} - d\right )}^{6}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^6,x, algorithm="maxima")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 - d)^6, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} - d\right )}^{6}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^6,x, algorithm="giac")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 - d)^6, x)
Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (d-e\,x^2\right )}^6} \,d x \] Input:
int((d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^6,x)
Output:
int((d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^6, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^6} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x +3 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{6}-12 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{5} e \,x^{2}+18 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{4} e^{2} x^{4}-12 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{3} e^{3} x^{6}+3 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{2} e^{4} x^{8}-\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{4} e^{2}+4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{3} e^{3} x^{2}-6 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{2} e^{4} x^{4}+4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) d \,e^{5} x^{6}-\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}+4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}-4 d^{5} e \,x^{2}+d^{6}}d x \right ) e^{6} x^{8}}{4 e^{4} x^{8}-16 d \,e^{3} x^{6}+24 d^{2} e^{2} x^{4}-16 d^{3} e \,x^{2}+4 d^{4}} \] Input:
int((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^6,x)
Output:
(sqrt(d**2 - e**2*x**4)*x + 3*int(sqrt(d**2 - e**2*x**4)/(d**6 - 4*d**5*e* x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e**5*x**10 - e**6*x**12), x)*d**6 - 12*int(sqrt(d**2 - e**2*x**4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e** 2*x**4 - 5*d**2*e**4*x**8 + 4*d*e**5*x**10 - e**6*x**12),x)*d**5*e*x**2 + 18*int(sqrt(d**2 - e**2*x**4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5 *d**2*e**4*x**8 + 4*d*e**5*x**10 - e**6*x**12),x)*d**4*e**2*x**4 - 12*int( sqrt(d**2 - e**2*x**4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e **4*x**8 + 4*d*e**5*x**10 - e**6*x**12),x)*d**3*e**3*x**6 + 3*int(sqrt(d** 2 - e**2*x**4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e**5*x**10 - e**6*x**12),x)*d**2*e**4*x**8 - int((sqrt(d**2 - e**2* x**4)*x**4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e**5*x**10 - e**6*x**12),x)*d**4*e**2 + 4*int((sqrt(d**2 - e**2*x**4)* x**4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e* *5*x**10 - e**6*x**12),x)*d**3*e**3*x**2 - 6*int((sqrt(d**2 - e**2*x**4)*x **4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e** 5*x**10 - e**6*x**12),x)*d**2*e**4*x**4 + 4*int((sqrt(d**2 - e**2*x**4)*x* *4)/(d**6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e**5 *x**10 - e**6*x**12),x)*d*e**5*x**6 - int((sqrt(d**2 - e**2*x**4)*x**4)/(d **6 - 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 + 4*d*e**5*x**10 - e**6*x**12),x)*e**6*x**8)/(4*(d**4 - 4*d**3*e*x**2 + 6*d**2*e**2*x**...