\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d-e x^2)^2} \, dx\) [98]

Optimal result

Integrand size = 27, antiderivative size = 144 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=-\frac {1}{3} x \sqrt {d^2-e^2 x^4}+\frac {2 d^{5/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {2 d^{5/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

-1/3*x*(-e^2*x^4+d^2)^(1/2)+2*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)-2/3*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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.47 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (-d^2+e^2 x^4\right )-6 i d^2 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )+2 i d^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{3 \sqrt {-\frac {e}{d}} \sqrt {d^2-e^2 x^4}} \] Input:

Integrate[(d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^2,x]
 

Output:

(Sqrt[-(e/d)]*x*(-d^2 + e^2*x^4) - (6*I)*d^2*Sqrt[1 - (e^2*x^4)/d^2]*Ellip 
ticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] + (2*I)*d^2*Sqrt[1 - (e^2*x^4)/d^2]*El 
lipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1])/(3*Sqrt[-(e/d)]*Sqrt[d^2 - e^2*x^4 
])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.45, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1396, 318, 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.

\(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1396

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\left (e x^2+d\right )^{3/2}}{\sqrt {d-e x^2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -\frac {2 d e \left (3 e x^2+2 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \int \frac {3 e x^2+2 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \left (3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx\right )-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \left (3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \left (\frac {3 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \left (\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \left (\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{3} d \left (\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {1}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[(d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^2,x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(-1/3*(x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) + (2*d*((3* 
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]) - (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))/(Sqrt[d - e*x^2]*Sqrt[d + e*x^2])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 289

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]
 

rule 318

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]
 

rule 327

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]
 

rule 329

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])
 

rule 399

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])))))
 

rule 762

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]
 

rule 765

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]
 

rule 1396

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])
 

Maple [A] (verified)

Time = 3.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {4 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {2 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(160\)
risch	\(-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d \left (\frac {2 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{3}\)	\(160\)
elliptic	\(-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {4 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {2 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(160\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/3*x*(-e^2*x^4+d^2)^(1/2)+4/3*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)-2*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))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=-\frac {6 \, \sqrt {-e^{2}} d^{2} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 2 \, {\left (3 \, d^{2} + 2 \, d e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (e^{2} x^{2} + 6 \, d e\right )}}{3 \, e^{2} x} \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^2,x, algorithm="fricas")
 

Output:

-1/3*(6*sqrt(-e^2)*d^2*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - 2 
*(3*d^2 + 2*d*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), -1 
) + sqrt(-e^2*x^4 + d^2)*(e^2*x^2 + 6*d*e))/(e^2*x)
 

Sympy [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=\int \frac {\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}{\left (- d + e x^{2}\right )^{2}}\, dx \] Input:

integrate((-e**2*x**4+d**2)**(3/2)/(-e*x**2+d)**2,x)
 

Output:

Integral((-(-d + e*x**2)*(d + e*x**2))**(3/2)/(-d + e*x**2)**2, x)
 

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} - d\right )}^{2}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^2,x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 - d)^2, x)
 

Giac [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} - d\right )}^{2}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^2,x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 - d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (d-e\,x^2\right )}^2} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^2,x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d - e*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^2} \, dx=\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e \,x^{2}+d}d x \right ) d +\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e \,x^{2}+d}d x \right ) e \] Input:

int((-e^2*x^4+d^2)^(3/2)/(-e*x^2+d)^2,x)
 

Output:

int(sqrt(d**2 - e**2*x**4)/(d - e*x**2),x)*d + int((sqrt(d**2 - e**2*x**4) 
*x**2)/(d - e*x**2),x)*e