3.466 \(\int x^2 \sqrt {-9+4 x^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac {9}{32} \sqrt {4 x^2-9} x-\frac {81}{64} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right )+\frac {1}{4} \sqrt {4 x^2-9} x^3 \]

[Out]

-81/64*arctanh(2*x/(4*x^2-9)^(1/2))-9/32*x*(4*x^2-9)^(1/2)+1/4*x^3*(4*x^2-9)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 54, 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 = {279, 321, 217, 206} \[ \frac {1}{4} \sqrt {4 x^2-9} x^3-\frac {9}{32} \sqrt {4 x^2-9} x-\frac {81}{64} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[-9 + 4*x^2],x]

[Out]

(-9*x*Sqrt[-9 + 4*x^2])/32 + (x^3*Sqrt[-9 + 4*x^2])/4 - (81*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/64

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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 279

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

Rule 321

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 x^2 \sqrt {-9+4 x^2} \, dx &=\frac {1}{4} x^3 \sqrt {-9+4 x^2}-\frac {9}{4} \int \frac {x^2}{\sqrt {-9+4 x^2}} \, dx\\ &=-\frac {9}{32} x \sqrt {-9+4 x^2}+\frac {1}{4} x^3 \sqrt {-9+4 x^2}-\frac {81}{32} \int \frac {1}{\sqrt {-9+4 x^2}} \, dx\\ &=-\frac {9}{32} x \sqrt {-9+4 x^2}+\frac {1}{4} x^3 \sqrt {-9+4 x^2}-\frac {81}{32} \operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x}{\sqrt {-9+4 x^2}}\right )\\ &=-\frac {9}{32} x \sqrt {-9+4 x^2}+\frac {1}{4} x^3 \sqrt {-9+4 x^2}-\frac {81}{64} \tanh ^{-1}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 46, normalized size = 0.85 \[ \sqrt {4 x^2-9} \left (\frac {x^3}{4}-\frac {9 x}{32}\right )-\frac {81}{64} \log \left (\sqrt {4 x^2-9}+2 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[-9 + 4*x^2],x]

[Out]

Sqrt[-9 + 4*x^2]*((-9*x)/32 + x^3/4) - (81*Log[2*x + Sqrt[-9 + 4*x^2]])/64

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fricas [A]  time = 0.85, size = 37, normalized size = 0.69 \[ \frac {1}{32} \, {\left (8 \, x^{3} - 9 \, x\right )} \sqrt {4 \, x^{2} - 9} + \frac {81}{64} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*x^3 - 9*x)*sqrt(4*x^2 - 9) + 81/64*log(-2*x + sqrt(4*x^2 - 9))

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giac [A]  time = 1.16, size = 37, normalized size = 0.69 \[ \frac {1}{32} \, {\left (8 \, x^{2} - 9\right )} \sqrt {4 \, x^{2} - 9} x + \frac {81}{64} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 9} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*x^2 - 9)*sqrt(4*x^2 - 9)*x + 81/64*log(abs(-2*x + sqrt(4*x^2 - 9)))

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maple [A]  time = 0.01, size = 47, normalized size = 0.87 \[ \frac {\left (4 x^{2}-9\right )^{\frac {3}{2}} x}{16}+\frac {9 \sqrt {4 x^{2}-9}\, x}{32}-\frac {81 \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-9}\right )}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*x^2-9)^(3/2)*x+9/32*(4*x^2-9)^(1/2)*x-81/128*4^(1/2)*ln(4^(1/2)*x+(4*x^2-9)^(1/2))

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maxima [A]  time = 2.89, size = 43, normalized size = 0.80 \[ \frac {1}{16} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x + \frac {9}{32} \, \sqrt {4 \, x^{2} - 9} x - \frac {81}{64} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*x^2 - 9)^(3/2)*x + 9/32*sqrt(4*x^2 - 9)*x - 81/64*log(8*x + 4*sqrt(4*x^2 - 9))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\sqrt {4\,x^2-9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(4*x^2 - 9)^(1/2),x)

[Out]

int(x^2*(4*x^2 - 9)^(1/2), x)

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sympy [A]  time = 2.80, size = 124, normalized size = 2.30 \[ \begin {cases} \frac {x^{5}}{\sqrt {4 x^{2} - 9}} - \frac {27 x^{3}}{8 \sqrt {4 x^{2} - 9}} + \frac {81 x}{32 \sqrt {4 x^{2} - 9}} - \frac {81 \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{64} & \text {for}\: \frac {4 \left |{x^{2}}\right |}{9} > 1 \\- \frac {i x^{5}}{\sqrt {9 - 4 x^{2}}} + \frac {27 i x^{3}}{8 \sqrt {9 - 4 x^{2}}} - \frac {81 i x}{32 \sqrt {9 - 4 x^{2}}} + \frac {81 i \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{64} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5/sqrt(4*x**2 - 9) - 27*x**3/(8*sqrt(4*x**2 - 9)) + 81*x/(32*sqrt(4*x**2 - 9)) - 81*acosh(2*x/3)
/64, 4*Abs(x**2)/9 > 1), (-I*x**5/sqrt(9 - 4*x**2) + 27*I*x**3/(8*sqrt(9 - 4*x**2)) - 81*I*x/(32*sqrt(9 - 4*x*
*2)) + 81*I*asin(2*x/3)/64, True))

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