\(\int \frac {1}{\sqrt {d-e x^2} (d^2-e^2 x^4)^{5/2}} \, dx\) [171]

Optimal result

Integrand size = 29, antiderivative size = 203 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x}{8 d^2 \sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}}+\frac {13 x \sqrt {d-e x^2}}{32 d^3 \left (d^2-e^2 x^4\right )^{3/2}}-\frac {37 x \left (d-e x^2\right )^{3/2}}{192 d^4 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {53 x \sqrt {d-e x^2}}{384 d^5 \sqrt {d^2-e^2 x^4}}+\frac {67 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{128 \sqrt {2} d^5 \sqrt {e}} \] Output:

1/8*x/d^2/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2)+13/32*x*(-e*x^2+d)^(1/2)/d 
^3/(-e^2*x^4+d^2)^(3/2)-37/192*x*(-e*x^2+d)^(3/2)/d^4/(-e^2*x^4+d^2)^(3/2) 
+53/384*x*(-e*x^2+d)^(1/2)/d^5/(-e^2*x^4+d^2)^(1/2)+67/256*arctanh(2^(1/2) 
*e^(1/2)*x*(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2))*2^(1/2)/d^5/e^(1/2)
 

Mathematica [A] (verified)

Time = 3.65 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d+e x^2} \left (183 d^3-61 d^2 e x^2-127 d e^2 x^4+53 e^3 x^6\right )+201 \sqrt {2} \left (d^2-e^2 x^4\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{768 d^5 \sqrt {e} \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[1/(Sqrt[d - e*x^2]*(d^2 - e^2*x^4)^(5/2)),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d + e*x^2]*(183*d^3 - 61*d^2*e*x^2 
- 127*d*e^2*x^4 + 53*e^3*x^6) + 201*Sqrt[2]*(d^2 - e^2*x^4)^2*ArcTanh[(Sqr 
t[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]))/(768*d^5*Sqrt[e]*(d - e*x^2)^(5/2)*(d + 
 e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1396, 316, 27, 402, 27, 402, 25, 27, 402, 27, 291, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1396

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (d-e x^2\right )^3 \left (e x^2+d\right )^{5/2}}dx}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (6 e x^2+7 d\right )}{\left (d-e x^2\right )^2 \left (e x^2+d\right )^{5/2}}dx}{8 d^2 e}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {6 e x^2+7 d}{\left (d-e x^2\right )^2 \left (e x^2+d\right )^{5/2}}dx}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {d e \left (52 e x^2+15 d\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{5/2}}dx}{4 d^2 e}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {52 e x^2+15 d}{\left (d-e x^2\right ) \left (e x^2+d\right )^{5/2}}dx}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {-\frac {\int -\frac {d e \left (74 e x^2+127 d\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {d e \left (74 e x^2+127 d\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {74 e x^2+127 d}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {53 x}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {201 d^2 e}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx}{2 d^2 e}}{6 d}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {201}{2} \int \frac {1}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx+\frac {53 x}{2 d \sqrt {d+e x^2}}}{6 d}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {201}{2} \int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {53 x}{2 d \sqrt {d+e x^2}}}{6 d}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {201 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {53 x}{2 d \sqrt {d+e x^2}}}{6 d}-\frac {37 x}{6 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {13 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[1/(Sqrt[d - e*x^2]*(d^2 - e^2*x^4)^(5/2)),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(8*d^2*(d - e*x^2)^2*(d + e*x^2)^(3/2) 
) + ((13*x)/(4*d*(d - e*x^2)*(d + e*x^2)^(3/2)) + ((-37*x)/(6*d*(d + e*x^2 
)^(3/2)) + ((53*x)/(2*d*Sqrt[d + e*x^2]) + (201*ArcTanh[(Sqrt[2]*Sqrt[e]*x 
)/Sqrt[d + e*x^2]])/(2*Sqrt[2]*d*Sqrt[e]))/(6*d))/(4*d))/(8*d^2)))/Sqrt[d^ 
2 - e^2*x^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 402

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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1659\) vs. \(2(165)=330\).

Time = 0.89 (sec) , antiderivative size = 1660, normalized size of antiderivative = 8.18

method	result	size
default	\(\text {Expression too large to display}\)	\(1660\)

Input:

