3.4.18 \(\int (7+5 x^2)^2 \sqrt {2+x^2-x^4} \, dx\) [318]

3.4.18.1 Optimal result

Integrand size = 24, antiderivative size = 74 \[ \int \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4} \, dx=\frac {1}{21} x \left (275+354 x^2\right ) \sqrt {2+x^2-x^4}-\frac {25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac {2045}{21} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {79}{7} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]

output

-25/7*x*(-x^4+x^2+2)^(3/2)+2045/21*EllipticE(1/2*x*2^(1/2),I*2^(1/2))-79/7 
*EllipticF(1/2*x*2^(1/2),I*2^(1/2))+1/21*x*(354*x^2+275)*(-x^4+x^2+2)^(1/2 
)
 

3.4.18.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38 \[ \int \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4} \, dx=\frac {250 x+683 x^3+304 x^5-204 x^7-75 x^9+2045 i \sqrt {4+2 x^2-2 x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )-2949 i \sqrt {4+2 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )}{21 \sqrt {2+x^2-x^4}} \]

input

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

output

(250*x + 683*x^3 + 304*x^5 - 204*x^7 - 75*x^9 + (2045*I)*Sqrt[4 + 2*x^2 - 
2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (2949*I)*Sqrt[4 + 2*x^2 - 2*x^4]*El 
lipticF[I*ArcSinh[x], -1/2])/(21*Sqrt[2 + x^2 - x^4])
 

3.4.18.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1518, 25, 1490, 27, 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.

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

\(\Big \downarrow \) 1518

\(\displaystyle -\frac {1}{7} \int -\left (\left (590 x^2+393\right ) \sqrt {-x^4+x^2+2}\right )dx-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{7} \int \left (590 x^2+393\right ) \sqrt {-x^4+x^2+2}dx-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{7} \left (\frac {1}{3} x \left (354 x^2+275\right ) \sqrt {-x^4+x^2+2}-\frac {1}{15} \int -\frac {5 \left (2045 x^2+1808\right )}{\sqrt {-x^4+x^2+2}}dx\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \frac {2045 x^2+1808}{\sqrt {-x^4+x^2+2}}dx+\frac {1}{3} x \sqrt {-x^4+x^2+2} \left (354 x^2+275\right )\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {1}{7} \left (\frac {2}{3} \int \frac {2045 x^2+1808}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {1}{3} x \sqrt {-x^4+x^2+2} \left (354 x^2+275\right )\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \frac {2045 x^2+1808}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {1}{3} x \sqrt {-x^4+x^2+2} \left (354 x^2+275\right )\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (2045 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx-237 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx\right )+\frac {1}{3} x \sqrt {-x^4+x^2+2} \left (354 x^2+275\right )\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (2045 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx-237 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )+\frac {1}{3} x \sqrt {-x^4+x^2+2} \left (354 x^2+275\right )\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (2045 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-237 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )+\frac {1}{3} x \sqrt {-x^4+x^2+2} \left (354 x^2+275\right )\right )-\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}\)

input

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

output

(-25*x*(2 + x^2 - x^4)^(3/2))/7 + ((x*(275 + 354*x^2)*Sqrt[2 + x^2 - x^4]) 
/3 + (2045*EllipticE[ArcSin[x/Sqrt[2]], -2] - 237*EllipticF[ArcSin[x/Sqrt[ 
2]], -2])/3)/7
 

3.4.18.3.1 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 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

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]
 

3.4.18.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (68 ) = 136\).

Time = 2.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.99

method	result	size
risch	\(-\frac {x \left (75 x^{4}+279 x^{2}+125\right ) \left (x^{4}-x^{2}-2\right )}{21 \sqrt {-x^{4}+x^{2}+2}}+\frac {904 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{21 \sqrt {-x^{4}+x^{2}+2}}-\frac {2045 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{42 \sqrt {-x^{4}+x^{2}+2}}\)	\(147\)
default	\(\frac {125 x \sqrt {-x^{4}+x^{2}+2}}{21}+\frac {904 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{21 \sqrt {-x^{4}+x^{2}+2}}-\frac {2045 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{42 \sqrt {-x^{4}+x^{2}+2}}+\frac {25 x^{5} \sqrt {-x^{4}+x^{2}+2}}{7}+\frac {93 x^{3} \sqrt {-x^{4}+x^{2}+2}}{7}\)	\(159\)
elliptic	\(\frac {125 x \sqrt {-x^{4}+x^{2}+2}}{21}+\frac {904 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{21 \sqrt {-x^{4}+x^{2}+2}}-\frac {2045 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{42 \sqrt {-x^{4}+x^{2}+2}}+\frac {25 x^{5} \sqrt {-x^{4}+x^{2}+2}}{7}+\frac {93 x^{3} \sqrt {-x^{4}+x^{2}+2}}{7}\)	\(159\)

input

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

output

-1/21*x*(75*x^4+279*x^2+125)*(x^4-x^2-2)/(-x^4+x^2+2)^(1/2)+904/21*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))-2045/42*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))-EllipticE(1/2*x*2^(1/2),I*2^(1/2)) 
)
 

3.4.18.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4} \, dx=\frac {-4090 i \, \sqrt {2} x E(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + 4994 i \, \sqrt {2} x F(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + {\left (75 \, x^{6} + 279 \, x^{4} + 125 \, x^{2} - 2045\right )} \sqrt {-x^{4} + x^{2} + 2}}{21 \, x} \]

input

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

output

1/21*(-4090*I*sqrt(2)*x*elliptic_e(arcsin(sqrt(2)/x), -1/2) + 4994*I*sqrt( 
2)*x*elliptic_f(arcsin(sqrt(2)/x), -1/2) + (75*x^6 + 279*x^4 + 125*x^2 - 2 
045)*sqrt(-x^4 + x^2 + 2))/x
 

3.4.18.6 Sympy [F]

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

input

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

output

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

3.4.18.7 Maxima [F]

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

input

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

output

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

3.4.18.8 Giac [F]

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

input

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

output

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

3.4.18.9 Mupad [F(-1)]

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

input

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

output

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