\(\int \frac {(d+e x^2)^3}{\sqrt {d^2-e^2 x^4}} \, dx\) [54]

Optimal result

Integrand size = 26, antiderivative size = 169 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {d^2-e^2 x^4}} \, dx=-d x \sqrt {d^2-e^2 x^4}-\frac {1}{5} e x^3 \sqrt {d^2-e^2 x^4}+\frac {18 d^{7/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{5 \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {8 d^{7/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{5 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

-d*x*(-e^2*x^4+d^2)^(1/2)-1/5*e*x^3*(-e^2*x^4+d^2)^(1/2)+18/5*d^(7/2)*(1-e 
^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1 
/2)-8/5*d^(7/2)*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/e^(1/ 
2)/(-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {d^2-e^2 x^4}} \, dx=\frac {-5 d^3 x-d^2 e x^3+5 d e^2 x^5+e^3 x^7+10 d^3 x \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+6 d^2 e x^3 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )}{5 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(-5*d^3*x - d^2*e*x^3 + 5*d*e^2*x^5 + e^3*x^7 + 10*d^3*x*Sqrt[1 - (e^2*x^4 
)/d^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (e^2*x^4)/d^2] + 6*d^2*e*x^3*Sqrt[ 
1 - (e^2*x^4)/d^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (e^2*x^4)/d^2])/(5*Sqr 
t[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.44, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {1396, 318, 27, 403, 25, 27, 399, 289, 329, 327, 765, 762}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 318

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {\int -\frac {6 d e \sqrt {e x^2+d} \left (2 e x^2+d\right )}{\sqrt {d-e x^2}}dx}{5 e}-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \int \frac {\sqrt {e x^2+d} \left (2 e x^2+d\right )}{\sqrt {d-e x^2}}dx-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (-\frac {\int -\frac {d e \left (9 e x^2+5 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {\int \frac {d e \left (9 e x^2+5 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \int \frac {9 e x^2+5 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \left (9 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-4 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx\right )-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \left (9 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {4 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \left (\frac {9 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {4 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \left (\frac {9 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {4 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \left (\frac {9 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {4 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {6}{5} d \left (\frac {1}{3} d \left (\frac {9 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {4 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )-\frac {2}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

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

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(-1/5*(x*Sqrt[d - e*x^2]*(d + e*x^2)^(3/2 
)) + (6*d*((-2*x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])/3 + (d*((9*d^(3/2)*Sqrt[ 
1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sq 
rt[d - e*x^2]*Sqrt[d + e*x^2]) - (4*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*Ellipt 
icF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e* 
x^2])))/3))/5))/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 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 329

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]))   Int[Sqrt[1 
+ b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
*c + a*d, 0] &&  !(LtQ[a*c, 0] && GtQ[a*b, 0])
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 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 = 7.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {x \left (e \,x^{2}+5 d \right ) \sqrt {-e^{2} x^{4}+d^{2}}}{5}+\frac {2 d^{2} \left (\frac {5 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {9 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{5}\)	\(171\)
elliptic	\(-\frac {e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}-d x \sqrt {-e^{2} x^{4}+d^{2}}+\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {18 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(181\)
default	\(\frac {d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+e^{3} \left (-\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5 e^{2}}-\frac {3 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 e^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d \,e^{2} \left (-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3 e^{2}}+\frac {d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-\frac {3 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(343\)

Input:

int((e*x^2+d)^3/(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5*x*(e*x^2+5*d)*(-e^2*x^4+d^2)^(1/2)+2/5*d^2*(5*d/(e/d)^(1/2)*(1-e*x^2/ 
d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2),I) 
-9*d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)* 
(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {d^2-e^2 x^4}} \, dx=-\frac {18 \, \sqrt {-e^{2}} d^{3} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 2 \, {\left (9 \, d^{3} + 5 \, d^{2} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + {\left (e^{3} x^{4} + 5 \, d e^{2} x^{2} + 18 \, d^{2} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{5 \, e^{2} x} \] Input:

integrate((e*x^2+d)^3/(-e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
 

Output:

-1/5*(18*sqrt(-e^2)*d^3*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - 
2*(9*d^3 + 5*d^2*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), 
 -1) + (e^3*x^4 + 5*d*e^2*x^2 + 18*d^2*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
 

Sympy [A] (verification not implemented)

Time = 1.84 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {d^2-e^2 x^4}} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {3 e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 d \Gamma \left (\frac {11}{4}\right )} \] Input:

integrate((e*x**2+d)**3/(-e**2*x**4+d**2)**(1/2),x)
 

Output:

d**2*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d* 
*2)/(4*gamma(5/4)) + 3*d*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), e**2* 
x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(7/4)) + 3*e**2*x**5*gamma(5/4)*hyper 
((1/2, 5/4), (9/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(9/4)) + e* 
*3*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), e**2*x**4*exp_polar(2*I*pi)/ 
d**2)/(4*d*gamma(11/4))
 

Maxima [F]

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

integrate((e*x^2+d)^3/(-e^2*x^4+d^2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x^2+d)^3/(-e^2*x^4+d^2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((d + e*x^2)^3/(d^2 - e^2*x^4)^(1/2),x)
 

Output:

int((d + e*x^2)^3/(d^2 - e^2*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {d^2-e^2 x^4}} \, dx=-\sqrt {-e^{2} x^{4}+d^{2}}\, d x -\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e \,x^{3}}{5}+2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{3}+\frac {18 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{2} e}{5} \] Input:

int((e*x^2+d)^3/(-e^2*x^4+d^2)^(1/2),x)
 

Output:

( - 5*sqrt(d**2 - e**2*x**4)*d*x - sqrt(d**2 - e**2*x**4)*e*x**3 + 10*int( 
sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**3 + 18*int((sqrt(d**2 - e* 
*2*x**4)*x**2)/(d**2 - e**2*x**4),x)*d**2*e)/5