\(\int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx\) [1291]

Optimal result

Integrand size = 27, antiderivative size = 105 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{8 x^8}+\frac {b^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}{16 a^2 x^4}+\frac {b^4 \text {arctanh}\left (\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a}\right )}{16 a^3} \] Output:

-1/8*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^8+1/16*b^2*(-b*x^2+a)^(1/2)*(b*x^2 
+a)^(1/2)/a^2/x^4+1/16*b^4*arctanh((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a)/a^3
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\frac {a \sqrt {a-b x^2} \sqrt {a+b x^2} \left (-2 a^2+b^2 x^4\right )+2 b^4 x^8 \text {arctanh}\left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right )}{16 a^3 x^8} \] Input:

Integrate[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^9,x]
 

Output:

(a*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*(-2*a^2 + b^2*x^4) + 2*b^4*x^8*ArcTanh[ 
Sqrt[a - b*x^2]/Sqrt[a + b*x^2]])/(16*a^3*x^8)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {338, 108, 25, 27, 114, 25, 27, 103, 221}

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.

\(\displaystyle \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx\)

\(\Big \downarrow \) 338

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {a-b x^2} \sqrt {b x^2+a}}{x^{10}}dx^2\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \int -\frac {b^2}{x^6 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} \int \frac {b^2}{x^6 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b^2 \int \frac {1}{x^6 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b^2 \left (-\frac {\int -\frac {b^2}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b^2 \left (\frac {\int \frac {b^2}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b^2 \left (\frac {b^2 \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 103

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b^2 \left (-\frac {b^3 \int \frac {1}{a^2 b-b x^4}d\left (\sqrt {a-b x^2} \sqrt {b x^2+a}\right )}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b^2 \left (-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a}\right )}{2 a^3}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^8}\right )\)

Input:

Int[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^9,x]
 

Output:

(-1/4*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^8 - (b^2*(-1/2*(Sqrt[a - b*x^2]* 
Sqrt[a + b*x^2])/(a^2*x^4) - (b^2*ArcTanh[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2] 
)/a])/(2*a^3)))/4)/2
 

Defintions of rubi rules used

rule 25

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

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 103

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

rule 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 338

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.67 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-\ln \left (\frac {2 a \left (\operatorname {csgn}\left (a \right ) \sqrt {-b^{2} x^{4}+a^{2}}+a \right )}{x^{2}}\right ) b^{4} x^{8}-b^{2} \sqrt {-b^{2} x^{4}+a^{2}}\, x^{4} \operatorname {csgn}\left (a \right ) a +2 \,\operatorname {csgn}\left (a \right ) a^{3} \sqrt {-b^{2} x^{4}+a^{2}}\right ) \operatorname {csgn}\left (a \right )}{16 a^{3} \sqrt {-b^{2} x^{4}+a^{2}}\, x^{8}}\)	\(125\)
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-b^{2} x^{4}+2 a^{2}\right )}{16 x^{8} a^{2}}+\frac {b^{4} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {-b^{2} x^{4}+a^{2}}}{x^{2}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{16 a^{2} \sqrt {a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(125\)
elliptic	\(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {\left (-b^{2} x^{4}+a^{2}\right )^{\frac {3}{2}}}{8 a^{2} x^{8}}-\frac {b^{2} \left (-b^{2} x^{4}+a^{2}\right )^{\frac {3}{2}}}{16 a^{4} x^{4}}-\frac {b^{4} \sqrt {-b^{2} x^{4}+a^{2}}}{16 a^{4}}+\frac {b^{4} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {-b^{2} x^{4}+a^{2}}}{x^{2}}\right )}{16 a^{2} \sqrt {a^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(150\)

Input:

int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^9,x,method=_RETURNVERBOSE)
 

Output:

-1/16*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a^3*(-ln(2*a*(csgn(a)*(-b^2*x^4+a^2 
)^(1/2)+a)/x^2)*b^4*x^8-b^2*(-b^2*x^4+a^2)^(1/2)*x^4*csgn(a)*a+2*csgn(a)*a 
^3*(-b^2*x^4+a^2)^(1/2))*csgn(a)/(-b^2*x^4+a^2)^(1/2)/x^8
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\frac {b^{4} x^{8} \log \left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} + a\right ) - b^{4} x^{8} \log \left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} - a\right ) + 2 \, {\left (a b^{2} x^{4} - 2 \, a^{3}\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{32 \, a^{3} x^{8}} \] Input:

integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^9,x, algorithm="fricas")
 

Output:

1/32*(b^4*x^8*log(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a) + a) - b^4*x^8*log(sqrt 
(b*x^2 + a)*sqrt(-b*x^2 + a) - a) + 2*(a*b^2*x^4 - 2*a^3)*sqrt(b*x^2 + a)* 
sqrt(-b*x^2 + a))/(a^3*x^8)
 

Sympy [F]

\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\int \frac {\sqrt {a - b x^{2}} \sqrt {a + b x^{2}}}{x^{9}}\, dx \] Input:

integrate((-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2)/x**9,x)
 

Output:

Integral(sqrt(a - b*x**2)*sqrt(a + b*x**2)/x**9, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\frac {b^{4} \log \left (\frac {2 \, a^{2}}{x^{2}} + \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} a}{x^{2}}\right )}{16 \, a^{3}} - \frac {\sqrt {-b^{2} x^{4} + a^{2}} b^{4}}{16 \, a^{4}} - \frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}} b^{2}}{16 \, a^{4} x^{4}} - \frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}}}{8 \, a^{2} x^{8}} \] Input:

integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^9,x, algorithm="maxima")
 

Output:

1/16*b^4*log(2*a^2/x^2 + 2*sqrt(-b^2*x^4 + a^2)*a/x^2)/a^3 - 1/16*sqrt(-b^ 
2*x^4 + a^2)*b^4/a^4 - 1/16*(-b^2*x^4 + a^2)^(3/2)*b^2/(a^4*x^4) - 1/8*(-b 
^2*x^4 + a^2)^(3/2)/(a^2*x^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (87) = 174\).

