\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^{15/2}} \, dx\) [147]

Optimal result

Integrand size = 28, antiderivative size = 230 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{5 \left (d+e x^2\right )^{11/2}}+\frac {7 x \sqrt {d^2-e^2 x^4}}{80 d \left (d+e x^2\right )^{9/2}}+\frac {33 x \sqrt {d^2-e^2 x^4}}{320 d^2 \left (d+e x^2\right )^{7/2}}+\frac {327 x \sqrt {d^2-e^2 x^4}}{2560 d^3 \left (d+e x^2\right )^{5/2}}+\frac {1887 x \sqrt {d^2-e^2 x^4}}{10240 d^4 \left (d+e x^2\right )^{3/2}}+\frac {609 \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2048 \sqrt {2} d^4 \sqrt {e}} \] Output:

1/5*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(11/2)+7/80*x*(-e^2*x^4+d^2)^(1/2)/d/ 
(e*x^2+d)^(9/2)+33/320*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)^(7/2)+327/2560 
*x*(-e^2*x^4+d^2)^(1/2)/d^3/(e*x^2+d)^(5/2)+1887/10240*x*(-e^2*x^4+d^2)^(1 
/2)/d^4/(e*x^2+d)^(3/2)+609/4096*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^4/e^(1/2)
 

Mathematica [A] (verified)

Time = 5.39 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (\frac {2 \left (7195 d^4 x+14480 d^3 e x^3+16302 d^2 e^2 x^5+8856 d e^3 x^7+1887 e^4 x^9\right )}{\left (d+e x^2\right )^5}+\frac {3045 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{\sqrt {e} \sqrt {d-e x^2}}\right )}{20480 d^4 \sqrt {d+e x^2}} \] Input:

Integrate[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(15/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((2*(7195*d^4*x + 14480*d^3*e*x^3 + 16302*d^2*e^2*x^5 
 + 8856*d*e^3*x^7 + 1887*e^4*x^9))/(d + e*x^2)^5 + (3045*Sqrt[2]*ArcTan[(S 
qrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]])/(Sqrt[e]*Sqrt[d - e*x^2])))/(20480*d^4 
*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1396, 315, 27, 402, 27, 402, 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.

\(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int \frac {2 d e \left (4 d-3 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^5}dx}{10 d e}+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}-\frac {\int -\frac {3 d e \left (19 d-14 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^4}dx}{16 d^2 e}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \int \frac {19 d-14 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^4}dx}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}-\frac {\int -\frac {3 d e \left (65 d-44 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^3}dx}{12 d^2 e}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\int \frac {65 d-44 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^3}dx}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}-\frac {\int -\frac {d e \left (411 d-218 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2 e}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\int \frac {d e \left (411 d-218 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2 e}+\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\int \frac {411 d-218 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d}+\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\frac {629 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}-\frac {\int -\frac {1015 d^2 e}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx}{4 d^2 e}}{8 d}+\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\frac {1015}{4} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx+\frac {629 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d}+\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\frac {1015}{4} \int \frac {1}{\frac {2 d e x^2}{d-e x^2}+d}d\frac {x}{\sqrt {d-e x^2}}+\frac {629 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d}+\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\frac {1015 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{4 \sqrt {2} d \sqrt {e}}+\frac {629 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d}+\frac {109 x \sqrt {d-e x^2}}{8 d \left (d+e x^2\right )^2}}{4 d}+\frac {11 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )^3}\right )}{16 d}+\frac {7 x \sqrt {d-e x^2}}{16 d \left (d+e x^2\right )^4}\right )+\frac {x \sqrt {d-e x^2}}{5 \left (d+e x^2\right )^5}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(15/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d - e*x^2])/(5*(d + e*x^2)^5) + ((7*x*Sqrt[d 
 - e*x^2])/(16*d*(d + e*x^2)^4) + (3*((11*x*Sqrt[d - e*x^2])/(4*d*(d + e*x 
^2)^3) + ((109*x*Sqrt[d - e*x^2])/(8*d*(d + e*x^2)^2) + ((629*x*Sqrt[d - e 
*x^2])/(4*d*(d + e*x^2)) + (1015*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2 
]])/(4*Sqrt[2]*d*Sqrt[e]))/(8*d))/(4*d)))/(16*d))/5))/(Sqrt[d - e*x^2]*Sqr 
t[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. \(883\) vs. \(2(186)=372\).

