\(\int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx\) [1304]

Optimal result

Integrand size = 27, antiderivative size = 169 \[ \int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}+\frac {2 a^{7/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{5 b^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {2 a^{7/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{5 b^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

1/5*x^3*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)+2/5*a^(7/2)*(1-b^2*x^4/a^2)^(1/2) 
*EllipticE(b^(1/2)*x/a^(1/2),I)/b^(3/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)-2 
/5*a^(7/2)*(1-b^2*x^4/a^2)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),I)/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 = 0.54 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.41 \[ \int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {x^3 \sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {b^2 x^4}{a^2}\right )}{3 \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:

Integrate[x^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^2],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {340, 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.

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

\(\Big \downarrow \) 340

\(\displaystyle \frac {2}{5} a^2 \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 344

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \int \frac {x^2}{\sqrt {a^2-b^2 x^4}}dx}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {a \int \frac {b x^2+a}{a \sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {1}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {b x^2+a}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\)

Input:

Int[x^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^2],x]
 

Output:

(x^3*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/5 + (2*a^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[ 
ArcSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(b^(3/2)*Sqrt[a^2 - b^2*x^4])))/(5*Sqrt[ 
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 340

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*((c + d*x^2)^p/(e*(m + 4*p 
 + 1))), x] + Simp[4*a*c*(p/(m + 4*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] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0] && 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 = 0.86 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.80

method	result	size
elliptic	\(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (\frac {x^{3} \sqrt {-b^{2} x^{4}+a^{2}}}{5}-\frac {2 a^{3} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{5 \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, b}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(136\)
risch	\(\frac {x^{3} \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{5}-\frac {2 a^{3} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{5 \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, b \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(144\)
default	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {\frac {b}{a}}\, b^{3} x^{7}-\sqrt {\frac {b}{a}}\, a^{2} b \,x^{3}+2 a^{3} \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-2 a^{3} \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{5 \left (-b^{2} x^{4}+a^{2}\right ) \sqrt {\frac {b}{a}}\, b}\)	\(165\)

Input:

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

Output:

(-b^2*x^4+a^2)^(1/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)*(1/5*x^3*(-b^2*x^4+a 
^2)^(1/2)-2/5*a^3/(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 = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.66 \[ \int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=-\frac {2 \, \sqrt {-b^{2}} a^{3} x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-1) - 2 \, \sqrt {-b^{2}} a^{3} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-1) - {\left (b^{3} x^{4} - 2 \, a^{2} b\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{5 \, b^{3} x} \] Input:

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

Output:

-1/5*(2*sqrt(-b^2)*a^3*x*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), -1) - 2 
*sqrt(-b^2)*a^3*x*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), -1) - (b^3*x^4 
 - 2*a^2*b)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a))/(b^3*x)
 

Sympy [F]

\[ \int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\int x^{2} \sqrt {a - b x^{2}} \sqrt {a + b x^{2}}\, dx \] Input:

integrate(x**2*(-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2),x)
 

Output:

Integral(x**2*sqrt(a - b*x**2)*sqrt(a + b*x**2), x)
 

Maxima [F]

\[ \int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} x^{2} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} x^{2} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a + b*x**2)*sqrt(a - b*x**2)*x**3 + 2*int((sqrt(a + b*x**2)*sqrt(a - 
 b*x**2)*x**2)/(a**2 - b**2*x**4),x)*a**2)/5