Time = 0.30 (sec) , antiderivative size = 512, normalized size of antiderivative = 4.88 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\frac {\frac {b^{5} \log \left ({\left | -\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} + \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}} + 2 \right |}\right )}{a^{3}} - \frac {b^{5} \log \left ({\left | -\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} + \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}} - 2 \right |}\right )}{a^{3}} + \frac {4 \, {\left (b^{5} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{7} + 28 \, b^{5} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{5} + 112 \, b^{5} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{3} + 64 \, b^{5} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{2} - 4\right )}^{4} a^{3}}}{16 \, b} \] Input:

integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^9,x, algorithm="giac")
 

Output:

1/16*(b^5*log(abs(-(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) + 
sqrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)) + 2))/a^3 - b^5*log(a 
bs(-(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) + sqrt(b*x^2 + a) 
/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)) - 2))/a^3 + 4*(b^5*((sqrt(2)*sqrt(a) 
 - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) - sqrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - 
sqrt(-b*x^2 + a)))^7 + 28*b^5*((sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b 
*x^2 + a) - sqrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)))^5 + 112* 
b^5*((sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) - sqrt(b*x^2 + a 
)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)))^3 + 64*b^5*((sqrt(2)*sqrt(a) - sqr 
t(-b*x^2 + a))/sqrt(b*x^2 + a) - sqrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(- 
b*x^2 + a))))/((((sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) - sq 
rt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)))^2 - 4)^4*a^3))/b
 

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 587, normalized size of antiderivative = 5.59 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\frac {b^4\,\ln \left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {a-b\,x^2}-\sqrt {a}}\right )}{16\,a^3}-\frac {b^4\,\ln \left (\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-1\right )}{16\,a^3}-\frac {\frac {2\,b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-\frac {b^4}{4}-\frac {11\,b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{2\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4}+\frac {7\,b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^6}+\frac {239\,b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8}{4\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^8}+\frac {b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{10}}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{10}}}{\frac {512\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4}-\frac {2048\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^6}+\frac {3072\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^8}-\frac {2048\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{10}}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{10}}+\frac {512\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{12}}}-\frac {b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{512\,a^3\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}+\frac {b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{2048\,a^3\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4} \] Input:

int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^9,x)
 

Output:

(b^4*log(((a + b*x^2)^(1/2) - a^(1/2))/((a - b*x^2)^(1/2) - a^(1/2))))/(16 
*a^3) - (b^4*log(((a + b*x^2)^(1/2) - a^(1/2))^2/((a - b*x^2)^(1/2) - a^(1 
/2))^2 - 1))/(16*a^3) - ((2*b^4*((a + b*x^2)^(1/2) - a^(1/2))^2)/((a - b*x 
^2)^(1/2) - a^(1/2))^2 - b^4/4 - (11*b^4*((a + b*x^2)^(1/2) - a^(1/2))^4)/ 
(2*((a - b*x^2)^(1/2) - a^(1/2))^4) + (7*b^4*((a + b*x^2)^(1/2) - a^(1/2)) 
^6)/((a - b*x^2)^(1/2) - a^(1/2))^6 + (239*b^4*((a + b*x^2)^(1/2) - a^(1/2 
))^8)/(4*((a - b*x^2)^(1/2) - a^(1/2))^8) + (b^4*((a + b*x^2)^(1/2) - a^(1 
/2))^10)/((a - b*x^2)^(1/2) - a^(1/2))^10)/((512*a^3*((a + b*x^2)^(1/2) - 
a^(1/2))^4)/((a - b*x^2)^(1/2) - a^(1/2))^4 - (2048*a^3*((a + b*x^2)^(1/2) 
 - a^(1/2))^6)/((a - b*x^2)^(1/2) - a^(1/2))^6 + (3072*a^3*((a + b*x^2)^(1 
/2) - a^(1/2))^8)/((a - b*x^2)^(1/2) - a^(1/2))^8 - (2048*a^3*((a + b*x^2) 
^(1/2) - a^(1/2))^10)/((a - b*x^2)^(1/2) - a^(1/2))^10 + (512*a^3*((a + b* 
x^2)^(1/2) - a^(1/2))^12)/((a - b*x^2)^(1/2) - a^(1/2))^12) - (b^4*((a + b 
*x^2)^(1/2) - a^(1/2))^2)/(512*a^3*((a - b*x^2)^(1/2) - a^(1/2))^2) + (b^4 
*((a + b*x^2)^(1/2) - a^(1/2))^4)/(2048*a^3*((a - b*x^2)^(1/2) - a^(1/2))^ 
4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^9} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{3}+\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a \,b^{2} x^{4}+2 \,\mathrm {log}\left (\sqrt {-b \,x^{2}+a}+\sqrt {b \,x^{2}+a}\right ) b^{4} x^{8}-2 \,\mathrm {log}\left (x \right ) b^{4} x^{8}}{16 a^{3} x^{8}} \] Input:

int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^9,x)
 

Output:

( - 2*sqrt(a + b*x**2)*sqrt(a - b*x**2)*a**3 + sqrt(a + b*x**2)*sqrt(a - b 
*x**2)*a*b**2*x**4 + 2*log(sqrt(a - b*x**2) + sqrt(a + b*x**2))*b**4*x**8 
- 2*log(x)*b**4*x**8)/(16*a**3*x**8)