\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^6} \, dx\) [52]

Optimal result

Integrand size = 26, antiderivative size = 254 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{7 \left (d+e x^2\right )^4}+\frac {4 x \sqrt {d^2-e^2 x^4}}{35 d \left (d+e x^2\right )^3}+\frac {11 x \sqrt {d^2-e^2 x^4}}{70 d^2 \left (d+e x^2\right )^2}+\frac {3 x \sqrt {d^2-e^2 x^4}}{10 d^3 \left (d+e x^2\right )}+\frac {3 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{10 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {11 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{70 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

2/7*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^4+4/35*x*(-e^2*x^4+d^2)^(1/2)/d/(e*x^ 
2+d)^3+11/70*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)^2+3/10*x*(-e^2*x^4+d^2)^ 
(1/2)/d^3/(e*x^2+d)+3/10*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2) 
,I)/d^(3/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)-11/70*(1-e^2*x^4/d^2)^(1/2)*Ellip 
ticF(e^(1/2)*x/d^(1/2),I)/d^(3/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\frac {-\frac {x \left (-d+e x^2\right ) \left (60 d^3+93 d^2 e x^2+74 d e^2 x^4+21 e^3 x^6\right )}{\left (d+e x^2\right )^3}+\frac {i e \sqrt {1-\frac {e^2 x^4}{d^2}} \left (21 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-11 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\left (-\frac {e}{d}\right )^{3/2}}}{70 d^3 \sqrt {d^2-e^2 x^4}} \] Input:

Integrate[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^6,x]
 

Output:

(-((x*(-d + e*x^2)*(60*d^3 + 93*d^2*e*x^2 + 74*d*e^2*x^4 + 21*e^3*x^6))/(d 
 + e*x^2)^3) + (I*e*Sqrt[1 - (e^2*x^4)/d^2]*(21*EllipticE[I*ArcSinh[Sqrt[- 
(e/d)]*x], -1] - 11*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/(-(e/d))^(3 
/2))/(70*d^3*Sqrt[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.26, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {1396, 315, 27, 402, 27, 402, 25, 27, 402, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}-\frac {\int -\frac {6 d e \left (7 d-4 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {d e \left (31 d-11 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\int \frac {d e \left (31 d-11 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\int \frac {31 d-11 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\int -\frac {2 d e \left (21 e x^2+10 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\int \frac {21 e x^2+10 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-11 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {21 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {11 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {21 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {11 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {11 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {11 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{7} \left (\frac {3 \left (\frac {\frac {\frac {21 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {11 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {21 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{6 d}+\frac {11 x \sqrt {d-e x^2}}{6 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {4 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{7 \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^6,x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((2*x*Sqrt[d - e*x^2])/(7*(d + e*x^2)^(7/2)) + ((4*x* 
Sqrt[d - e*x^2])/(5*d*(d + e*x^2)^(5/2)) + (3*((11*x*Sqrt[d - e*x^2])/(6*d 
*(d + e*x^2)^(3/2)) + ((21*x*Sqrt[d - e*x^2])/(d*Sqrt[d + e*x^2]) + ((21*d 
^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1]) 
/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) - (11*d^(3/2)*Sqrt[1 - (e^2*x^4 
)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2 
]*Sqrt[d + e*x^2]))/d)/(6*d)))/(5*d))/7))/(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 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 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 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 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 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 = 5.31 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{7 e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {4 x \sqrt {-e^{2} x^{4}+d^{2}}}{35 d \,e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 e^{2} d^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {3 \left (-e^{2} x^{2}+d e \right ) x}{10 e \,d^{3} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(287\)
elliptic	\(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{7 e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {4 x \sqrt {-e^{2} x^{4}+d^{2}}}{35 d \,e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 e^{2} d^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {3 \left (-e^{2} x^{2}+d e \right ) x}{10 e \,d^{3} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(287\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^6,x,method=_RETURNVERBOSE)
 

Output:

