\(\int \frac {(a-b x^2)^{3/2} (a+b x^2)^{3/2}}{x^2} \, dx\) [1314]

Optimal result

Integrand size = 27, antiderivative size = 200 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x^2} \, dx=-\frac {6}{5} b^2 x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}-\frac {12 a^{7/2} \sqrt {b} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {12 a^{7/2} \sqrt {b} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

-6/5*b^2*x^3*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)-(-b*x^2+a)^(3/2)*(b*x^2+a)^( 
3/2)/x-12/5*a^(7/2)*b^(1/2)*(1-b^2*x^4/a^2)^(1/2)*EllipticE(b^(1/2)*x/a^(1 
/2),I)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)+12/5*a^(7/2)*b^(1/2)*(1-b^2*x^4/a^ 
2)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),I)/(-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 = 3.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.35 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x^2} \, dx=-\frac {a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};\frac {b^2 x^4}{a^2}\right )}{x \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {339, 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.

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

\(\Big \downarrow \) 339

\(\displaystyle -6 b^2 \int x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}dx-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 340

\(\displaystyle -6 b^2 \left (\frac {2}{5} a^2 \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 344

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \int \frac {x^2}{\sqrt {a^2-b^2 x^4}}dx}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 836

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {a \int \frac {b x^2+a}{a \sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 765

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {1}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 762

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 1390

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {b x^2+a}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 1389

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

\(\Big \downarrow \) 327

\(\displaystyle -6 b^2 \left (\frac {2 a^2 \sqrt {a^2-b^2 x^4} \left (\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {1}{5} x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x}\)

Input:

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

Output:

-(((a - b*x^2)^(3/2)*(a + b*x^2)^(3/2))/x) - 6*b^2*((x^3*Sqrt[a - b*x^2]*S 
qrt[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)/Sqr 
t[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 339

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 + 1))) 
, x] - Simp[4*b*d*(p/(e^4*(m + 1)))   Int[(e*x)^(m + 4)*(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] && LtQ[m, -1]
 

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 = 1.78 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (b^{2} x^{4}+5 a^{2}\right )}{5 x}+\frac {12 a^{3} b \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{5 \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(155\)
elliptic	\(\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-\frac {a^{2} \sqrt {-b^{2} x^{4}+a^{2}}}{x}-\frac {b^{2} x^{3} \sqrt {-b^{2} x^{4}+a^{2}}}{5}+\frac {12 a^{3} b \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{5 \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}}\right )}{\sqrt {-b^{2} x^{4}+a^{2}}}\)	\(159\)
default	\(\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {\frac {b}{a}}\, b^{4} x^{8}+4 \sqrt {\frac {b}{a}}\, a^{2} b^{2} x^{4}+12 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right ) a^{3} b x -12 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right ) a^{3} b x -5 \sqrt {\frac {b}{a}}\, a^{4}\right )}{5 \left (-b^{2} x^{4}+a^{2}\right ) x \sqrt {\frac {b}{a}}}\)	\(183\)

Input:

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

Output:

-1/5*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)*(b^2*x^4+5*a^2)/x+12/5*a^3*b/(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)*(EllipticF(x 
*(b/a)^(1/2),I)-EllipticE(x*(b/a)^(1/2),I))*((-b*x^2+a)*(b*x^2+a))^(1/2)/( 
-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)
 

Fricas [F]

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

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

Output:

integral(-(b^2*x^4 - a^2)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/x^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\frac {7 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{2}-\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, b^{2} x^{4}+12 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} x^{6}+a^{2} x^{2}}d x \right ) a^{4} x}{5 x} \] Input:

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

Output:

(7*sqrt(a + b*x**2)*sqrt(a - b*x**2)*a**2 - sqrt(a + b*x**2)*sqrt(a - b*x* 
*2)*b**2*x**4 + 12*int((sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a**2*x**2 - b* 
*2*x**6),x)*a**4*x)/(5*x)