Integrand size = 27, antiderivative size = 94 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{3 \left (d-e x^2\right )^2}+\frac {\sqrt {d} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/3*x*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^2+1/3*d^(1/2)*(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 complex when optimal does not.
Time = 10.59 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\frac {\frac {2 x \left (d+e x^2\right )}{d-e x^2}-\frac {i \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{\sqrt {-\frac {e}{d}}}}{3 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[(d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^4,x]
Output:
((2*x*(d + e*x^2))/(d - e*x^2) - (I*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*Ar cSinh[Sqrt[-(e/d)]*x], -1])/Sqrt[-(e/d)])/(3*Sqrt[d^2 - e^2*x^4])
Time = 0.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.47, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1396, 315, 25, 27, 289, 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.
\(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\left (e x^2+d\right )^{3/2}}{\left (d-e x^2\right )^{5/2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2 x \sqrt {d+e x^2}}{3 \left (d-e x^2\right )^{3/2}}-\frac {\int -\frac {d e}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 d e}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int \frac {d e}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 d e}+\frac {2 x \sqrt {d+e x^2}}{3 \left (d-e x^2\right )^{3/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{3} \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+\frac {2 x \sqrt {d+e x^2}}{3 \left (d-e x^2\right )^{3/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 289 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{3 \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{3 \left (d-e x^2\right )^{3/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{3 \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{3 \left (d-e x^2\right )^{3/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\sqrt {d} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{3 \left (d-e x^2\right )^{3/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^4,x]
Output:
(Sqrt[d^2 - e^2*x^4]*((2*x*Sqrt[d + e*x^2])/(3*(d - e*x^2)^(3/2)) + (Sqrt[ d]*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(3* Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])))/(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[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 = 3.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{3 e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(94\) |
elliptic | \(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{3 e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(94\) |
Input:
int((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^4,x,method=_RETURNVERBOSE)
Output:
2/3*x/e^2*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^2+1/3/(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)
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} e x + {\left (e^{2} x^{4} - 2 \, d e x^{2} + d^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1)}{3 \, {\left (e^{3} x^{4} - 2 \, d e^{2} x^{2} + d^{2} e\right )}} \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^4,x, algorithm="fricas")
Output:
1/3*(2*sqrt(-e^2*x^4 + d^2)*e*x + (e^2*x^4 - 2*d*e*x^2 + d^2)*sqrt(e/d)*el liptic_f(arcsin(x*sqrt(e/d)), -1))/(e^3*x^4 - 2*d*e^2*x^2 + d^2*e)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\int \frac {\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}{\left (- d + e x^{2}\right )^{4}}\, dx \] Input:
integrate((-e**2*x**4+d**2)**(3/2)/(-e*x**2+d)**4,x)
Output:
Integral((-(-d + e*x**2)*(d + e*x**2))**(3/2)/(-d + e*x**2)**4, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} - d\right )}^{4}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^4,x, algorithm="maxima")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 - d)^4, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} - d\right )}^{4}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^4,x, algorithm="giac")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 - d)^4, x)
Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (d-e\,x^2\right )}^4} \,d x \] Input:
int((d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^4,x)
Output:
int((d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^4, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^4} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x +\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}+2 d \,e^{3} x^{6}-2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{4}-2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}+2 d \,e^{3} x^{6}-2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{3} e \,x^{2}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}+2 d \,e^{3} x^{6}-2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2} x^{4}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}+2 d \,e^{3} x^{6}-2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2}-2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}+2 d \,e^{3} x^{6}-2 d^{3} e \,x^{2}+d^{4}}d x \right ) d \,e^{3} x^{2}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}+2 d \,e^{3} x^{6}-2 d^{3} e \,x^{2}+d^{4}}d x \right ) e^{4} x^{4}}{2 e^{2} x^{4}-4 d e \,x^{2}+2 d^{2}} \] Input:
int((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^4,x)
Output:
(sqrt(d**2 - e**2*x**4)*x + int(sqrt(d**2 - e**2*x**4)/(d**4 - 2*d**3*e*x* *2 + 2*d*e**3*x**6 - e**4*x**8),x)*d**4 - 2*int(sqrt(d**2 - e**2*x**4)/(d* *4 - 2*d**3*e*x**2 + 2*d*e**3*x**6 - e**4*x**8),x)*d**3*e*x**2 + int(sqrt( d**2 - e**2*x**4)/(d**4 - 2*d**3*e*x**2 + 2*d*e**3*x**6 - e**4*x**8),x)*d* *2*e**2*x**4 + int((sqrt(d**2 - e**2*x**4)*x**4)/(d**4 - 2*d**3*e*x**2 + 2 *d*e**3*x**6 - e**4*x**8),x)*d**2*e**2 - 2*int((sqrt(d**2 - e**2*x**4)*x** 4)/(d**4 - 2*d**3*e*x**2 + 2*d*e**3*x**6 - e**4*x**8),x)*d*e**3*x**2 + int ((sqrt(d**2 - e**2*x**4)*x**4)/(d**4 - 2*d**3*e*x**2 + 2*d*e**3*x**6 - e** 4*x**8),x)*e**4*x**4)/(2*(d**2 - 2*d*e*x**2 + e**2*x**4))