\(\int (d+e x^2)^{5/2} (d^2-e^2 x^4)^{3/2} \, dx\) [137]

Optimal result

Integrand size = 28, antiderivative size = 276 \[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {185 d^5 x \sqrt {d^2-e^2 x^4}}{1024 \sqrt {d+e x^2}}+\frac {403 d^4 e x^3 \sqrt {d^2-e^2 x^4}}{512 \sqrt {d+e x^2}}+\frac {55 d^3 e^2 x^5 \sqrt {d^2-e^2 x^4}}{128 \sqrt {d+e x^2}}-\frac {13 d^2 e^3 x^7 \sqrt {d^2-e^2 x^4}}{64 \sqrt {d+e x^2}}-\frac {7 d e^4 x^9 \sqrt {d^2-e^2 x^4}}{24 \sqrt {d+e x^2}}-\frac {e^5 x^{11} \sqrt {d^2-e^2 x^4}}{12 \sqrt {d+e x^2}}+\frac {839 d^6 \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{1024 \sqrt {e}} \] Output:

185/1024*d^5*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+403/512*d^4*e*x^3*(-e^ 
2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+55/128*d^3*e^2*x^5*(-e^2*x^4+d^2)^(1/2)/( 
e*x^2+d)^(1/2)-13/64*d^2*e^3*x^7*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)-7/24 
*d*e^4*x^9*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)-1/12*e^5*x^11*(-e^2*x^4+d^ 
2)^(1/2)/(e*x^2+d)^(1/2)+839/1024*d^6*arctan(e^(1/2)*x*(e*x^2+d)^(1/2)/(-e 
^2*x^4+d^2)^(1/2))/e^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.51 \[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {\frac {x \sqrt {d^2-e^2 x^4} \left (555 d^5+2418 d^4 e x^2+1320 d^3 e^2 x^4-624 d^2 e^3 x^6-896 d e^4 x^8-256 e^5 x^{10}\right )}{\sqrt {d+e x^2}}+\frac {2517 i d^6 \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}}{3072} \] Input:

Integrate[(d + e*x^2)^(5/2)*(d^2 - e^2*x^4)^(3/2),x]
 

Output:

((x*Sqrt[d^2 - e^2*x^4]*(555*d^5 + 2418*d^4*e*x^2 + 1320*d^3*e^2*x^4 - 624 
*d^2*e^3*x^6 - 896*d*e^4*x^8 - 256*e^5*x^10))/Sqrt[d + e*x^2] + ((2517*I)* 
d^6*Log[(-2*I)*Sqrt[e]*x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2]])/Sqrt[ 
e])/3072
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1396, 318, 25, 27, 403, 27, 403, 25, 27, 299, 211, 211, 224, 216}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 318

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^2 \left (25 e x^2+13 d\right )dx}{12 e}-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^2 \left (25 e x^2+13 d\right )dx}{12 e}-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^2 \left (25 e x^2+13 d\right )dx-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (-\frac {\int -5 d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right ) \left (71 e x^2+31 d\right )dx}{10 e}-\frac {5}{2} x \left (d+e x^2\right )^2 \left (d-e x^2\right )^{5/2}\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right ) \left (71 e x^2+31 d\right )dx-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \left (603 e x^2+319 d\right )dx}{8 e}-\frac {71}{8} x \left (d+e x^2\right ) \left (d-e x^2\right )^{5/2}\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \left (603 e x^2+319 d\right )dx}{8 e}-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {1}{8} d \int \left (d-e x^2\right )^{3/2} \left (603 e x^2+319 d\right )dx-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {1}{8} d \left (\frac {839}{2} d \int \left (d-e x^2\right )^{3/2}dx-\frac {201}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {1}{8} d \left (\frac {839}{2} d \left (\frac {3}{4} d \int \sqrt {d-e x^2}dx+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {201}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {1}{8} d \left (\frac {839}{2} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {d-e x^2}}dx+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {201}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {1}{8} d \left (\frac {839}{2} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {201}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{12} d \left (\frac {1}{2} d \left (\frac {1}{8} d \left (\frac {839}{2} d \left (\frac {3}{4} d \left (\frac {d \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {201}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {71}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {5}{2} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )-\frac {1}{12} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^3\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[(d + e*x^2)^(5/2)*(d^2 - e^2*x^4)^(3/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(-1/12*(x*(d - e*x^2)^(5/2)*(d + e*x^2)^3) + (d*((-5* 
x*(d - e*x^2)^(5/2)*(d + e*x^2)^2)/2 + (d*((-71*x*(d - e*x^2)^(5/2)*(d + e 
*x^2))/8 + (d*((-201*x*(d - e*x^2)^(5/2))/2 + (839*d*((x*(d - e*x^2)^(3/2) 
)/4 + (3*d*((x*Sqrt[d - e*x^2])/2 + (d*ArcTan[(Sqrt[e]*x)/Sqrt[d - e*x^2]] 
)/(2*Sqrt[e])))/4))/2))/8))/2))/12))/(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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[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 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 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 = 0.25 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-256 e^{\frac {11}{2}} x^{11} \sqrt {-e \,x^{2}+d}-896 d \,e^{\frac {9}{2}} x^{9} \sqrt {-e \,x^{2}+d}-624 d^{2} e^{\frac {7}{2}} x^{7} \sqrt {-e \,x^{2}+d}+1320 d^{3} e^{\frac {5}{2}} x^{5} \sqrt {-e \,x^{2}+d}+2418 d^{4} e^{\frac {3}{2}} x^{3} \sqrt {-e \,x^{2}+d}+555 \sqrt {-e \,x^{2}+d}\, \sqrt {e}\, d^{5} x +2517 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{6}\right )}{3072 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\)	\(180\)
risch	\(\frac {x \left (-256 e^{5} x^{10}-896 d \,e^{4} x^{8}-624 d^{2} e^{3} x^{6}+1320 d^{3} e^{2} x^{4}+2418 d^{4} e \,x^{2}+555 d^{5}\right ) \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{3072 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {839 d^{6} \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{1024 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(187\)