Time = 0.55 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.84

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{5} \left (3045 \sqrt {2}\, \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 ) e^{5} x^{10} \sqrt {d}-3045 \sqrt {2}\, \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 ) e^{5} x^{10} \sqrt {d}+15225 \sqrt {2}\, \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 ) d^{\frac {3}{2}} e^{4} x^{8}-15225 \sqrt {2}\, \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 ) d^{\frac {3}{2}} e^{4} x^{8}+7548 e^{4} \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x^{9}+30450 \sqrt {2}\, \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 ) d^{\frac {5}{2}} e^{3} x^{6}-30450 \sqrt {2}\, \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 ) d^{\frac {5}{2}} e^{3} x^{6}+35424 e^{3} d \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x^{7}+30450 \sqrt {2}\, \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 ) d^{\frac {7}{2}} e^{2} x^{4}-30450 \sqrt {2}\, \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 ) d^{\frac {7}{2}} e^{2} x^{4}+65208 e^{2} d^{2} \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x^{5}+15225 \sqrt {2}\, \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 ) d^{\frac {9}{2}} e \,x^{2}-15225 \sqrt {2}\, \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 ) d^{\frac {9}{2}} e \,x^{2}+57920 e \,d^{3} \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x^{3}+3045 \sqrt {2}\, \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 ) d^{\frac {11}{2}}-3045 \sqrt {2}\, \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 ) d^{\frac {11}{2}}+28780 d^{4} \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x \right )}{40960 d^{4} \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (e x -\sqrt {-d e}\right )^{5} \left (e x +\sqrt {-d e}\right )^{5} \sqrt {-d e}}\)	\(884\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(15/2),x,method=_RETURNVERBOSE)
 

Output:

1/40960*(-e^2*x^4+d^2)^(1/2)*e^5/d^4*(3045*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)))*e^5*x^10*d^(1/2)-3 
045*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)))*e^5*x^10*d^(1/2)+15225*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^4*x^8-15225 
*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^4*x^8+7548*e^4*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*x^9+3 
0450*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^3*x^6-30450*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^3*x^6+35424 
*e^3*d*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*x^7+30450*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^(7/2)*e^2*x^ 
4-30450*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^(7/2)*e^2*x^4+65208*e^2*d^2*(-d*e)^(1/2)*(-e*x^2+d) 
^(1/2)*x^5+15225*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^(9/2)*e*x^2-15225*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^(9/2)*e*x 
^2+57920*e*d^3*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*x^3+3045*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^(11/2 
)-3045*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)-(-d*e)^(1/2)*x+...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.45 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\left [-\frac {3045 \, \sqrt {2} {\left (e^{6} x^{12} + 6 \, d e^{5} x^{10} + 15 \, d^{2} e^{4} x^{8} + 20 \, d^{3} e^{3} x^{6} + 15 \, d^{4} e^{2} x^{4} + 6 \, d^{5} e x^{2} + d^{6}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 4 \, {\left (1887 \, e^{5} x^{9} + 8856 \, d e^{4} x^{7} + 16302 \, d^{2} e^{3} x^{5} + 14480 \, d^{3} e^{2} x^{3} + 7195 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{40960 \, {\left (d^{4} e^{7} x^{12} + 6 \, d^{5} e^{6} x^{10} + 15 \, d^{6} e^{5} x^{8} + 20 \, d^{7} e^{4} x^{6} + 15 \, d^{8} e^{3} x^{4} + 6 \, d^{9} e^{2} x^{2} + d^{10} e\right )}}, -\frac {3045 \, \sqrt {2} {\left (e^{6} x^{12} + 6 \, d e^{5} x^{10} + 15 \, d^{2} e^{4} x^{8} + 20 \, d^{3} e^{3} x^{6} + 15 \, d^{4} e^{2} x^{4} + 6 \, d^{5} e x^{2} + d^{6}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, {\left (1887 \, e^{5} x^{9} + 8856 \, d e^{4} x^{7} + 16302 \, d^{2} e^{3} x^{5} + 14480 \, d^{3} e^{2} x^{3} + 7195 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{20480 \, {\left (d^{4} e^{7} x^{12} + 6 \, d^{5} e^{6} x^{10} + 15 \, d^{6} e^{5} x^{8} + 20 \, d^{7} e^{4} x^{6} + 15 \, d^{8} e^{3} x^{4} + 6 \, d^{9} e^{2} x^{2} + d^{10} e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(15/2),x, algorithm="fricas")
 

