\(\int \frac {(d-e x^2)^{13/2}}{(d^2-e^2 x^4)^{5/2}} \, dx\) [164]

Optimal result

Integrand size = 29, antiderivative size = 191 \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {16 d^3 x \left (d-e x^2\right )^{3/2}}{3 \left (d^2-e^2 x^4\right )^{3/2}}-\frac {64 d^2 x \sqrt {d-e x^2}}{3 \sqrt {d^2-e^2 x^4}}-\frac {27 d x \sqrt {d^2-e^2 x^4}}{8 \sqrt {d-e x^2}}+\frac {e x^3 \sqrt {d^2-e^2 x^4}}{4 \sqrt {d-e x^2}}+\frac {163 d^2 \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{8 \sqrt {e}} \] Output:

16/3*d^3*x*(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(3/2)-64/3*d^2*x*(-e*x^2+d)^(1/ 
2)/(-e^2*x^4+d^2)^(1/2)-27/8*d*x*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^(1/2)+1/4 
*e*x^3*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^(1/2)+163/8*d^2*arctanh(e^(1/2)*x*( 
-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2))/e^(1/2)
 

Mathematica [A] (verified)

Time = 5.76 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.81 \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {1}{24} \left (-\frac {x \sqrt {d^2-e^2 x^4} \left (465 d^3+668 d^2 e x^2+69 d e^2 x^4-6 e^3 x^6\right )}{\sqrt {d-e x^2} \left (d+e x^2\right )^2}-\frac {489 d^2 \log \left (-d+e x^2\right )}{\sqrt {e}}+\frac {489 d^2 \log \left (d e x-e^2 x^3+\sqrt {e} \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}\right )}{\sqrt {e}}\right ) \] Input:

Integrate[(d - e*x^2)^(13/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

(-((x*Sqrt[d^2 - e^2*x^4]*(465*d^3 + 668*d^2*e*x^2 + 69*d*e^2*x^4 - 6*e^3* 
x^6))/(Sqrt[d - e*x^2]*(d + e*x^2)^2)) - (489*d^2*Log[-d + e*x^2])/Sqrt[e] 
 + (489*d^2*Log[d*e*x - e^2*x^3 + Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d^2 - e^2*x 
^4]])/Sqrt[e])/24
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1396, 315, 27, 401, 25, 27, 403, 27, 299, 224, 219}

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.

\(\displaystyle \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {d e \left (d-e x^2\right )^2 \left (11 e x^2+d\right )}{\left (e x^2+d\right )^{3/2}}dx}{3 d e}+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \int \frac {\left (d-e x^2\right )^2 \left (11 e x^2+d\right )}{\left (e x^2+d\right )^{3/2}}dx+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (-\frac {\int -\frac {d e \left (11 d-51 e x^2\right ) \left (d-e x^2\right )}{\sqrt {e x^2+d}}dx}{d e}-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (11 d-51 e x^2\right ) \left (d-e x^2\right )}{\sqrt {e x^2+d}}dx}{d e}-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\int \frac {\left (11 d-51 e x^2\right ) \left (d-e x^2\right )}{\sqrt {e x^2+d}}dx-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (55 d-379 e x^2\right )}{\sqrt {e x^2+d}}dx}{4 e}-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\frac {1}{4} x \left (11 d-51 e x^2\right ) \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {1}{4} d \int \frac {55 d-379 e x^2}{\sqrt {e x^2+d}}dx-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\frac {1}{4} x \left (11 d-51 e x^2\right ) \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {1}{4} d \left (\frac {489}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {379}{2} x \sqrt {d+e x^2}\right )-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\frac {1}{4} x \left (11 d-51 e x^2\right ) \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {1}{4} d \left (\frac {489}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {379}{2} x \sqrt {d+e x^2}\right )-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\frac {1}{4} x \left (11 d-51 e x^2\right ) \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {1}{4} d \left (\frac {489 d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}-\frac {379}{2} x \sqrt {d+e x^2}\right )-\frac {10 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\frac {1}{4} x \left (11 d-51 e x^2\right ) \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^3}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[(d - e*x^2)^(13/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((2*x*(d - e*x^2)^3)/(3*(d + e*x^2)^(3/2) 
) + ((-10*x*(d - e*x^2)^2)/Sqrt[d + e*x^2] - (x*(11*d - 51*e*x^2)*Sqrt[d + 
 e*x^2])/4 + (d*((-379*x*Sqrt[d + e*x^2])/2 + (489*d*ArcTanh[(Sqrt[e]*x)/S 
qrt[d + e*x^2]])/(2*Sqrt[e])))/4)/3))/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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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 = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.74

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (6 x^{7} e^{\frac {7}{2}}-69 e^{\frac {5}{2}} d \,x^{5}+489 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d^{2} e \,x^{2} \sqrt {e \,x^{2}+d}-668 e^{\frac {3}{2}} d^{2} x^{3}+489 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d^{3} \sqrt {e \,x^{2}+d}-465 \sqrt {e}\, d^{3} x \right )}{24 \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right )^{2} \sqrt {e}}\)	\(141\)
risch	\(\frac {x \left (-2 e \,x^{2}+27 d \right ) \sqrt {e \,x^{2}+d}\, \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{8 \sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\left (\frac {163 d^{2} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{8 \sqrt {e}}+\frac {4 d^{3} \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{3 e \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}}-\frac {32 d^{2} \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{3 e \left (x -\frac {\sqrt {-d e}}{e}\right )}-\frac {4 d^{3} \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{3 e \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}}-\frac {32 d^{2} \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{3 e \left (x +\frac {\sqrt {-d e}}{e}\right )}\right ) \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{\sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(447\)