2/7*x/e^4*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^4+4/35/d*x/e^3*(-e^2*x^4+d^2)^(1/ 
2)/(x^2+d/e)^3+11/70/e^2/d^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+3/10*(-e^2 
*x^2+d*e)/e/d^3*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/7/d^2/(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)-3/10/d^2/(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 = 0.10 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\frac {21 \, {\left (e^{5} x^{8} + 4 \, d e^{4} x^{6} + 6 \, d^{2} e^{3} x^{4} + 4 \, d^{3} e^{2} x^{2} + d^{4} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (10 \, d e^{4} - 21 \, e^{5}\right )} x^{8} + 4 \, {\left (10 \, d^{2} e^{3} - 21 \, d e^{4}\right )} x^{6} + 10 \, d^{5} - 21 \, d^{4} e + 6 \, {\left (10 \, d^{3} e^{2} - 21 \, d^{2} e^{3}\right )} x^{4} + 4 \, {\left (10 \, d^{4} e - 21 \, d^{3} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (21 \, e^{4} x^{7} + 74 \, d e^{3} x^{5} + 93 \, d^{2} e^{2} x^{3} + 60 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{70 \, {\left (d^{3} e^{5} x^{8} + 4 \, d^{4} e^{4} x^{6} + 6 \, d^{5} e^{3} x^{4} + 4 \, d^{6} e^{2} x^{2} + d^{7} e\right )}} \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^6,x, algorithm="fricas")
 

Output:

1/70*(21*(e^5*x^8 + 4*d*e^4*x^6 + 6*d^2*e^3*x^4 + 4*d^3*e^2*x^2 + d^4*e)*s 
qrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), -1) + ((10*d*e^4 - 21*e^5)*x^8 + 
4*(10*d^2*e^3 - 21*d*e^4)*x^6 + 10*d^5 - 21*d^4*e + 6*(10*d^3*e^2 - 21*d^2 
*e^3)*x^4 + 4*(10*d^4*e - 21*d^3*e^2)*x^2)*sqrt(e/d)*elliptic_f(arcsin(x*s 
qrt(e/d)), -1) + (21*e^4*x^7 + 74*d*e^3*x^5 + 93*d^2*e^2*x^3 + 60*d^3*e*x) 
*sqrt(-e^2*x^4 + d^2))/(d^3*e^5*x^8 + 4*d^4*e^4*x^6 + 6*d^5*e^3*x^4 + 4*d^ 
6*e^2*x^2 + d^7*e)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\text {Timed out} \] Input:

integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**6,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{6}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^6,x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^6, x)
 

Giac [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{6}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^6,x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^6} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^6,x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^6, x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^6} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x +3 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{6}+12 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{5} e \,x^{2}+18 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{4} e^{2} x^{4}+12 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{3} e^{3} x^{6}+3 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{2} e^{4} x^{8}-\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{4} e^{2}-4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{3} e^{3} x^{2}-6 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{2} e^{4} x^{4}-4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d \,e^{5} x^{6}-\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) e^{6} x^{8}}{4 e^{4} x^{8}+16 d \,e^{3} x^{6}+24 d^{2} e^{2} x^{4}+16 d^{3} e \,x^{2}+4 d^{4}} \] Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^6,x)
 

Output:

(sqrt(d**2 - e**2*x**4)*x + 3*int(sqrt(d**2 - e**2*x**4)/(d**6 + 4*d**5*e* 
x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x**10 - e**6*x**12), 
x)*d**6 + 12*int(sqrt(d**2 - e**2*x**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e** 
2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x**10 - e**6*x**12),x)*d**5*e*x**2 + 
18*int(sqrt(d**2 - e**2*x**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5 
*d**2*e**4*x**8 - 4*d*e**5*x**10 - e**6*x**12),x)*d**4*e**2*x**4 + 12*int( 
sqrt(d**2 - e**2*x**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e 
**4*x**8 - 4*d*e**5*x**10 - e**6*x**12),x)*d**3*e**3*x**6 + 3*int(sqrt(d** 
2 - e**2*x**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 
 - 4*d*e**5*x**10 - e**6*x**12),x)*d**2*e**4*x**8 - int((sqrt(d**2 - e**2* 
x**4)*x**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 
4*d*e**5*x**10 - e**6*x**12),x)*d**4*e**2 - 4*int((sqrt(d**2 - e**2*x**4)* 
x**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e* 
*5*x**10 - e**6*x**12),x)*d**3*e**3*x**2 - 6*int((sqrt(d**2 - e**2*x**4)*x 
**4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e** 
5*x**10 - e**6*x**12),x)*d**2*e**4*x**4 - 4*int((sqrt(d**2 - e**2*x**4)*x* 
*4)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5 
*x**10 - e**6*x**12),x)*d*e**5*x**6 - int((sqrt(d**2 - e**2*x**4)*x**4)/(d 
**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x**10 
 - e**6*x**12),x)*e**6*x**8)/(4*(d**4 + 4*d**3*e*x**2 + 6*d**2*e**2*x**...