\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^{13/2}} \, dx\) [146]

Optimal result

Integrand size = 28, antiderivative size = 195 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{4 \left (d+e x^2\right )^{9/2}}+\frac {5 x \sqrt {d^2-e^2 x^4}}{48 d \left (d+e x^2\right )^{7/2}}+\frac {17 x \sqrt {d^2-e^2 x^4}}{128 d^2 \left (d+e x^2\right )^{5/2}}+\frac {299 x \sqrt {d^2-e^2 x^4}}{1536 d^3 \left (d+e x^2\right )^{3/2}}+\frac {163 \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{512 \sqrt {2} d^3 \sqrt {e}} \] Output:

1/4*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(9/2)+5/48*x*(-e^2*x^4+d^2)^(1/2)/d/( 
e*x^2+d)^(7/2)+17/128*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)^(5/2)+299/1536* 
x*(-e^2*x^4+d^2)^(1/2)/d^3/(e*x^2+d)^(3/2)+163/1024*arctan(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^3/e^(1/2)
 

Mathematica [A] (verified)

Time = 4.99 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d-e x^2} \left (1047 d^3+1465 d^2 e x^2+1101 d e^2 x^4+299 e^3 x^6\right )+489 \sqrt {2} \left (d+e x^2\right )^4 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )\right )}{3072 d^3 \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{9/2}} \] Input:

Integrate[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(13/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d - e*x^2]*(1047*d^3 + 1465*d^2*e*x 
^2 + 1101*d*e^2*x^4 + 299*e^3*x^6) + 489*Sqrt[2]*(d + e*x^2)^4*ArcTan[(Sqr 
t[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]]))/(3072*d^3*Sqrt[e]*Sqrt[d - e*x^2]*(d + 
e*x^2)^(9/2))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {1396, 315, 27, 402, 25, 27, 402, 25, 27, 402, 27, 291, 218}

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.

\(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}-\frac {\int -\frac {d e \left (31 d-20 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^3}dx}{12 d^2 e}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\int \frac {d e \left (31 d-20 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^3}dx}{12 d^2 e}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\int \frac {31 d-20 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^3}dx}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}-\frac {\int -\frac {d e \left (197 d-102 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2 e}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {d e \left (197 d-102 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2 e}+\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {197 d-102 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d}+\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {\frac {299 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}-\frac {\int -\frac {489 d^2 e}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx}{4 d^2 e}}{8 d}+\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {\frac {489}{4} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx+\frac {299 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d}+\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {\frac {489}{4} \int \frac {1}{\frac {2 d e x^2}{d-e x^2}+d}d\frac {x}{\sqrt {d-e x^2}}+\frac {299 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d}+\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{4} \left (\frac {\frac {\frac {489 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{4 \sqrt {2} d \sqrt {e}}+\frac {299 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d}+\frac {51 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{12 d}+\frac {5 x \sqrt {d-e x^2}}{12 d \left (d+e x^2\right )^3}\right )+\frac {x \sqrt {d-e x^2}}{4 \left (d+e x^2\right )^4}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(13/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d - e*x^2])/(4*(d + e*x^2)^4) + ((5*x*Sqrt[d 
 - e*x^2])/(12*d*(d + e*x^2)^3) + ((51*x*Sqrt[d - e*x^2])/(8*d*(d + e*x^2) 
^2) + ((299*x*Sqrt[d - e*x^2])/(4*d*(d + e*x^2)) + (489*ArcTan[(Sqrt[2]*Sq 
rt[e]*x)/Sqrt[d - e*x^2]])/(4*Sqrt[2]*d*Sqrt[e]))/(8*d))/(12*d))/4))/(Sqrt 
[d - e*x^2]*Sqrt[d + e*x^2])
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 315

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]
 

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. \(739\) vs. \(2(157)=314\).

