Integrand size = 27, antiderivative size = 69 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^4}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a}\right )}{4 a} \] Output:
-1/4*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^4+1/4*b^2*arctanh((-b*x^2+a)^(1/2) *(b*x^2+a)^(1/2)/a)/a
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 x^4}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right )}{2 a} \] Input:
Integrate[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^5,x]
Output:
-1/4*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^4 + (b^2*ArcTanh[Sqrt[a - b*x^2]/ Sqrt[a + b*x^2]])/(2*a)
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.58, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {338, 105, 105, 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.
\(\displaystyle \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx\) |
\(\Big \downarrow \) 338 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {a-b x^2} \sqrt {b x^2+a}}{x^6}dx^2\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \int \frac {\sqrt {b x^2+a}}{x^4 \sqrt {a-b x^2}}dx^2-\frac {\sqrt {a-b x^2} \left (a+b x^2\right )^{3/2}}{2 a x^4}\right )\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \left (b \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a-b x^2} \left (a+b x^2\right )^{3/2}}{2 a x^4}\right )\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \left (b^2 \left (-\int \frac {1}{a^2 b-b x^4}d\left (\sqrt {a-b x^2} \sqrt {b x^2+a}\right )\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a-b x^2} \left (a+b x^2\right )^{3/2}}{2 a x^4}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a}\right )}{a}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a-b x^2} \left (a+b x^2\right )^{3/2}}{2 a x^4}\right )\) |
Input:
Int[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^5,x]
Output:
(-1/2*(Sqrt[a - b*x^2]*(a + b*x^2)^(3/2))/(a*x^4) - (b*(-((Sqrt[a - b*x^2] *Sqrt[a + b*x^2])/(a*x^2)) - (b*ArcTanh[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/ a])/a))/2)/2
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]
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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.58 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41
method | result | size |
default | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-\ln \left (\frac {2 a \left (\operatorname {csgn}\left (a \right ) \sqrt {-b^{2} x^{4}+a^{2}}+a \right )}{x^{2}}\right ) b^{2} x^{4}+\operatorname {csgn}\left (a \right ) a \sqrt {-b^{2} x^{4}+a^{2}}\right ) \operatorname {csgn}\left (a \right )}{4 a \sqrt {-b^{2} x^{4}+a^{2}}\, x^{4}}\) | \(97\) |
risch | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{4 x^{4}}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {-b^{2} x^{4}+a^{2}}}{x^{2}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{4 \sqrt {a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(105\) |
elliptic | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {\left (-b^{2} x^{4}+a^{2}\right )^{\frac {3}{2}}}{4 a^{2} x^{4}}-\frac {b^{2} \sqrt {-b^{2} x^{4}+a^{2}}}{4 a^{2}}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {-b^{2} x^{4}+a^{2}}}{x^{2}}\right )}{4 \sqrt {a^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(122\) |
Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/4*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a*(-ln(2*a*(csgn(a)*(-b^2*x^4+a^2)^( 1/2)+a)/x^2)*b^2*x^4+csgn(a)*a*(-b^2*x^4+a^2)^(1/2))*csgn(a)/(-b^2*x^4+a^2 )^(1/2)/x^4
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=\frac {b^{2} x^{4} \log \left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} + a\right ) - b^{2} x^{4} \log \left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} - a\right ) - 2 \, \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} a}{8 \, a x^{4}} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^5,x, algorithm="fricas")
Output:
1/8*(b^2*x^4*log(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a) + a) - b^2*x^4*log(sqrt( b*x^2 + a)*sqrt(-b*x^2 + a) - a) - 2*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)*a)/( a*x^4)
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=\int \frac {\sqrt {a - b x^{2}} \sqrt {a + b x^{2}}}{x^{5}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2)/x**5,x)
Output:
Integral(sqrt(a - b*x**2)*sqrt(a + b*x**2)/x**5, x)
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=\frac {b^{2} \log \left (\frac {2 \, a^{2}}{x^{2}} + \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} a}{x^{2}}\right )}{4 \, a} - \frac {\sqrt {-b^{2} x^{4} + a^{2}} b^{2}}{4 \, a^{2}} - \frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}}}{4 \, a^{2} x^{4}} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^5,x, algorithm="maxima")
Output:
1/4*b^2*log(2*a^2/x^2 + 2*sqrt(-b^2*x^4 + a^2)*a/x^2)/a - 1/4*sqrt(-b^2*x^ 4 + a^2)*b^2/a^2 - 1/4*(-b^2*x^4 + a^2)^(3/2)/(a^2*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (57) = 114\).
