3.4.32 \(\int \frac {(7+5 x^2)^3}{\sqrt {2+x^2-x^4}} \, dx\) [332]

3.4.32.1 Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=-\frac {625}{3} x \sqrt {2+x^2-x^4}-25 x^3 \sqrt {2+x^2-x^4}+\frac {3905}{3} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-542 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]

output

3905/3*EllipticE(1/2*x*2^(1/2),I*2^(1/2))-542*EllipticF(1/2*x*2^(1/2),I*2^ 
(1/2))-625/3*x*(-x^4+x^2+2)^(1/2)-25*x^3*(-x^4+x^2+2)^(1/2)
 

3.4.32.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=\frac {-2500 x-1550 x^3+1100 x^5+150 x^7+7810 i \sqrt {4+2 x^2-2 x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )-10089 i \sqrt {4+2 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )}{6 \sqrt {2+x^2-x^4}} \]

input

Integrate[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]
 

output

(-2500*x - 1550*x^3 + 1100*x^5 + 150*x^7 + (7810*I)*Sqrt[4 + 2*x^2 - 2*x^4 
]*EllipticE[I*ArcSinh[x], -1/2] - (10089*I)*Sqrt[4 + 2*x^2 - 2*x^4]*Ellipt 
icF[I*ArcSinh[x], -1/2])/(6*Sqrt[2 + x^2 - x^4])
 

3.4.32.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1518, 27, 2207, 25, 1494, 27, 399, 321, 327}

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.

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

\(\Big \downarrow \) 1518

\(\displaystyle -\frac {1}{5} \int -\frac {5 \left (625 x^4+885 x^2+343\right )}{\sqrt {-x^4+x^2+2}}dx-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {625 x^4+885 x^2+343}{\sqrt {-x^4+x^2+2}}dx-25 x^3 \sqrt {-x^4+x^2+2}\)

\(\Big \downarrow \) 2207

\(\displaystyle -\frac {1}{3} \int -\frac {3905 x^2+2279}{\sqrt {-x^4+x^2+2}}dx-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \int \frac {3905 x^2+2279}{\sqrt {-x^4+x^2+2}}dx-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {2}{3} \int \frac {3905 x^2+2279}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3905 x^2+2279}{\sqrt {2-x^2} \sqrt {x^2+1}}dx-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{3} \left (3905 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx-1626 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx\right )-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{3} \left (3905 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx-1626 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{3} \left (3905 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-1626 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )-\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3\)

input

Int[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]
 

output

(-625*x*Sqrt[2 + x^2 - x^4])/3 - 25*x^3*Sqrt[2 + x^2 - x^4] + (3905*Ellipt 
icE[ArcSin[x/Sqrt[2]], -2] - 1626*EllipticF[ArcSin[x/Sqrt[2]], -2])/3
 

3.4.32.3.1 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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

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 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 1494

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c]   Int[(d + e*x^2)/(Sqr 
t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e 
}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
 

rule 1518

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> Simp[e^q*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q 
 + 1))), x] + Simp[1/(c*(4*p + 2*q + 1))   Int[(a + b*x^2 + c*x^4)^p*Expand 
ToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2* 
p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; 
 FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 
 a*e^2, 0] && IGtQ[q, 1]
 

rule 2207

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = 
 Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( 
a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p 
+ 1))   Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 
*n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) 
*x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ 
Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] &&  !LtQ[p, -1]
 

3.4.32.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (63 ) = 126\).

Time = 5.36 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.18

method	result	size
default	\(\frac {2279 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{6 \sqrt {-x^{4}+x^{2}+2}}-25 x^{3} \sqrt {-x^{4}+x^{2}+2}-\frac {625 x \sqrt {-x^{4}+x^{2}+2}}{3}-\frac {3905 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {-x^{4}+x^{2}+2}}\)	\(142\)
risch	\(\frac {25 x \left (3 x^{2}+25\right ) \left (x^{4}-x^{2}-2\right )}{3 \sqrt {-x^{4}+x^{2}+2}}+\frac {2279 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{6 \sqrt {-x^{4}+x^{2}+2}}-\frac {3905 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {-x^{4}+x^{2}+2}}\)	\(142\)
elliptic	\(\frac {2279 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{6 \sqrt {-x^{4}+x^{2}+2}}-25 x^{3} \sqrt {-x^{4}+x^{2}+2}-\frac {625 x \sqrt {-x^{4}+x^{2}+2}}{3}-\frac {3905 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {-x^{4}+x^{2}+2}}\)	\(142\)

input

int((5*x^2+7)^3/(-x^4+x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

output

2279/6*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticF 
(1/2*x*2^(1/2),I*2^(1/2))-25*x^3*(-x^4+x^2+2)^(1/2)-625/3*x*(-x^4+x^2+2)^( 
1/2)-3905/6*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*(Ell 
ipticF(1/2*x*2^(1/2),I*2^(1/2))-EllipticE(1/2*x*2^(1/2),I*2^(1/2)))
 

3.4.32.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=\frac {-15620 i \, \sqrt {2} x E(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + 17899 i \, \sqrt {2} x F(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) - 10 \, {\left (15 \, x^{4} + 125 \, x^{2} + 781\right )} \sqrt {-x^{4} + x^{2} + 2}}{6 \, x} \]

input

integrate((5*x^2+7)^3/(-x^4+x^2+2)^(1/2),x, algorithm="fricas")
 

output

1/6*(-15620*I*sqrt(2)*x*elliptic_e(arcsin(sqrt(2)/x), -1/2) + 17899*I*sqrt 
(2)*x*elliptic_f(arcsin(sqrt(2)/x), -1/2) - 10*(15*x^4 + 125*x^2 + 781)*sq 
rt(-x^4 + x^2 + 2))/x
 

3.4.32.6 Sympy [F]

\[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{3}}{\sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )}}\, dx \]

input

integrate((5*x**2+7)**3/(-x**4+x**2+2)**(1/2),x)
 

output

Integral((5*x**2 + 7)**3/sqrt(-(x**2 - 2)*(x**2 + 1)), x)
 

3.4.32.7 Maxima [F]

\[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt {-x^{4} + x^{2} + 2}} \,d x } \]

input

integrate((5*x^2+7)^3/(-x^4+x^2+2)^(1/2),x, algorithm="maxima")
 

output

integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2), x)
 

3.4.32.8 Giac [F]

\[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt {-x^{4} + x^{2} + 2}} \,d x } \]

input

integrate((5*x^2+7)^3/(-x^4+x^2+2)^(1/2),x, algorithm="giac")
 

output

integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2), x)
 

3.4.32.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^3}{\sqrt {-x^4+x^2+2}} \,d x \]

input

int((5*x^2 + 7)^3/(x^2 - x^4 + 2)^(1/2),x)
 

output

int((5*x^2 + 7)^3/(x^2 - x^4 + 2)^(1/2), x)