Integrand size = 15, antiderivative size = 57 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=-\frac {\sqrt {-9+4 x^2}}{4 x^4}+\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {2}{27} \arctan \left (\frac {1}{3} \sqrt {-9+4 x^2}\right ) \] Output:
-1/4*(4*x^2-9)^(1/2)/x^4+1/18*(4*x^2-9)^(1/2)/x^2+2/27*arctan(1/3*(4*x^2-9 )^(1/2))
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=\frac {\left (-9+2 x^2\right ) \sqrt {-9+4 x^2}}{36 x^4}+\frac {2}{27} \arctan \left (\frac {1}{3} \sqrt {-9+4 x^2}\right ) \] Input:
Integrate[Sqrt[-9 + 4*x^2]/x^5,x]
Output:
((-9 + 2*x^2)*Sqrt[-9 + 4*x^2])/(36*x^4) + (2*ArcTan[Sqrt[-9 + 4*x^2]/3])/ 27
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.07, 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, 216}
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.
\(\displaystyle \int \frac {\sqrt {4 x^2-9}}{x^5} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {4 x^2-9}}{x^6}dx^2\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{x^4 \sqrt {4 x^2-9}}dx^2-\frac {\sqrt {4 x^2-9}}{2 x^4}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{9} \int \frac {1}{x^2 \sqrt {4 x^2-9}}dx^2+\frac {\sqrt {4 x^2-9}}{9 x^2}-\frac {\sqrt {4 x^2-9}}{2 x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \int \frac {1}{\frac {x^4}{4}+\frac {9}{4}}d\sqrt {4 x^2-9}+\frac {\sqrt {4 x^2-9}}{9 x^2}-\frac {\sqrt {4 x^2-9}}{2 x^4}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (\frac {4}{27} \arctan \left (\frac {1}{3} \sqrt {4 x^2-9}\right )+\frac {\sqrt {4 x^2-9}}{9 x^2}-\frac {\sqrt {4 x^2-9}}{2 x^4}\right )\) |
Input:
Int[Sqrt[-9 + 4*x^2]/x^5,x]
Output:
(-1/2*Sqrt[-9 + 4*x^2]/x^4 + Sqrt[-9 + 4*x^2]/(9*x^2) + (4*ArcTan[Sqrt[-9 + 4*x^2]/3])/27)/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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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 = 0.48 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {8 x^{4}-54 x^{2}+81}{36 x^{4} \sqrt {4 x^{2}-9}}-\frac {2 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{27}\) | \(42\) |
pseudoelliptic | \(\frac {8 \arctan \left (\frac {\sqrt {4 x^{2}-9}}{3}\right ) x^{4}+6 x^{2} \sqrt {4 x^{2}-9}-27 \sqrt {4 x^{2}-9}}{108 x^{4}}\) | \(49\) |
default | \(\frac {\left (4 x^{2}-9\right )^{\frac {3}{2}}}{36 x^{4}}+\frac {\left (4 x^{2}-9\right )^{\frac {3}{2}}}{162 x^{2}}-\frac {2 \sqrt {4 x^{2}-9}}{81}-\frac {2 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{27}\) | \(55\) |
trager | \(\frac {\left (2 x^{2}-9\right ) \sqrt {4 x^{2}-9}}{36 x^{4}}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\sqrt {4 x^{2}-9}}{x}\right )}{27}\) | \(57\) |
meijerg | \(-\frac {4 \sqrt {\operatorname {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (\frac {81 \sqrt {\pi }}{16 x^{4}}-\frac {9 \sqrt {\pi }}{4 x^{2}}+\frac {\left (\frac {1}{2}+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }}{4}-\frac {81 \sqrt {\pi }\, \left (\frac {16}{81} x^{4}-\frac {32}{9} x^{2}+8\right )}{128 x^{4}}+\frac {81 \sqrt {\pi }\, \left (-\frac {16 x^{2}}{9}+8\right ) \sqrt {-\frac {4 x^{2}}{9}+1}}{128 x^{4}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {4 x^{2}}{9}+1}}{2}\right )}{2}\right )}{27 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\) | \(127\) |
Input:
int((4*x^2-9)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
1/36*(8*x^4-54*x^2+81)/x^4/(4*x^2-9)^(1/2)-2/27*arctan(3/(4*x^2-9)^(1/2))
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=\frac {16 \, x^{4} \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) + 3 \, \sqrt {4 \, x^{2} - 9} {\left (2 \, x^{2} - 9\right )}}{108 \, x^{4}} \] Input:
integrate((4*x^2-9)^(1/2)/x^5,x, algorithm="fricas")
Output:
1/108*(16*x^4*arctan(-2/3*x + 1/3*sqrt(4*x^2 - 9)) + 3*sqrt(4*x^2 - 9)*(2* x^2 - 9))/x^4
Result contains complex when optimal does not.