Input:

int((-e*x^2+d)^(13/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/24*(-e^2*x^4+d^2)^(1/2)*(6*x^7*e^(7/2)-69*e^(5/2)*d*x^5+489*ln(x*e^(1/2) 
+(e*x^2+d)^(1/2))*d^2*e*x^2*(e*x^2+d)^(1/2)-668*e^(3/2)*d^2*x^3+489*ln(x*e 
^(1/2)+(e*x^2+d)^(1/2))*d^3*(e*x^2+d)^(1/2)-465*e^(1/2)*d^3*x)/(-e*x^2+d)^ 
(1/2)/(e*x^2+d)^2/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.06 \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\left [\frac {489 \, {\left (d^{2} e^{3} x^{6} + d^{3} e^{2} x^{4} - d^{4} e x^{2} - d^{5}\right )} \sqrt {e} \log \left (\frac {2 \, e^{2} x^{4} - d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e x^{2} - d}\right ) - 2 \, {\left (6 \, e^{4} x^{7} - 69 \, d e^{3} x^{5} - 668 \, d^{2} e^{2} x^{3} - 465 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{48 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )}}, \frac {489 \, {\left (d^{2} e^{3} x^{6} + d^{3} e^{2} x^{4} - d^{4} e x^{2} - d^{5}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) - {\left (6 \, e^{4} x^{7} - 69 \, d e^{3} x^{5} - 668 \, d^{2} e^{2} x^{3} - 465 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{24 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(13/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/48*(489*(d^2*e^3*x^6 + d^3*e^2*x^4 - d^4*e*x^2 - d^5)*sqrt(e)*log((2*e^ 
2*x^4 - d*e*x^2 - 2*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(e)*x - d^2) 
/(e*x^2 - d)) - 2*(6*e^4*x^7 - 69*d*e^3*x^5 - 668*d^2*e^2*x^3 - 465*d^3*e* 
x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(e^4*x^6 + d*e^3*x^4 - d^2*e^2*x 
^2 - d^3*e), 1/24*(489*(d^2*e^3*x^6 + d^3*e^2*x^4 - d^4*e*x^2 - d^5)*sqrt( 
-e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d^2 
)) - (6*e^4*x^7 - 69*d*e^3*x^5 - 668*d^2*e^2*x^3 - 465*d^3*e*x)*sqrt(-e^2* 
x^4 + d^2)*sqrt(-e*x^2 + d))/(e^4*x^6 + d*e^3*x^4 - d^2*e^2*x^2 - d^3*e)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((-e*x**2+d)**(13/2)/(-e**2*x**4+d**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {13}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-e*x^2+d)^(13/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-e*x^2 + d)^(13/2)/(-e^2*x^4 + d^2)^(5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.40 \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {1}{8} \, {\left (2 \, e x^{2} - 27 \, d\right )} \sqrt {e x^{2} + d} x - \frac {163 \, d^{2} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{8 \, \sqrt {e}} - \frac {16 \, {\left (4 \, d^{2} e x^{2} + 3 \, d^{3}\right )} x}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \] Input:

integrate((-e*x^2+d)^(13/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
 

Output:

1/8*(2*e*x^2 - 27*d)*sqrt(e*x^2 + d)*x - 163/8*d^2*log(abs(-sqrt(e)*x + sq 
rt(e*x^2 + d)))/sqrt(e) - 16/3*(4*d^2*e*x^2 + 3*d^3)*x/(e*x^2 + d)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^{13/2}}{{\left (d^2-e^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d - e*x^2)^(13/2)/(d^2 - e^2*x^4)^(5/2),x)
 

Output:

int((d - e*x^2)^(13/2)/(d^2 - e^2*x^4)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d-e x^2\right )^{13/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {-3720 \sqrt {e \,x^{2}+d}\, d^{3} e x -5344 \sqrt {e \,x^{2}+d}\, d^{2} e^{2} x^{3}-552 \sqrt {e \,x^{2}+d}\, d \,e^{3} x^{5}+48 \sqrt {e \,x^{2}+d}\, e^{4} x^{7}+3912 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{4}+7824 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{3} e \,x^{2}+3912 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2} e^{2} x^{4}+111 \sqrt {e}\, d^{4}+222 \sqrt {e}\, d^{3} e \,x^{2}+111 \sqrt {e}\, d^{2} e^{2} x^{4}}{192 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((-e*x^2+d)^(13/2)/(-e^2*x^4+d^2)^(5/2),x)
 

Output:

( - 3720*sqrt(d + e*x**2)*d**3*e*x - 5344*sqrt(d + e*x**2)*d**2*e**2*x**3 
- 552*sqrt(d + e*x**2)*d*e**3*x**5 + 48*sqrt(d + e*x**2)*e**4*x**7 + 3912* 
sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**4 + 7824*sqrt(e)*lo 
g((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**3*e*x**2 + 3912*sqrt(e)*log(( 
sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**2*e**2*x**4 + 111*sqrt(e)*d**4 + 
 222*sqrt(e)*d**3*e*x**2 + 111*sqrt(e)*d**2*e**2*x**4)/(192*e*(d**2 + 2*d* 
e*x**2 + e**2*x**4))