int(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/48*(-e^2*x^4+d^2)^(1/2)*e^(19/2)*(252*d^4*x*(e*x^2+d)^(1/2)*(d*e)^(1/2)* 
e^(1/2)-288*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x)/e^(1/2))*d*e^4*x^8*(d*e)^(1/2 
)+288*ln((e^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+e*x)/e 
^(1/2))*d*e^4*x^8*(d*e)^(1/2)+576*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x)/e^(1/2) 
)*d^2*e^3*x^6*(d*e)^(1/2)-576*ln((e^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d 
*e)^(1/2)))^(1/2)+e*x)/e^(1/2))*d^2*e^3*x^6*(d*e)^(1/2)+576*ln(((e*x^2+d)^ 
(1/2)*e^(1/2)+e*x)/e^(1/2))*d^3*e^2*x^4*(d*e)^(1/2)-576*ln((e^(1/2)*(-(e*x 
+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+e*x)/e^(1/2))*d^3*e^2*x^4*(d*e 
)^(1/2)-288*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x)/e^(1/2))*d^4*e*x^2*(d*e)^(1/2 
)+288*ln((e^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+e*x)/e 
^(1/2))*d^4*e*x^2*(d*e)^(1/2)+201*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d 
)^(1/2)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*d^(3/2)*e^(9/2)*x^8-201*2^(1/2 
)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^(1/2 
)))*d^(3/2)*e^(9/2)*x^8-402*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2 
)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*d^(5/2)*e^(7/2)*x^6+64*(d*e)^(1/2)*d 
*e^(7/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)*x^7-896*d^2*e^( 
5/2)*(d*e)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)*x^5-64* 
d^3*e^(3/2)*(d*e)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)* 
x^3+402*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/( 
e*x+(d*e)^(1/2)))*d^(5/2)*e^(7/2)*x^6-204*e^(9/2)*x^9*(e*x^2+d)^(1/2)*(...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.48 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\left [\frac {201 \, \sqrt {2} {\left (e^{5} x^{10} - d e^{4} x^{8} - 2 \, d^{2} e^{3} x^{6} + 2 \, d^{3} e^{2} x^{4} + d^{4} e x^{2} - d^{5}\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) - 4 \, {\left (53 \, e^{4} x^{7} - 127 \, d e^{3} x^{5} - 61 \, d^{2} e^{2} x^{3} + 183 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{1536 \, {\left (d^{5} e^{6} x^{10} - d^{6} e^{5} x^{8} - 2 \, d^{7} e^{4} x^{6} + 2 \, d^{8} e^{3} x^{4} + d^{9} e^{2} x^{2} - d^{10} e\right )}}, \frac {201 \, \sqrt {2} {\left (e^{5} x^{10} - d e^{4} x^{8} - 2 \, d^{2} e^{3} x^{6} + 2 \, d^{3} e^{2} x^{4} + d^{4} e x^{2} - d^{5}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, {\left (53 \, e^{4} x^{7} - 127 \, d e^{3} x^{5} - 61 \, d^{2} e^{2} x^{3} + 183 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{768 \, {\left (d^{5} e^{6} x^{10} - d^{6} e^{5} x^{8} - 2 \, d^{7} e^{4} x^{6} + 2 \, d^{8} e^{3} x^{4} + d^{9} e^{2} x^{2} - d^{10} e\right )}}\right ] \] Input:

integrate(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/1536*(201*sqrt(2)*(e^5*x^10 - d*e^4*x^8 - 2*d^2*e^3*x^6 + 2*d^3*e^2*x^4 
 + d^4*e*x^2 - d^5)*sqrt(e)*log(-(3*e^2*x^4 - 2*d*e*x^2 - 2*sqrt(2)*sqrt(- 
e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(e)*x - d^2)/(e^2*x^4 - 2*d*e*x^2 + d^ 
2)) - 4*(53*e^4*x^7 - 127*d*e^3*x^5 - 61*d^2*e^2*x^3 + 183*d^3*e*x)*sqrt(- 
e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(d^5*e^6*x^10 - d^6*e^5*x^8 - 2*d^7*e^4*x 
^6 + 2*d^8*e^3*x^4 + d^9*e^2*x^2 - d^10*e), 1/768*(201*sqrt(2)*(e^5*x^10 - 
 d*e^4*x^8 - 2*d^2*e^3*x^6 + 2*d^3*e^2*x^4 + d^4*e*x^2 - d^5)*sqrt(-e)*arc 
tan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d^ 
2)) - 2*(53*e^4*x^7 - 127*d*e^3*x^5 - 61*d^2*e^2*x^3 + 183*d^3*e*x)*sqrt(- 
e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(d^5*e^6*x^10 - d^6*e^5*x^8 - 2*d^7*e^4*x 
^6 + 2*d^8*e^3*x^4 + d^9*e^2*x^2 - d^10*e)]
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}} \sqrt {d - e x^{2}}}\, dx \] Input:

integrate(1/(-e*x**2+d)**(1/2)/(-e**2*x**4+d**2)**(5/2),x)
 

Output:

Integral(1/((-(-d + e*x**2)*(d + e*x**2))**(5/2)*sqrt(d - e*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} \sqrt {-e x^{2} + d}} \,d x } \] Input:

integrate(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((-e^2*x^4 + d^2)^(5/2)*sqrt(-e*x^2 + d)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} \sqrt {-e x^{2} + d}} \,d x } \] Input:

integrate(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((-e^2*x^4 + d^2)^(5/2)*sqrt(-e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{5/2}\,\sqrt {d-e\,x^2}} \,d x \] Input:

int(1/((d^2 - e^2*x^4)^(5/2)*(d - e*x^2)^(1/2)),x)
 

Output:

int(1/((d^2 - e^2*x^4)^(5/2)*(d - e*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.98 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(5/2),x)
 

Output:

(2196*sqrt(d + e*x**2)*d**3*e*x - 732*sqrt(d + e*x**2)*d**2*e**2*x**3 - 15 
24*sqrt(d + e*x**2)*d*e**3*x**5 + 636*sqrt(d + e*x**2)*e**4*x**7 - 603*sqr 
t(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x 
)/sqrt(d))*d**4 + 1206*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqr 
t(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d**2*e**2*x**4 - 603*sqrt(e)*sqrt(2)* 
log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e* 
*4*x**8 + 603*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sq 
rt(d) + sqrt(e)*x)/sqrt(d))*d**4 - 1206*sqrt(e)*sqrt(2)*log((sqrt(d + e*x* 
*2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**2*e**2*x**4 + 603 
*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt( 
e)*x)/sqrt(d))*e**4*x**8 + 603*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqr 
t(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d**4 - 1206*sqrt(e)*sqrt(2)*l 
og((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d** 
2*e**2*x**4 + 603*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) 
- sqrt(d) + sqrt(e)*x)/sqrt(d))*e**4*x**8 - 603*sqrt(e)*sqrt(2)*log((sqrt( 
d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**4 + 1206* 
sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e 
)*x)/sqrt(d))*d**2*e**2*x**4 - 603*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + 
 sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*e**4*x**8 - 1172*sqrt(e)* 
d**4 + 2344*sqrt(e)*d**2*e**2*x**4 - 1172*sqrt(e)*e**4*x**8)/(4608*d**5...