Time = 0.29 (sec) , antiderivative size = 370, normalized size of antiderivative = 5.36 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=\frac {\frac {b^{3} \log \left ({\left | -\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} + \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}} + 2 \right |}\right )}{a} - \frac {b^{3} \log \left ({\left | -\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} + \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}} - 2 \right |}\right )}{a} + \frac {4 \, {\left (b^{3} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{3} + 4 \, b^{3} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{2} - 4\right )}^{2} a}}{4 \, b} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^5,x, algorithm="giac")
Output:
1/4*(b^3*log(abs(-(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) + s qrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)) + 2))/a - b^3*log(abs( -(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) + sqrt(b*x^2 + a)/(s qrt(2)*sqrt(a) - sqrt(-b*x^2 + a)) - 2))/a + 4*(b^3*((sqrt(2)*sqrt(a) - sq rt(-b*x^2 + a))/sqrt(b*x^2 + a) - sqrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt( -b*x^2 + a)))^3 + 4*b^3*((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))))/((((sqrt(2)*s qrt(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 - 4)^2*a))/b
Time = 4.84 (sec) , antiderivative size = 338, normalized size of antiderivative = 4.90 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=\frac {b^2\,\ln \left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {a-b\,x^2}-\sqrt {a}}\right )}{4\,a}-\frac {b^2\,\ln \left (\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-1\right )}{4\,a}-\frac {\frac {b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-\frac {b^2}{2}+\frac {15\,b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{2\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4}}{\frac {32\,a\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-\frac {64\,a\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4}+\frac {32\,a\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^6}}+\frac {b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{64\,a\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2} \] Input:
int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^5,x)
Output:
(b^2*log(((a + b*x^2)^(1/2) - a^(1/2))/((a - b*x^2)^(1/2) - a^(1/2))))/(4* a) - (b^2*log(((a + b*x^2)^(1/2) - a^(1/2))^2/((a - b*x^2)^(1/2) - a^(1/2) )^2 - 1))/(4*a) - ((b^2*((a + b*x^2)^(1/2) - a^(1/2))^2)/((a - b*x^2)^(1/2 ) - a^(1/2))^2 - b^2/2 + (15*b^2*((a + b*x^2)^(1/2) - a^(1/2))^4)/(2*((a - b*x^2)^(1/2) - a^(1/2))^4))/((32*a*((a + b*x^2)^(1/2) - a^(1/2))^2)/((a - b*x^2)^(1/2) - a^(1/2))^2 - (64*a*((a + b*x^2)^(1/2) - a^(1/2))^4)/((a - b*x^2)^(1/2) - a^(1/2))^4 + (32*a*((a + b*x^2)^(1/2) - a^(1/2))^6)/((a - b *x^2)^(1/2) - a^(1/2))^6) + (b^2*((a + b*x^2)^(1/2) - a^(1/2))^2)/(64*a*(( a - b*x^2)^(1/2) - a^(1/2))^2)
Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^5} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a +2 \,\mathrm {log}\left (\sqrt {-b \,x^{2}+a}+\sqrt {b \,x^{2}+a}\right ) b^{2} x^{4}-2 \,\mathrm {log}\left (x \right ) b^{2} x^{4}}{4 a \,x^{4}} \] Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^5,x)
Output:
( - sqrt(a + b*x**2)*sqrt(a - b*x**2)*a + 2*log(sqrt(a - b*x**2) + sqrt(a + b*x**2))*b**2*x**4 - 2*log(x)*b**2*x**4)/(4*a*x**4)