Time = 2.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=\begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {i}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} + \frac {3 i}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {9 i}{8 x^{5} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {4}{9} \\- \frac {2 \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} + \frac {1}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} - \frac {3}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {9}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \] Input:
integrate((4*x**2-9)**(1/2)/x**5,x)
Output:
Piecewise((2*I*acosh(3/(2*x))/27 - I/(9*x*sqrt(-1 + 9/(4*x**2))) + 3*I/(4* x**3*sqrt(-1 + 9/(4*x**2))) - 9*I/(8*x**5*sqrt(-1 + 9/(4*x**2))), 1/Abs(x* *2) > 4/9), (-2*asin(3/(2*x))/27 + 1/(9*x*sqrt(1 - 9/(4*x**2))) - 3/(4*x** 3*sqrt(1 - 9/(4*x**2))) + 9/(8*x**5*sqrt(1 - 9/(4*x**2))), True))
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=-\frac {2}{81} \, \sqrt {4 \, x^{2} - 9} + \frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{162 \, x^{2}} + \frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{36 \, x^{4}} - \frac {2}{27} \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \] Input:
integrate((4*x^2-9)^(1/2)/x^5,x, algorithm="maxima")
Output:
-2/81*sqrt(4*x^2 - 9) + 1/162*(4*x^2 - 9)^(3/2)/x^2 + 1/36*(4*x^2 - 9)^(3/ 2)/x^4 - 2/27*arcsin(3/2/abs(x))
Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=\frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} - 9 \, \sqrt {4 \, x^{2} - 9}}{72 \, x^{4}} + \frac {2}{27} \, \arctan \left (\frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \] Input:
integrate((4*x^2-9)^(1/2)/x^5,x, algorithm="giac")
Output:
1/72*((4*x^2 - 9)^(3/2) - 9*sqrt(4*x^2 - 9))/x^4 + 2/27*arctan(1/3*sqrt(4* x^2 - 9))
Time = 0.39 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=\frac {2\,\mathrm {atan}\left (\frac {\sqrt {4\,x^2-9}}{3}\right )}{27}-\frac {\frac {\sqrt {4\,x^2-9}}{8}-\frac {{\left (4\,x^2-9\right )}^{3/2}}{72}}{x^4} \] Input:
int((4*x^2 - 9)^(1/2)/x^5,x)
Output:
(2*atan((4*x^2 - 9)^(1/2)/3))/27 - ((4*x^2 - 9)^(1/2)/8 - (4*x^2 - 9)^(3/2 )/72)/x^4
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx=\frac {16 \mathit {atan} \left (\frac {\sqrt {4 x^{2}-9}}{3}+\frac {2 x}{3}\right ) x^{4}+6 \sqrt {4 x^{2}-9}\, x^{2}-27 \sqrt {4 x^{2}-9}}{108 x^{4}} \] Input:
int((4*x^2-9)^(1/2)/x^5,x)
Output:
(16*atan((sqrt(4*x**2 - 9) + 2*x)/3)*x**4 + 6*sqrt(4*x**2 - 9)*x**2 - 27*s qrt(4*x**2 - 9))/(108*x**4)