\(\int \frac {x^2}{(a-b x^2)^{3/2} (a+b x^2)^{3/2}} \, dx\) [1346]

Optimal result

Integrand size = 27, antiderivative size = 172 \[ \int \frac {x^2}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{2 \sqrt {a} b^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

1/2*x^3/a^2/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)-1/2*(1-b^2*x^4/a^2)^(1/2)*Ell 
ipticE(b^(1/2)*x/a^(1/2),I)/a^(1/2)/b^(3/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/ 
2)+1/2*(1-b^2*x^4/a^2)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),I)/a^(1/2)/b^(3/2 
)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 2.97 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.42 \[ \int \frac {x^2}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {x^3 \sqrt {1-\frac {b^2 x^4}{a^2}} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};\frac {b^2 x^4}{a^2}\right )}{3 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Input:

Integrate[x^2/((a - b*x^2)^(3/2)*(a + b*x^2)^(3/2)),x]
 

Output:

(x^3*Sqrt[1 - (b^2*x^4)/a^2]*HypergeometricPFQ[{3/4, 3/2}, {7/4}, (b^2*x^4 
)/a^2])/(3*a^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 344, 836, 27, 765, 762, 1390, 1389, 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.

\(\displaystyle \int \frac {x^2}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 342

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\int \frac {x^2}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx}{2 a^2}\)

\(\Big \downarrow \) 344

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \int \frac {x^2}{\sqrt {a^2-b^2 x^4}}dx}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {a \int \frac {b x^2+a}{a \sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {1}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {b x^2+a}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x^3}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a^2-b^2 x^4} \left (\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{2 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}\)

Input:

Int[x^2/((a - b*x^2)^(3/2)*(a + b*x^2)^(3/2)),x]
 

Output:

x^3/(2*a^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]) - (Sqrt[a^2 - b^2*x^4]*((a^(3/ 
2)*Sqrt[1 - (b^2*x^4)/a^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(b^ 
(3/2)*Sqrt[a^2 - b^2*x^4]) - (a^(3/2)*Sqrt[1 - (b^2*x^4)/a^2]*EllipticF[Ar 
cSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(b^(3/2)*Sqrt[a^2 - b^2*x^4])))/(2*a^2*Sqr 
t[a - b*x^2]*Sqrt[a + b*x^2])
 

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 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 342

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(-(e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((c + d*x^2)^(p + 
 1)/(4*a*c*e*(p + 1))), x] + Simp[(m + 4*p + 5)/(4*a*c*(p + 1))   Int[(e*x) 
^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m}, x] && EqQ[b*c + a*d, 0] && LtQ[p, -1] && IntegerQ[2*m]
 

rule 344

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c 
+ b*d*x^4)^FracPart[p])   Int[(e*x)^m*(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a 
, b, c, d, e, m, p}, x] && EqQ[b*c + a*d, 0] &&  !IntegerQ[p]
 

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 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [A] (verified)

Time = 3.46 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.84

method	result	size
elliptic	\(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (\frac {x^{3}}{2 a^{2} \sqrt {-\left (x^{4}-\frac {a^{2}}{b^{2}}\right ) b^{2}}}+\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{2 a \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, b}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(144\)
default	\(\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (b \,x^{3} \sqrt {\frac {b}{a}}+\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right ) a \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right ) a \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\right )}{2 b \left (-b^{2} x^{4}+a^{2}\right ) a^{2} \sqrt {\frac {b}{a}}}\)	\(145\)

Input:

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

Output:

(-b^2*x^4+a^2)^(1/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)*(1/2/a^2*x^3/(-(x^4- 
a^2/b^2)*b^2)^(1/2)+1/2/a/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+b*x^2/a)^(1/2)/ 
(-b^2*x^4+a^2)^(1/2)/b*(EllipticF(x*(b/a)^(1/2),I)-EllipticE(x*(b/a)^(1/2) 
,I)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} b x^{3} + {\left (b^{2} x^{4} - a^{2}\right )} \sqrt {\frac {b}{a}} E(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-1) - {\left (b^{2} x^{4} - a^{2}\right )} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-1)}{2 \, {\left (a^{2} b^{3} x^{4} - a^{4} b\right )}} \] Input:

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

Output:

-1/2*(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)*b*x^3 + (b^2*x^4 - a^2)*sqrt(b/a)*e 
lliptic_e(arcsin(x*sqrt(b/a)), -1) - (b^2*x^4 - a^2)*sqrt(b/a)*elliptic_f( 
arcsin(x*sqrt(b/a)), -1))/(a^2*b^3*x^4 - a^4*b)
 

Sympy [A] (verification not implemented)

Time = 28.88 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.59 \[ \int \frac {x^2}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {1}{4}, \frac {5}{4}, \frac {7}{4} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {7}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{4}}} \right )}}{4 \pi ^{\frac {3}{2}} a^{\frac {3}{2}} b^{\frac {3}{2}}} + \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2} & 1 \\0, \frac {1}{2}, 0 & - \frac {3}{4}, - \frac {1}{4}, \frac {3}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{4}}} \right )}}{4 \pi ^{\frac {3}{2}} a^{\frac {3}{2}} b^{\frac {3}{2}}} \] Input:

integrate(x**2/(-b*x**2+a)**(3/2)/(b*x**2+a)**(3/2),x)
 

Output:

-I*meijerg(((1/2, 1, 1), (1/4, 5/4, 7/4)), ((1/2, 3/4, 1, 5/4, 7/4), (0,)) 
, a**2/(b**2*x**4))/(4*pi**(3/2)*a**(3/2)*b**(3/2)) + I*meijerg(((-3/4, -1 
/4, 0, 1/4, 1/2), (1,)), ((0, 1/2, 0), (-3/4, -1/4, 3/4)), a**2*exp_polar( 
-2*I*pi)/(b**2*x**4))/(4*pi**(3/2)*a**(3/2)*b**(3/2))
 

Maxima [F]

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

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

Output:

integrate(x^2/((b*x^2 + a)^(3/2)*(-b*x^2 + a)^(3/2)), x)
 

Giac [F]

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

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

Output:

integrate(x^2/((b*x^2 + a)^(3/2)*(-b*x^2 + a)^(3/2)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{b^{4} x^{8}-2 a^{2} b^{2} x^{4}+a^{4}}d x \] Input:

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

Output:

int((sqrt(a + b*x**2)*sqrt(a - b*x**2)*x**2)/(a**4 - 2*a**2*b**2*x**4 + b* 
*4*x**8),x)