Time = 0.52 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.79

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{4} \left (489 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, e^{4} x^{8} \sqrt {d}-489 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, e^{4} x^{8} \sqrt {d}+1956 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {3}{2}} e^{3} x^{6}-1956 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {3}{2}} e^{3} x^{6}-1196 \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, e^{3} x^{7}+2934 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {5}{2}} e^{2} x^{4}-2934 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {5}{2}} e^{2} x^{4}-4404 d \,e^{2} \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x^{5}+1956 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {7}{2}} e \,x^{2}-1956 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {7}{2}} e \,x^{2}-5860 d^{2} e \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x^{3}+489 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {9}{2}}-489 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {9}{2}}-4188 d^{3} \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x \right )}{6144 d^{3} \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (e x -\sqrt {-d e}\right )^{4} \left (e x +\sqrt {-d e}\right )^{4} \sqrt {-d e}}\)	\(740\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(13/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6144*(-e^2*x^4+d^2)^(1/2)/d^3*e^4*(489*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)))*2^(1/2)*e^4*x^8*d^(1/2)-489 
*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)))*2^(1/2)*e^4*x^8*d^(1/2)+1956*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)))*2^(1/2)*d^(3/2)*e^3*x^6-1956*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)))* 
2^(1/2)*d^(3/2)*e^3*x^6-1196*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*e^3*x^7+2934*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) 
))*2^(1/2)*d^(5/2)*e^2*x^4-2934*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)))*2^(1/2)*d^(5/2)*e^2*x^4-4404*d*e^2*(- 
d*e)^(1/2)*(-e*x^2+d)^(1/2)*x^5+1956*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)))*2^(1/2)*d^(7/2)*e*x^2-1956*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)))* 
2^(1/2)*d^(7/2)*e*x^2-5860*d^2*e*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*x^3+489*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)) 
)*2^(1/2)*d^(9/2)-489*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)))*2^(1/2)*d^(9/2)-4188*d^3*(-d*e)^(1/2)*(-e*x^2+d 
)^(1/2)*x)/(e*x^2+d)^(1/2)/(-e*x^2+d)^(1/2)/(e*x-(-d*e)^(1/2))^4/(e*x+(-d* 
e)^(1/2))^4/(-d*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.55 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\left [-\frac {489 \, \sqrt {2} {\left (e^{5} x^{10} + 5 \, d e^{4} x^{8} + 10 \, d^{2} e^{3} x^{6} + 10 \, d^{3} e^{2} x^{4} + 5 \, d^{4} e x^{2} + d^{5}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 4 \, {\left (299 \, e^{4} x^{7} + 1101 \, d e^{3} x^{5} + 1465 \, d^{2} e^{2} x^{3} + 1047 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{6144 \, {\left (d^{3} e^{6} x^{10} + 5 \, d^{4} e^{5} x^{8} + 10 \, d^{5} e^{4} x^{6} + 10 \, d^{6} e^{3} x^{4} + 5 \, d^{7} e^{2} x^{2} + d^{8} e\right )}}, -\frac {489 \, \sqrt {2} {\left (e^{5} x^{10} + 5 \, d e^{4} x^{8} + 10 \, d^{2} e^{3} x^{6} + 10 \, d^{3} e^{2} x^{4} + 5 \, d^{4} e x^{2} + d^{5}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, {\left (299 \, e^{4} x^{7} + 1101 \, d e^{3} x^{5} + 1465 \, d^{2} e^{2} x^{3} + 1047 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{3072 \, {\left (d^{3} e^{6} x^{10} + 5 \, d^{4} e^{5} x^{8} + 10 \, d^{5} e^{4} x^{6} + 10 \, d^{6} e^{3} x^{4} + 5 \, d^{7} e^{2} x^{2} + d^{8} e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(13/2),x, algorithm="fricas")
 

Output:

[-1/6144*(489*sqrt(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)*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*(299*e^4*x^7 + 1101*d*e^3*x^5 + 1465*d^2*e^2*x^3 + 1047*d^ 
3*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(d^3*e^6*x^10 + 5*d^4*e^5*x^8 
 + 10*d^5*e^4*x^6 + 10*d^6*e^3*x^4 + 5*d^7*e^2*x^2 + d^8*e), -1/3072*(489* 
sqrt(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)*sqrt(e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*s 
qrt(e)*x/(e^2*x^4 - d^2)) - 2*(299*e^4*x^7 + 1101*d*e^3*x^5 + 1465*d^2*e^2 
*x^3 + 1047*d^3*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(d^3*e^6*x^10 + 
 5*d^4*e^5*x^8 + 10*d^5*e^4*x^6 + 10*d^6*e^3*x^4 + 5*d^7*e^2*x^2 + d^8*e)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(13/2),x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(13/2), x)
 

Giac [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(13/2),x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(13/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{13/2}} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(13/2),x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(13/2), x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {423 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d^{3} x +335 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} e \,x^{3}+180 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d \,e^{2} x^{5}+40 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e^{3} x^{7}+1956 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{7} x^{14}-5 d \,e^{6} x^{12}-9 d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}+5 d^{4} e^{3} x^{6}+9 d^{5} e^{2} x^{4}+5 d^{6} e \,x^{2}+d^{7}}d x \right ) d^{9} e +9780 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{7} x^{14}-5 d \,e^{6} x^{12}-9 d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}+5 d^{4} e^{3} x^{6}+9 d^{5} e^{2} x^{4}+5 d^{6} e \,x^{2}+d^{7}}d x \right ) d^{8} e^{2} x^{2}+19560 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{7} x^{14}-5 d \,e^{6} x^{12}-9 d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}+5 d^{4} e^{3} x^{6}+9 d^{5} e^{2} x^{4}+5 d^{6} e \,x^{2}+d^{7}}d x \right ) d^{7} e^{3} x^{4}+19560 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{7} x^{14}-5 d \,e^{6} x^{12}-9 d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}+5 d^{4} e^{3} x^{6}+9 d^{5} e^{2} x^{4}+5 d^{6} e \,x^{2}+d^{7}}d x \right ) d^{6} e^{4} x^{6}+9780 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{7} x^{14}-5 d \,e^{6} x^{12}-9 d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}+5 d^{4} e^{3} x^{6}+9 d^{5} e^{2} x^{4}+5 d^{6} e \,x^{2}+d^{7}}d x \right ) d^{5} e^{5} x^{8}+1956 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{7} x^{14}-5 d \,e^{6} x^{12}-9 d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}+5 d^{4} e^{3} x^{6}+9 d^{5} e^{2} x^{4}+5 d^{6} e \,x^{2}+d^{7}}d x \right ) d^{4} e^{6} x^{10}}{423 d^{3} \left (e^{5} x^{10}+5 d \,e^{4} x^{8}+10 d^{2} e^{3} x^{6}+10 d^{3} e^{2} x^{4}+5 d^{4} e \,x^{2}+d^{5}\right )} \] Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(13/2),x)
 

Output:

(423*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*d**3*x + 335*sqrt(d + e*x**2) 
*sqrt(d**2 - e**2*x**4)*d**2*e*x**3 + 180*sqrt(d + e*x**2)*sqrt(d**2 - e** 
2*x**4)*d*e**2*x**5 + 40*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*e**3*x**7 
 + 1956*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**7 + 5*d**6* 
e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e 
**5*x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**9*e + 9780*int((sqrt(d + e* 
x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x** 
4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x** 
12 - e**7*x**14),x)*d**8*e**2*x**2 + 19560*int((sqrt(d + e*x**2)*sqrt(d**2 
 - e**2*x**4)*x**2)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3 
*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x**14 
),x)*d**7*e**3*x**4 + 19560*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x 
**2)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3* 
e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**6*e**4* 
x**6 + 9780*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**7 + 5*d 
**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d* 
*2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**5*e**5*x**8 + 1956*int( 
(sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**7 + 5*d**6*e*x**2 + 9*d 
**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x**10 - 
5*d*e**6*x**12 - e**7*x**14),x)*d**4*e**6*x**10)/(423*d**3*(d**5 + 5*d*...