Input:

int((e*x^2+d)^(5/2)*(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/3072*(-e^2*x^4+d^2)^(1/2)*(-256*e^(11/2)*x^11*(-e*x^2+d)^(1/2)-896*d*e^( 
9/2)*x^9*(-e*x^2+d)^(1/2)-624*d^2*e^(7/2)*x^7*(-e*x^2+d)^(1/2)+1320*d^3*e^ 
(5/2)*x^5*(-e*x^2+d)^(1/2)+2418*d^4*e^(3/2)*x^3*(-e*x^2+d)^(1/2)+555*(-e*x 
^2+d)^(1/2)*e^(1/2)*d^5*x+2517*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2))*d^6)/(e* 
x^2+d)^(1/2)/(-e*x^2+d)^(1/2)/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.24 \[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\left [-\frac {2517 \, {\left (d^{6} e x^{2} + d^{7}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right ) + 2 \, {\left (256 \, e^{6} x^{11} + 896 \, d e^{5} x^{9} + 624 \, d^{2} e^{4} x^{7} - 1320 \, d^{3} e^{3} x^{5} - 2418 \, d^{4} e^{2} x^{3} - 555 \, d^{5} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{6144 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {2517 \, {\left (d^{6} e x^{2} + d^{7}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) + {\left (256 \, e^{6} x^{11} + 896 \, d e^{5} x^{9} + 624 \, d^{2} e^{4} x^{7} - 1320 \, d^{3} e^{3} x^{5} - 2418 \, d^{4} e^{2} x^{3} - 555 \, d^{5} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{3072 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:

integrate((e*x^2+d)^(5/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

[-1/6144*(2517*(d^6*e*x^2 + d^7)*sqrt(-e)*log(-(2*e^2*x^4 + d*e*x^2 - 2*sq 
rt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(-e)*x - d^2)/(e*x^2 + d)) + 2*(256 
*e^6*x^11 + 896*d*e^5*x^9 + 624*d^2*e^4*x^7 - 1320*d^3*e^3*x^5 - 2418*d^4* 
e^2*x^3 - 555*d^5*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(e^2*x^2 + d* 
e), -1/3072*(2517*(d^6*e*x^2 + d^7)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2)*sq 
rt(e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)) + (256*e^6*x^11 + 896*d*e^5*x^9 + 
 624*d^2*e^4*x^7 - 1320*d^3*e^3*x^5 - 2418*d^4*e^2*x^3 - 555*d^5*e*x)*sqrt 
(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(e^2*x^2 + d*e)]
 

Sympy [F]

\[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int \left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((e*x**2+d)**(5/2)*(-e**2*x**4+d**2)**(3/2),x)
 

Output:

Integral((-(-d + e*x**2)*(d + e*x**2))**(3/2)*(d + e*x**2)**(5/2), x)
 

Maxima [F]

\[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((e*x^2+d)^(5/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^(5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.40 \[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {839 \, d^{6} \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{1024 \, \sqrt {-e}} + \frac {1}{3072} \, {\left (555 \, d^{5} + 2 \, {\left (1209 \, d^{4} e + 4 \, {\left (165 \, d^{3} e^{2} - 2 \, {\left (39 \, d^{2} e^{3} + 8 \, {\left (2 \, e^{5} x^{2} + 7 \, d e^{4}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {-e x^{2} + d} x \] Input:

integrate((e*x^2+d)^(5/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

-839/1024*d^6*log(abs(-sqrt(-e)*x + sqrt(-e*x^2 + d)))/sqrt(-e) + 1/3072*( 
555*d^5 + 2*(1209*d^4*e + 4*(165*d^3*e^2 - 2*(39*d^2*e^3 + 8*(2*e^5*x^2 + 
7*d*e^4)*x^2)*x^2)*x^2)*x^2)*sqrt(-e*x^2 + d)*x
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int {\left (d^2-e^2\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^{5/2} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^(5/2),x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.48 \[ \int \left (d+e x^2\right )^{5/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {2517 \sqrt {e}\, \mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right ) d^{6}+555 \sqrt {-e \,x^{2}+d}\, d^{5} e x +2418 \sqrt {-e \,x^{2}+d}\, d^{4} e^{2} x^{3}+1320 \sqrt {-e \,x^{2}+d}\, d^{3} e^{3} x^{5}-624 \sqrt {-e \,x^{2}+d}\, d^{2} e^{4} x^{7}-896 \sqrt {-e \,x^{2}+d}\, d \,e^{5} x^{9}-256 \sqrt {-e \,x^{2}+d}\, e^{6} x^{11}}{3072 e} \] Input:

int((e*x^2+d)^(5/2)*(-e^2*x^4+d^2)^(3/2),x)
 

Output:

(2517*sqrt(e)*asin((sqrt(e)*x)/sqrt(d))*d**6 + 555*sqrt(d - e*x**2)*d**5*e 
*x + 2418*sqrt(d - e*x**2)*d**4*e**2*x**3 + 1320*sqrt(d - e*x**2)*d**3*e** 
3*x**5 - 624*sqrt(d - e*x**2)*d**2*e**4*x**7 - 896*sqrt(d - e*x**2)*d*e**5 
*x**9 - 256*sqrt(d - e*x**2)*e**6*x**11)/(3072*e)