Integrand size = 15, antiderivative size = 57 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=-\frac {\sqrt {9-4 x^2}}{4 x^4}+\frac {\sqrt {9-4 x^2}}{18 x^2}+\frac {2}{27} \text {arctanh}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \] Output:
-1/4*(-4*x^2+9)^(1/2)/x^4+1/18*(-4*x^2+9)^(1/2)/x^2+2/27*arctanh(1/3*(-4*x ^2+9)^(1/2))
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=\frac {\sqrt {9-4 x^2} \left (-9+2 x^2\right )}{36 x^4}+\frac {2}{27} \text {arctanh}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \] Input:
Integrate[Sqrt[9 - 4*x^2]/x^5,x]
Output:
(Sqrt[9 - 4*x^2]*(-9 + 2*x^2))/(36*x^4) + (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/2 7
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, 219}
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.
\(\displaystyle \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {9-4 x^2}}{x^6}dx^2\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {1}{x^4 \sqrt {9-4 x^2}}dx^2-\frac {\sqrt {9-4 x^2}}{2 x^4}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{9} \int \frac {1}{x^2 \sqrt {9-4 x^2}}dx^2+\frac {\sqrt {9-4 x^2}}{9 x^2}-\frac {\sqrt {9-4 x^2}}{2 x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \int \frac {1}{\frac {9}{4}-\frac {x^4}{4}}d\sqrt {9-4 x^2}+\frac {\sqrt {9-4 x^2}}{9 x^2}-\frac {\sqrt {9-4 x^2}}{2 x^4}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {4}{27} \text {arctanh}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )+\frac {\sqrt {9-4 x^2}}{9 x^2}-\frac {\sqrt {9-4 x^2}}{2 x^4}\right )\) |
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*ArcTanh[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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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 = 0.43 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.72
method | result | size |
trager | \(\frac {\left (2 x^{2}-9\right ) \sqrt {-4 x^{2}+9}}{36 x^{4}}+\frac {2 \ln \left (\frac {\sqrt {-4 x^{2}+9}+3}{x}\right )}{27}\) | \(41\) |
risch | \(-\frac {8 x^{4}-54 x^{2}+81}{36 x^{4} \sqrt {-4 x^{2}+9}}+\frac {2 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{27}\) | \(42\) |
default | \(-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{36 x^{4}}-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{162 x^{2}}-\frac {2 \sqrt {-4 x^{2}+9}}{81}+\frac {2 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{27}\) | \(55\) |
pseudoelliptic | \(\frac {\frac {16 \ln \left (\sqrt {-4 x^{2}+9}+3\right ) x^{4}}{27}-\frac {16 \ln \left (-3+\sqrt {-4 x^{2}+9}\right ) x^{4}}{27}+\frac {8 x^{2} \sqrt {-4 x^{2}+9}}{9}-4 \sqrt {-4 x^{2}+9}}{\left (\sqrt {-4 x^{2}+9}+3\right )^{2} \left (-3+\sqrt {-4 x^{2}+9}\right )^{2}}\) | \(89\) |
meijerg | \(-\frac {4 \left (\frac {81 \sqrt {\pi }}{16 x^{4}}-\frac {9 \sqrt {\pi }}{4 x^{2}}+\frac {\left (\frac {1}{2}+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }}{4}-\frac {81 \sqrt {\pi }\, \left (\frac {16}{81} x^{4}-\frac {32}{9} x^{2}+8\right )}{128 x^{4}}+\frac {81 \sqrt {\pi }\, \left (-\frac {16 x^{2}}{9}+8\right ) \sqrt {-\frac {4 x^{2}}{9}+1}}{128 x^{4}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {4 x^{2}}{9}+1}}{2}\right )}{2}\right )}{27 \sqrt {\pi }}\) | \(105\) |
Input:
int((-4*x^2+9)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
1/36*(2*x^2-9)/x^4*(-4*x^2+9)^(1/2)+2/27*ln(((-4*x^2+9)^(1/2)+3)/x)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=-\frac {8 \, x^{4} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, {\left (2 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9}}{108 \, x^{4}} \] Input:
integrate((-4*x^2+9)^(1/2)/x^5,x, algorithm="fricas")
Output:
-1/108*(8*x^4*log((sqrt(-4*x^2 + 9) - 3)/x) - 3*(2*x^2 - 9)*sqrt(-4*x^2 + 9))/x^4
Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=\begin {cases} \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {1}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} + \frac {3}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {9}{8 x^{5} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {4}{9} \\- \frac {2 i \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} + \frac {i}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} - \frac {3 i}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {9 i}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \] Input:
integrate((-4*x**2+9)**(1/2)/x**5,x)
Output:
Piecewise((2*acosh(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))), 1/Abs(x**2) > 4/9), (-2*I*asin(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))), True))
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=-\frac {2}{81} \, \sqrt {-4 \, x^{2} + 9} - \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{162 \, x^{2}} - \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{36 \, x^{4}} + \frac {2}{27} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \] 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*log(6*sqrt(-4*x^2 + 9)/abs(x) + 18/abs(x))
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=-\frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} + 9 \, \sqrt {-4 \, x^{2} + 9}}{72 \, x^{4}} + \frac {1}{27} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) - \frac {1}{27} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \] 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 + 1/27*log(sqrt(-4*x^2 + 9) + 3) - 1/27*log(-sqrt(-4*x^2 + 9) + 3)
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=\frac {\sqrt {\frac {9}{4}-x^2}}{9\,x^2}-\frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\frac {3\,\sqrt {\frac {1}{x^2}}}{2}\right )}{27}-\frac {\sqrt {\frac {9}{4}-x^2}}{2\,x^4} \] Input:
int((9 - 4*x^2)^(1/2)/x^5,x)
Output:
(9/4 - x^2)^(1/2)/(9*x^2) - (2*log((9/(4*x^2) - 1)^(1/2) - (3*(1/x^2)^(1/2 ))/2))/27 - (9/4 - x^2)^(1/2)/(2*x^4)
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx=\frac {6 \sqrt {-4 x^{2}+9}\, x^{2}-27 \sqrt {-4 x^{2}+9}-8 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {2 x}{3}\right )}{2}\right )\right ) x^{4}}{108 x^{4}} \] Input:
int((-4*x^2+9)^(1/2)/x^5,x)
Output:
(6*sqrt( - 4*x**2 + 9)*x**2 - 27*sqrt( - 4*x**2 + 9) - 8*log(tan(asin((2*x )/3)/2))*x**4)/(108*x**4)