\(\int \frac {1}{(2+5 x^2-7 x^4)^{3/2}} \, dx\) [308]

Optimal result

Integrand size = 16, antiderivative size = 59 \[ \int \frac {1}{\left (2+5 x^2-7 x^4\right )^{3/2}} \, dx=\frac {x \left (53-35 x^2\right )}{162 \sqrt {2+5 x^2-7 x^4}}+\frac {5 E\left (\arcsin (x)\left |-\frac {7}{2}\right .\right )}{81 \sqrt {2}}+\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {7}{2}\right )}{9 \sqrt {2}} \] Output:

1/162*x*(-35*x^2+53)/(-7*x^4+5*x^2+2)^(1/2)+5/162*EllipticE(x,1/2*I*14^(1/ 
2))*2^(1/2)+1/18*EllipticF(x,1/2*I*14^(1/2))*2^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\left (2+5 x^2-7 x^4\right )^{3/2}} \, dx=\frac {1}{162} \left (\frac {53 x}{\sqrt {2+5 x^2-7 x^4}}-\frac {35 x^3}{\sqrt {2+5 x^2-7 x^4}}+5 i \sqrt {7} E\left (i \text {arcsinh}\left (\sqrt {\frac {7}{2}} x\right )|-\frac {2}{7}\right )-9 i \sqrt {7} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {7}{2}} x\right ),-\frac {2}{7}\right )\right ) \] Input:

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

Output:

((53*x)/Sqrt[2 + 5*x^2 - 7*x^4] - (35*x^3)/Sqrt[2 + 5*x^2 - 7*x^4] + (5*I) 
*Sqrt[7]*EllipticE[I*ArcSinh[Sqrt[7/2]*x], -2/7] - (9*I)*Sqrt[7]*EllipticF 
[I*ArcSinh[Sqrt[7/2]*x], -2/7])/162
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1405, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}-\frac {1}{162} \int -\frac {7 \left (5 x^2+4\right )}{\sqrt {-7 x^4+5 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{162} \int \frac {5 x^2+4}{\sqrt {-7 x^4+5 x^2+2}}dx+\frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {7}{81} \sqrt {7} \int \frac {5 x^2+4}{2 \sqrt {7} \sqrt {1-x^2} \sqrt {7 x^2+2}}dx+\frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{162} \int \frac {5 x^2+4}{\sqrt {1-x^2} \sqrt {7 x^2+2}}dx+\frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {7}{162} \left (\frac {18}{7} \int \frac {1}{\sqrt {1-x^2} \sqrt {7 x^2+2}}dx+\frac {5}{7} \int \frac {\sqrt {7 x^2+2}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {7}{162} \left (\frac {5}{7} \int \frac {\sqrt {7 x^2+2}}{\sqrt {1-x^2}}dx+\frac {9}{7} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),-\frac {7}{2}\right )\right )+\frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {7}{162} \left (\frac {9}{7} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),-\frac {7}{2}\right )+\frac {5}{7} \sqrt {2} E\left (\arcsin (x)\left |-\frac {7}{2}\right .\right )\right )+\frac {x \left (53-35 x^2\right )}{162 \sqrt {-7 x^4+5 x^2+2}}\)

Input:

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

Output:

(x*(53 - 35*x^2))/(162*Sqrt[2 + 5*x^2 - 7*x^4]) + (7*((5*Sqrt[2]*EllipticE 
[ArcSin[x], -7/2])/7 + (9*Sqrt[2]*EllipticF[ArcSin[x], -7/2])/7))/162
 

Defintions of rubi rules used

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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (51 ) = 102\).

Time = 2.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.05

method	result	size
risch	\(-\frac {x \left (35 x^{2}-53\right )}{162 \sqrt {-7 x^{4}+5 x^{2}+2}}+\frac {7 \sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {14}}{2}\right )}{81 \sqrt {-7 x^{4}+5 x^{2}+2}}-\frac {5 \sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {14}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {14}}{2}\right )\right )}{162 \sqrt {-7 x^{4}+5 x^{2}+2}}\)	\(121\)
default	\(\frac {\frac {53}{162} x -\frac {35}{162} x^{3}}{\sqrt {-7 x^{4}+5 x^{2}+2}}+\frac {7 \sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {14}}{2}\right )}{81 \sqrt {-7 x^{4}+5 x^{2}+2}}-\frac {5 \sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {14}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {14}}{2}\right )\right )}{162 \sqrt {-7 x^{4}+5 x^{2}+2}}\)	\(122\)
elliptic	\(\frac {\frac {53}{162} x -\frac {35}{162} x^{3}}{\sqrt {-7 x^{4}+5 x^{2}+2}}+\frac {7 \sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {14}}{2}\right )}{81 \sqrt {-7 x^{4}+5 x^{2}+2}}-\frac {5 \sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {14}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {14}}{2}\right )\right )}{162 \sqrt {-7 x^{4}+5 x^{2}+2}}\)	\(122\)

Input:

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

Output:

-1/162*x*(35*x^2-53)/(-7*x^4+5*x^2+2)^(1/2)+7/81*(-x^2+1)^(1/2)*(14*x^2+4) 
^(1/2)/(-7*x^4+5*x^2+2)^(1/2)*EllipticF(x,1/2*I*14^(1/2))-5/162*(-x^2+1)^( 
1/2)*(14*x^2+4)^(1/2)/(-7*x^4+5*x^2+2)^(1/2)*(EllipticF(x,1/2*I*14^(1/2))- 
EllipticE(x,1/2*I*14^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\left (2+5 x^2-7 x^4\right )^{3/2}} \, dx=\frac {5 \, \sqrt {2} {\left (7 \, x^{4} - 5 \, x^{2} - 2\right )} E(\arcsin \left (x\right )\,|\,-\frac {7}{2}) + 9 \, \sqrt {2} {\left (7 \, x^{4} - 5 \, x^{2} - 2\right )} F(\arcsin \left (x\right )\,|\,-\frac {7}{2}) + \sqrt {-7 \, x^{4} + 5 \, x^{2} + 2} {\left (35 \, x^{3} - 53 \, x\right )}}{162 \, {\left (7 \, x^{4} - 5 \, x^{2} - 2\right )}} \] Input:

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

Output:

1/162*(5*sqrt(2)*(7*x^4 - 5*x^2 - 2)*elliptic_e(arcsin(x), -7/2) + 9*sqrt( 
2)*(7*x^4 - 5*x^2 - 2)*elliptic_f(arcsin(x), -7/2) + sqrt(-7*x^4 + 5*x^2 + 
 2)*(35*x^3 - 53*x))/(7*x^4 - 5*x^2 - 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (2+5 x^2-7 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-7\,x^4+5\,x^2+2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (2+5 x^2-7 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-7 x^{4}+5 x^{2}+2}}{49 x^{8}-70 x^{6}-3 x^{4}+20 x^{2}+4}d x \] Input:

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

Output:

int(sqrt( - 7*x**4 + 5*x**2 + 2)/(49*x**8 - 70*x**6 - 3*x**4 + 20*x**2 + 4 
),x)