Output:

[-1/40960*(3045*sqrt(2)*(e^6*x^12 + 6*d*e^5*x^10 + 15*d^2*e^4*x^8 + 20*d^3 
*e^3*x^6 + 15*d^4*e^2*x^4 + 6*d^5*e*x^2 + d^6)*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*(1887*e^5*x^9 + 8856*d*e^4*x^7 + 16302 
*d^2*e^3*x^5 + 14480*d^3*e^2*x^3 + 7195*d^4*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt 
(e*x^2 + d))/(d^4*e^7*x^12 + 6*d^5*e^6*x^10 + 15*d^6*e^5*x^8 + 20*d^7*e^4* 
x^6 + 15*d^8*e^3*x^4 + 6*d^9*e^2*x^2 + d^10*e), -1/20480*(3045*sqrt(2)*(e^ 
6*x^12 + 6*d*e^5*x^10 + 15*d^2*e^4*x^8 + 20*d^3*e^3*x^6 + 15*d^4*e^2*x^4 + 
 6*d^5*e*x^2 + d^6)*sqrt(e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 
 + d)*sqrt(e)*x/(e^2*x^4 - d^2)) - 2*(1887*e^5*x^9 + 8856*d*e^4*x^7 + 1630 
2*d^2*e^3*x^5 + 14480*d^3*e^2*x^3 + 7195*d^4*e*x)*sqrt(-e^2*x^4 + d^2)*sqr 
t(e*x^2 + d))/(d^4*e^7*x^12 + 6*d^5*e^6*x^10 + 15*d^6*e^5*x^8 + 20*d^7*e^4 
*x^6 + 15*d^8*e^3*x^4 + 6*d^9*e^2*x^2 + d^10*e)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**(15/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {15}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(15/2),x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(15/2), x)
 

Giac [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {15}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(15/2),x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(15/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{15/2}} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(15/2),x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(15/2), x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{15/2}} \, dx =\text {Too large to display} \] Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(15/2),x)
 

Output:

(285*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*d**4*x + 275*sqrt(d + e*x**2) 
*sqrt(d**2 - e**2*x**4)*d**3*e*x**3 + 206*sqrt(d + e*x**2)*sqrt(d**2 - e** 
2*x**4)*d**2*e**2*x**5 + 88*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*d*e**3 
*x**7 + 16*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*e**4*x**9 + 1740*int((s 
qrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**8 + 6*d**7*e*x**2 + 14*d* 
*6*e**2*x**4 + 14*d**5*e**3*x**6 - 14*d**3*e**5*x**10 - 14*d**2*e**6*x**12 
 - 6*d*e**7*x**14 - e**8*x**16),x)*d**11*e + 10440*int((sqrt(d + e*x**2)*s 
qrt(d**2 - e**2*x**4)*x**2)/(d**8 + 6*d**7*e*x**2 + 14*d**6*e**2*x**4 + 14 
*d**5*e**3*x**6 - 14*d**3*e**5*x**10 - 14*d**2*e**6*x**12 - 6*d*e**7*x**14 
 - e**8*x**16),x)*d**10*e**2*x**2 + 26100*int((sqrt(d + e*x**2)*sqrt(d**2 
- e**2*x**4)*x**2)/(d**8 + 6*d**7*e*x**2 + 14*d**6*e**2*x**4 + 14*d**5*e** 
3*x**6 - 14*d**3*e**5*x**10 - 14*d**2*e**6*x**12 - 6*d*e**7*x**14 - e**8*x 
**16),x)*d**9*e**3*x**4 + 34800*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x** 
4)*x**2)/(d**8 + 6*d**7*e*x**2 + 14*d**6*e**2*x**4 + 14*d**5*e**3*x**6 - 1 
4*d**3*e**5*x**10 - 14*d**2*e**6*x**12 - 6*d*e**7*x**14 - e**8*x**16),x)*d 
**8*e**4*x**6 + 26100*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/( 
d**8 + 6*d**7*e*x**2 + 14*d**6*e**2*x**4 + 14*d**5*e**3*x**6 - 14*d**3*e** 
5*x**10 - 14*d**2*e**6*x**12 - 6*d*e**7*x**14 - e**8*x**16),x)*d**7*e**5*x 
**8 + 10440*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**8 + 6*d 
**7*e*x**2 + 14*d**6*e**2*x**4 + 14*d**5*e**3*x**6 - 14*d**3*e**5*x**10...