3.6.53 \(\int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx\) [553]

Optimal. Leaf size=57 \[ -\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \]

[Out]

-2/81*arctanh(1/3*(-4*x^2+9)^(1/2))-1/36*(-4*x^2+9)^(1/2)/x^4-1/54*(-4*x^2+9)^(1/2)/x^2

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65, 212} \begin {gather*} -\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )-\frac {\sqrt {9-4 x^2}}{36 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[9 - 4*x^2]),x]

[Out]

-1/36*Sqrt[9 - 4*x^2]/x^4 - Sqrt[9 - 4*x^2]/(54*x^2) - (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/81

Rule 44

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x^3} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}+\frac {1}{27} \text {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {1}{54} \text {Subst}\left (\int \frac {1}{\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {9-4 x^2}\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.81 \begin {gather*} \frac {\sqrt {9-4 x^2} \left (-3-2 x^2\right )}{108 x^4}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[9 - 4*x^2]),x]

[Out]

(Sqrt[9 - 4*x^2]*(-3 - 2*x^2))/(108*x^4) - (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/81

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Maple [A]
time = 0.09, size = 44, normalized size = 0.77

method	result	size
trager	\(-\frac {\left (2 x^{2}+3\right ) \sqrt {-4 x^{2}+9}}{108 x^{4}}+\frac {2 \ln \left (\frac {\sqrt {-4 x^{2}+9}-3}{x}\right )}{81}\)	\(41\)
risch	\(\frac {8 x^{4}-6 x^{2}-27}{108 x^{4} \sqrt {-4 x^{2}+9}}-\frac {2 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{81}\)	\(42\)
default	\(-\frac {\sqrt {-4 x^{2}+9}}{36 x^{4}}-\frac {\sqrt {-4 x^{2}+9}}{54 x^{2}}-\frac {2 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{81}\)	\(44\)
meijerg	\(\frac {\frac {\sqrt {\pi }\, \left (-\frac {112}{81} x^{4}+\frac {32}{9} x^{2}+8\right )}{96 x^{4}}-\frac {\sqrt {\pi }\, \left (8+\frac {16 x^{2}}{3}\right ) \sqrt {1-\frac {4 x^{2}}{9}}}{96 x^{4}}-\frac {2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {4 x^{2}}{9}}}{2}\right )}{81}+\frac {\left (\frac {7}{6}+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }}{81}-\frac {\sqrt {\pi }}{12 x^{4}}-\frac {\sqrt {\pi }}{27 x^{2}}}{\sqrt {\pi }}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(-4*x^2+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/36*(-4*x^2+9)^(1/2)/x^4-1/54*(-4*x^2+9)^(1/2)/x^2-2/81*arctanh(3/(-4*x^2+9)^(1/2))

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Maxima [A]
time = 0.49, size = 54, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-4 \, x^{2} + 9}}{54 \, x^{2}} - \frac {\sqrt {-4 \, x^{2} + 9}}{36 \, x^{4}} - \frac {2}{81} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/54*sqrt(-4*x^2 + 9)/x^2 - 1/36*sqrt(-4*x^2 + 9)/x^4 - 2/81*log(6*sqrt(-4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [A]
time = 1.43, size = 45, normalized size = 0.79 \begin {gather*} \frac {8 \, x^{4} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, {\left (2 \, x^{2} + 3\right )} \sqrt {-4 \, x^{2} + 9}}{324 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/324*(8*x^4*log((sqrt(-4*x^2 + 9) - 3)/x) - 3*(2*x^2 + 3)*sqrt(-4*x^2 + 9))/x^4

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Sympy [C] Result contains complex when optimal does not.
time = 2.54, size = 136, normalized size = 2.39 \begin {gather*} \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{81} + \frac {1}{27 x \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{36 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{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 )}}{81} - \frac {i}{27 x \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{36 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(-4*x**2+9)**(1/2),x)

[Out]

Piecewise((-2*acosh(3/(2*x))/81 + 1/(27*x*sqrt(-1 + 9/(4*x**2))) - 1/(36*x**3*sqrt(-1 + 9/(4*x**2))) - 1/(8*x*
*5*sqrt(-1 + 9/(4*x**2))), 1/Abs(x**2) > 4/9), (2*I*asin(3/(2*x))/81 - I/(27*x*sqrt(1 - 9/(4*x**2))) + I/(36*x
**3*sqrt(1 - 9/(4*x**2))) + I/(8*x**5*sqrt(1 - 9/(4*x**2))), True))

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Giac [A]
time = 0.97, size = 57, normalized size = 1.00 \begin {gather*} \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} - 15 \, \sqrt {-4 \, x^{2} + 9}}{216 \, x^{4}} - \frac {1}{81} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) + \frac {1}{81} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/216*((-4*x^2 + 9)^(3/2) - 15*sqrt(-4*x^2 + 9))/x^4 - 1/81*log(sqrt(-4*x^2 + 9) + 3) + 1/81*log(-sqrt(-4*x^2
+ 9) + 3)

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Mupad [B]
time = 4.51, size = 49, normalized size = 0.86 \begin {gather*} \frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\sqrt {\frac {9}{4\,x^2}}\right )}{81}-\frac {\sqrt {\frac {9}{4}-x^2}\,\left (\frac {2}{27\,x^2}+\frac {1}{9\,x^4}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(9 - 4*x^2)^(1/2)),x)

[Out]

(2*log((9/(4*x^2) - 1)^(1/2) - (9/(4*x^2))^(1/2)))/81 - ((9/4 - x^2)^(1/2)*(2/(27*x^2) + 1/(9*x^4)))/2

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