Integrand size = 15, antiderivative size = 71 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=-\frac {\sqrt {a+b x^2}}{4 x^4}-\frac {b \sqrt {a+b x^2}}{8 a x^2}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Output:
-1/4*(b*x^2+a)^(1/2)/x^4-1/8*b*(b*x^2+a)^(1/2)/a/x^2+1/8*b^2*arctanh((b*x^ 2+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=\frac {\left (-2 a-b x^2\right ) \sqrt {a+b x^2}}{8 a x^4}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Input:
Integrate[Sqrt[a + b*x^2]/x^5,x]
Output:
((-2*a - b*x^2)*Sqrt[a + b*x^2])/(8*a*x^4) + (b^2*ArcTanh[Sqrt[a + b*x^2]/ Sqrt[a]])/(8*a^(3/2))
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {243, 51, 52, 73, 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.
\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^5} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {b x^2+a}}{x^6}dx^2\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} b \int \frac {1}{x^4 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{2 x^4}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{2 a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a+b x^2}}{2 x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a+b x^2}}{2 x^4}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a+b x^2}}{2 x^4}\right )\) |
Input:
Int[Sqrt[a + b*x^2]/x^5,x]
Output:
(-1/2*Sqrt[a + b*x^2]/x^4 + (b*(-(Sqrt[a + b*x^2]/(a*x^2)) + (b*ArcTanh[Sq rt[a + b*x^2]/Sqrt[a]])/a^(3/2)))/4)/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) b^{2} x^{4}-\sqrt {b \,x^{2}+a}\, \left (b \,x^{2} \sqrt {a}+2 a^{\frac {3}{2}}\right )}{8 a^{\frac {3}{2}} x^{4}}\) | \(56\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (b \,x^{2}+2 a \right )}{8 x^{4} a}+\frac {b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{8 a^{\frac {3}{2}}}\) | \(59\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\) | \(87\) |
Input:
int((b*x^2+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
1/8/a^(3/2)*(arctanh((b*x^2+a)^(1/2)/a^(1/2))*b^2*x^4-(b*x^2+a)^(1/2)*(b*x ^2*a^(1/2)+2*a^(3/2)))/x^4
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=\left [\frac {\sqrt {a} b^{2} x^{4} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{16 \, a^{2} x^{4}}, -\frac {\sqrt {-a} b^{2} x^{4} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{8 \, a^{2} x^{4}}\right ] \] Input:
integrate((b*x^2+a)^(1/2)/x^5,x, algorithm="fricas")
Output:
[1/16*(sqrt(a)*b^2*x^4*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*b*x^2 + 2*a^2)*sqrt(b*x^2 + a))/(a^2*x^4), -1/8*(sqrt(-a)*b^2*x^4* arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (a*b*x^2 + 2*a^2)*sqrt(b*x^2 + a))/(a ^2*x^4)]
Time = 1.91 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=- \frac {a}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {b^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} \] Input:
integrate((b*x**2+a)**(1/2)/x**5,x)
Output:
-a/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(8*x**3*sqrt(a/(b*x** 2) + 1)) - b**(3/2)/(8*a*x*sqrt(a/(b*x**2) + 1)) + b**2*asinh(sqrt(a)/(sqr t(b)*x))/(8*a**(3/2))
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=\frac {b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} b^{2}}{8 \, a^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{4 \, a x^{4}} \] Input:
integrate((b*x^2+a)^(1/2)/x^5,x, algorithm="maxima")
Output:
1/8*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 1/8*sqrt(b*x^2 + a)*b^2/a^ 2 + 1/8*(b*x^2 + a)^(3/2)*b/(a^2*x^2) - 1/4*(b*x^2 + a)^(3/2)/(a*x^4)
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=-\frac {\frac {b^{3} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} + \sqrt {b x^{2} + a} a b^{3}}{a b^{2} x^{4}}}{8 \, b} \] Input:
integrate((b*x^2+a)^(1/2)/x^5,x, algorithm="giac")
Output:
-1/8*(b^3*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + ((b*x^2 + a)^(3/ 2)*b^3 + sqrt(b*x^2 + a)*a*b^3)/(a*b^2*x^4))/b
Time = 0.54 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{3/2}}-\frac {\sqrt {b\,x^2+a}}{8\,x^4}-\frac {{\left (b\,x^2+a\right )}^{3/2}}{8\,a\,x^4} \] Input:
int((a + b*x^2)^(1/2)/x^5,x)
Output:
(b^2*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(3/2)) - (a + b*x^2)^(1/2)/(8* x^4) - (a + b*x^2)^(3/2)/(8*a*x^4)
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {a+b x^2}}{x^5} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a^{2}-\sqrt {b \,x^{2}+a}\, a b \,x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}}{8 a^{2} x^{4}} \] Input:
int((b*x^2+a)^(1/2)/x^5,x)
Output:
( - 2*sqrt(a + b*x**2)*a**2 - sqrt(a + b*x**2)*a*b*x**2 - sqrt(a)*log((sqr t(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4 + sqrt(a)*log((sqr t(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4)/(8*a**2*x**4)