3.6.40 \(\int \frac {1}{x^3 \sqrt {9+4 x^2}} \, dx\) [540]

Optimal. Leaf size=39 \[ -\frac {\sqrt {9+4 x^2}}{18 x^2}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right ) \]

[Out]

2/27*arctanh(1/3*(4*x^2+9)^(1/2))-1/18*(4*x^2+9)^(1/2)/x^2

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65, 213} \begin {gather*} \frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right )-\frac {\sqrt {4 x^2+9}}{18 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[9 + 4*x^2]),x]

[Out]

-1/18*Sqrt[9 + 4*x^2]/x^2 + (2*ArcTanh[Sqrt[9 + 4*x^2]/3])/27

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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^3 \sqrt {9+4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9+4 x^2}}{18 x^2}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{x \sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9+4 x^2}}{18 x^2}-\frac {1}{18} \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}}{18 x^2}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 39, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {9+4 x^2}}{18 x^2}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[9 + 4*x^2]),x]

[Out]

-1/18*Sqrt[9 + 4*x^2]/x^2 + (2*ArcTanh[Sqrt[9 + 4*x^2]/3])/27

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Maple [A]
time = 0.08, size = 30, normalized size = 0.77

method	result	size
default	\(-\frac {\sqrt {4 x^{2}+9}}{18 x^{2}}+\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{27}\)	\(30\)
risch	\(-\frac {\sqrt {4 x^{2}+9}}{18 x^{2}}+\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{27}\)	\(30\)
trager	\(-\frac {\sqrt {4 x^{2}+9}}{18 x^{2}}+\frac {2 \ln \left (\frac {\sqrt {4 x^{2}+9}+3}{x}\right )}{27}\)	\(34\)
meijerg	\(\frac {\frac {\sqrt {\pi }\, \left (8+\frac {16 x^{2}}{9}\right )}{48 x^{2}}-\frac {\sqrt {\pi }\, \sqrt {1+\frac {4 x^{2}}{9}}}{6 x^{2}}+\frac {2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {4 x^{2}}{9}}}{2}\right )}{27}-\frac {\left (1+2 \ln \left (x \right )-2 \ln \left (3\right )\right ) \sqrt {\pi }}{27}-\frac {\sqrt {\pi }}{6 x^{2}}}{\sqrt {\pi }}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(4*x^2+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/18*(4*x^2+9)^(1/2)/x^2+2/27*arctanh(3/(4*x^2+9)^(1/2))

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Maxima [A]
time = 0.50, size = 24, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {4 \, x^{2} + 9}}{18 \, x^{2}} + \frac {2}{27} \, \operatorname {arsinh}\left (\frac {3}{2 \, {\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/18*sqrt(4*x^2 + 9)/x^2 + 2/27*arcsinh(3/2/abs(x))

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Fricas [A]
time = 1.04, size = 57, normalized size = 1.46 \begin {gather*} \frac {4 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} + 3\right ) - 4 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} - 3\right ) - 3 \, \sqrt {4 \, x^{2} + 9}}{54 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/54*(4*x^2*log(-2*x + sqrt(4*x^2 + 9) + 3) - 4*x^2*log(-2*x + sqrt(4*x^2 + 9) - 3) - 3*sqrt(4*x^2 + 9))/x^2

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Sympy [A]
time = 1.05, size = 44, normalized size = 1.13 \begin {gather*} \frac {2 \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {1}{9 x \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {1}{4 x^{3} \sqrt {1 + \frac {9}{4 x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(4*x**2+9)**(1/2),x)

[Out]

2*asinh(3/(2*x))/27 - 1/(9*x*sqrt(1 + 9/(4*x**2))) - 1/(4*x**3*sqrt(1 + 9/(4*x**2)))

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Giac [A]
time = 1.93, size = 43, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {4 \, x^{2} + 9}}{18 \, x^{2}} + \frac {1}{27} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) - \frac {1}{27} \, \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^3/(4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

-1/18*sqrt(4*x^2 + 9)/x^2 + 1/27*log(sqrt(4*x^2 + 9) + 3) - 1/27*log(sqrt(4*x^2 + 9) - 3)

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.64 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{27}-\frac {\sqrt {x^2+\frac {9}{4}}}{9\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(4*x^2 + 9)^(1/2)),x)

[Out]

(2*atanh((2*(x^2 + 9/4)^(1/2))/3))/27 - (x^2 + 9/4)^(1/2)/(9*x^2)

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