\(\int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx\) [1201]

Optimal result

Integrand size = 26, antiderivative size = 70 \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{9 \sqrt {3}} \] Output:

-1/9*x*(-3*x^2+2)^(1/2)*(3*x^2-1)^(1/2)-1/9*EllipticE(1/2*(-6*x^2+4)^(1/2) 
,2^(1/2))*3^(1/2)-1/27*InverseJacobiAM(arccos(1/2*x*6^(1/2)),2^(1/2))*3^(1 
/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {3 x \left (1-3 x^2\right ) \sqrt {2-3 x^2}-3 \sqrt {6-18 x^2} E\left (\arcsin \left (\sqrt {3} x\right )|\frac {1}{2}\right )+2 \sqrt {6-18 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} x\right ),\frac {1}{2}\right )}{27 \sqrt {-1+3 x^2}} \] Input:

Integrate[(x^2*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]
 

Output:

(3*x*(1 - 3*x^2)*Sqrt[2 - 3*x^2] - 3*Sqrt[6 - 18*x^2]*EllipticE[ArcSin[Sqr 
t[3]*x], 1/2] + 2*Sqrt[6 - 18*x^2]*EllipticF[ArcSin[Sqrt[3]*x], 1/2])/(27* 
Sqrt[-1 + 3*x^2])
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {380, 25, 399, 322, 328}

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.

\(\displaystyle \int \frac {x^2 \sqrt {3 x^2-1}}{\sqrt {2-3 x^2}} \, dx\)

\(\Big \downarrow \) 380

\(\displaystyle \frac {1}{9} \int -\frac {2-9 x^2}{\sqrt {2-3 x^2} \sqrt {3 x^2-1}}dx-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {3 x^2-1}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \int \frac {2-9 x^2}{\sqrt {2-3 x^2} \sqrt {3 x^2-1}}dx-\frac {1}{9} \sqrt {2-3 x^2} \sqrt {3 x^2-1} x\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{9} \left (\int \frac {1}{\sqrt {2-3 x^2} \sqrt {3 x^2-1}}dx+3 \int \frac {\sqrt {3 x^2-1}}{\sqrt {2-3 x^2}}dx\right )-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {3 x^2-1}\)

\(\Big \downarrow \) 322

\(\displaystyle \frac {1}{9} \left (3 \int \frac {\sqrt {3 x^2-1}}{\sqrt {2-3 x^2}}dx-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{\sqrt {3}}\right )-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {3 x^2-1}\)

\(\Big \downarrow \) 328

\(\displaystyle \frac {1}{9} \left (-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{\sqrt {3}}-\sqrt {3} E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )\right )-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {3 x^2-1}\)

Input:

Int[(x^2*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]
 

Output:

-1/9*(x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2]) + (-(Sqrt[3]*EllipticE[ArcCos[Sq 
rt[3/2]*x], 2]) - EllipticF[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])/9
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 322

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a - b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/ 
c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && 
GtQ[c, 0] && GtQ[a - b*(c/d), 0]
 

rule 328

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

rule 380

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* 
(m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1))   Int[(e*x)^(m 
 - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 
*q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c 
- a*d, 0] && GtQ[q, 0] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, 
 q, x]
 

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])))))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(58)=116\).

Time = 2.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.84

method	result	size
default	\(-\frac {\sqrt {3 x^{2}-1}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \left (54 x^{5}+\sqrt {3}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, x}{2}, \sqrt {2}\right )-3 \sqrt {3}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, x}{2}, \sqrt {2}\right )-54 x^{3}+12 x \right )}{108 \left (9 x^{4}-9 x^{2}+2\right )}\)	\(129\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (-\frac {x \sqrt {-9 x^{4}+9 x^{2}-2}}{9}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, x}{2}, \sqrt {2}\right )}{27 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6}\, x}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6}\, x}{2}, \sqrt {2}\right )\right )}{18 \sqrt {-9 x^{4}+9 x^{2}-2}}\right )}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\)	\(163\)
risch	\(\frac {x \left (3 x^{2}-2\right ) \sqrt {3 x^{2}-1}\, \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{9 \sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \sqrt {-3 x^{2}+2}}+\frac {\left (-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, x}{2}, \sqrt {2}\right )}{27 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6}\, x}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6}\, x}{2}, \sqrt {2}\right )\right )}{18 \sqrt {-9 x^{4}+9 x^{2}-2}}\right ) \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\)	\(209\)

Input:

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

Output:

-1/108*(3*x^2-1)^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(54*x^5+3^(1/2)*2^(1/2)*(- 
6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*EllipticF(1/2*3^(1/2)*2^(1/2)*x,2^(1/2))-3 
*3^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*EllipticE(1/2*3^(1/2)*2 
^(1/2)*x,2^(1/2))-54*x^3+12*x)/(9*x^4-9*x^2+2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {-2 i \, \sqrt {\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {\frac {2}{3}}}{x}\right )\,|\,\frac {1}{2}) + i \, \sqrt {\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {\frac {2}{3}}}{x}\right )\,|\,\frac {1}{2}) - \sqrt {3 \, x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {-3 \, x^{2} + 2}}{9 \, x} \] Input:

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

Output:

1/9*(-2*I*sqrt(2/3)*x*elliptic_e(arcsin(sqrt(2/3)/x), 1/2) + I*sqrt(2/3)*x 
*elliptic_f(arcsin(sqrt(2/3)/x), 1/2) - sqrt(3*x^2 - 1)*(x^2 + 1)*sqrt(-3* 
x^2 + 2))/x
 

Sympy [F]

\[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {x^{2} \sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {\sqrt {3 \, x^{2} - 1} x^{2}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:

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

Output:

integrate(sqrt(3*x^2 - 1)*x^2/sqrt(-3*x^2 + 2), x)
 

Giac [F]

\[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {\sqrt {3 \, x^{2} - 1} x^{2}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:

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

Output:

integrate(sqrt(3*x^2 - 1)*x^2/sqrt(-3*x^2 + 2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {x^2\,\sqrt {3\,x^2-1}}{\sqrt {2-3\,x^2}} \,d x \] Input:

int((x^2*(3*x^2 - 1)^(1/2))/(2 - 3*x^2)^(1/2),x)
 

Output:

int((x^2*(3*x^2 - 1)^(1/2))/(2 - 3*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}\, x}{9}-\left (\int \frac {\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}\, x^{2}}{9 x^{4}-9 x^{2}+2}d x \right )+\frac {2 \left (\int \frac {\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}}{9 x^{4}-9 x^{2}+2}d x \right )}{9} \] Input:

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

Output:

( - sqrt(3*x**2 - 1)*sqrt( - 3*x**2 + 2)*x - 9*int((sqrt(3*x**2 - 1)*sqrt( 
 - 3*x**2 + 2)*x**2)/(9*x**4 - 9*x**2 + 2),x) + 2*int((sqrt(3*x**2 - 1)*sq 
rt( - 3*x**2 + 2))/(9*x**4 - 9*x**2 + 2),x))/9