3.6.57 \(\int \frac {x^2}{\sqrt {-9+4 x^2}} \, dx\) [557]

Optimal. Leaf size=36 \[ \frac {1}{8} x \sqrt {-9+4 x^2}+\frac {9}{16} \tanh ^{-1}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right ) \]

[Out]

9/16*arctanh(2*x/(4*x^2-9)^(1/2))+1/8*x*(4*x^2-9)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {327, 223, 212} \begin {gather*} \frac {1}{8} \sqrt {4 x^2-9} x+\frac {9}{16} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-9 + 4*x^2],x]

[Out]

(x*Sqrt[-9 + 4*x^2])/8 + (9*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/16

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {-9+4 x^2}} \, dx &=\frac {1}{8} x \sqrt {-9+4 x^2}+\frac {9}{8} \int \frac {1}{\sqrt {-9+4 x^2}} \, dx\\ &=\frac {1}{8} x \sqrt {-9+4 x^2}+\frac {9}{8} \text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x}{\sqrt {-9+4 x^2}}\right )\\ &=\frac {1}{8} x \sqrt {-9+4 x^2}+\frac {9}{16} \tanh ^{-1}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 37, normalized size = 1.03 \begin {gather*} \frac {1}{8} x \sqrt {-9+4 x^2}-\frac {9}{16} \log \left (-2 x+\sqrt {-9+4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-9 + 4*x^2],x]

[Out]

(x*Sqrt[-9 + 4*x^2])/8 - (9*Log[-2*x + Sqrt[-9 + 4*x^2]])/16

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Maple [A]
time = 0.09, size = 35, normalized size = 0.97

method	result	size
trager	\(\frac {x \sqrt {4 x^{2}-9}}{8}-\frac {9 \ln \left (-\sqrt {4 x^{2}-9}+2 x \right )}{16}\)	\(32\)
default	\(\frac {x \sqrt {4 x^{2}-9}}{8}+\frac {9 \ln \left (x \sqrt {4}+\sqrt {4 x^{2}-9}\right ) \sqrt {4}}{32}\)	\(35\)
risch	\(\frac {x \sqrt {4 x^{2}-9}}{8}+\frac {9 \ln \left (x \sqrt {4}+\sqrt {4 x^{2}-9}\right ) \sqrt {4}}{32}\)	\(35\)
meijerg	\(\frac {9 i \sqrt {-\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (\frac {2 i \sqrt {\pi }\, x \sqrt {1-\frac {4 x^{2}}{9}}}{3}-i \sqrt {\pi }\, \arcsin \left (\frac {2 x}{3}\right )\right )}{16 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x*(4*x^2-9)^(1/2)+9/32*ln(x*4^(1/2)+(4*x^2-9)^(1/2))*4^(1/2)

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Maxima [A]
time = 0.50, size = 31, normalized size = 0.86 \begin {gather*} \frac {1}{8} \, \sqrt {4 \, x^{2} - 9} x + \frac {9}{16} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} - 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*x^2 - 9)*x + 9/16*log(8*x + 4*sqrt(4*x^2 - 9))

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Fricas [A]
time = 1.36, size = 29, normalized size = 0.81 \begin {gather*} \frac {1}{8} \, \sqrt {4 \, x^{2} - 9} x - \frac {9}{16} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*x^2 - 9)*x - 9/16*log(-2*x + sqrt(4*x^2 - 9))

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Sympy [A]
time = 0.07, size = 22, normalized size = 0.61 \begin {gather*} \frac {x \sqrt {4 x^{2} - 9}}{8} + \frac {9 \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(4*x**2 - 9)/8 + 9*acosh(2*x/3)/16

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Giac [A]
time = 0.80, size = 30, normalized size = 0.83 \begin {gather*} \frac {1}{8} \, \sqrt {4 \, x^{2} - 9} x - \frac {9}{16} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 9} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*x^2 - 9)*x - 9/16*log(abs(-2*x + sqrt(4*x^2 - 9)))

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Mupad [B]
time = 0.10, size = 29, normalized size = 0.81 \begin {gather*} \frac {9\,\ln \left (x+\frac {\sqrt {4\,x^2-9}}{2}\right )}{16}+\frac {x\,\sqrt {4\,x^2-9}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(4*x^2 - 9)^(1/2),x)

[Out]

(9*log(x + (4*x^2 - 9)^(1/2)/2))/16 + (x*(4*x^2 - 9)^(1/2))/8

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