3.2.72 \(\int \frac {x^4}{(9+x^2)^3} \, dx\) [172]

Optimal. Leaf size=37 \[ -\frac {x^3}{4 \left (9+x^2\right )^2}-\frac {3 x}{8 \left (9+x^2\right )}+\frac {1}{8} \tan ^{-1}\left (\frac {x}{3}\right ) \]

[Out]

-1/4*x^3/(x^2+9)^2-3/8*x/(x^2+9)+1/8*arctan(1/3*x)

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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {294, 209} \begin {gather*} \frac {1}{8} \text {ArcTan}\left (\frac {x}{3}\right )-\frac {3 x}{8 \left (x^2+9\right )}-\frac {x^3}{4 \left (x^2+9\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(9 + x^2)^3,x]

[Out]

-1/4*x^3/(9 + x^2)^2 - (3*x)/(8*(9 + x^2)) + ArcTan[x/3]/8

Rule 209

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

Rule 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^4}{\left (9+x^2\right )^3} \, dx &=-\frac {x^3}{4 \left (9+x^2\right )^2}+\frac {3}{4} \int \frac {x^2}{\left (9+x^2\right )^2} \, dx\\ &=-\frac {x^3}{4 \left (9+x^2\right )^2}-\frac {3 x}{8 \left (9+x^2\right )}+\frac {3}{8} \int \frac {1}{9+x^2} \, dx\\ &=-\frac {x^3}{4 \left (9+x^2\right )^2}-\frac {3 x}{8 \left (9+x^2\right )}+\frac {1}{8} \tan ^{-1}\left (\frac {x}{3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.76 \begin {gather*} \frac {1}{8} \left (-\frac {x \left (27+5 x^2\right )}{\left (9+x^2\right )^2}+\tan ^{-1}\left (\frac {x}{3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(9 + x^2)^3,x]

[Out]

(-((x*(27 + 5*x^2))/(9 + x^2)^2) + ArcTan[x/3])/8

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Maple [A]
time = 0.05, size = 25, normalized size = 0.68

method	result	size
default	\(\frac {-\frac {5}{8} x^{3}-\frac {27}{8} x}{\left (x^{2}+9\right )^{2}}+\frac {\arctan \left (\frac {x}{3}\right )}{8}\)	\(25\)
risch	\(\frac {-\frac {5}{8} x^{3}-\frac {27}{8} x}{\left (x^{2}+9\right )^{2}}+\frac {\arctan \left (\frac {x}{3}\right )}{8}\)	\(25\)
meijerg	\(-\frac {x \left (\frac {25 x^{2}}{9}+15\right )}{360 \left (\frac {x^{2}}{9}+1\right )^{2}}+\frac {\arctan \left (\frac {x}{3}\right )}{8}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5/8*x^3-27/8*x)/(x^2+9)^2+1/8*arctan(1/3*x)

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Maxima [A]
time = 2.26, size = 30, normalized size = 0.81 \begin {gather*} -\frac {5 \, x^{3} + 27 \, x}{8 \, {\left (x^{4} + 18 \, x^{2} + 81\right )}} + \frac {1}{8} \, \arctan \left (\frac {1}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(5*x^3 + 27*x)/(x^4 + 18*x^2 + 81) + 1/8*arctan(1/3*x)

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Fricas [A]
time = 1.80, size = 39, normalized size = 1.05 \begin {gather*} -\frac {5 \, x^{3} - {\left (x^{4} + 18 \, x^{2} + 81\right )} \arctan \left (\frac {1}{3} \, x\right ) + 27 \, x}{8 \, {\left (x^{4} + 18 \, x^{2} + 81\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(5*x^3 - (x^4 + 18*x^2 + 81)*arctan(1/3*x) + 27*x)/(x^4 + 18*x^2 + 81)

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Sympy [A]
time = 0.04, size = 27, normalized size = 0.73 \begin {gather*} \frac {- 5 x^{3} - 27 x}{8 x^{4} + 144 x^{2} + 648} + \frac {\operatorname {atan}{\left (\frac {x}{3} \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5*x**3 - 27*x)/(8*x**4 + 144*x**2 + 648) + atan(x/3)/8

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Giac [A]
time = 0.48, size = 25, normalized size = 0.68 \begin {gather*} -\frac {5 \, x^{3} + 27 \, x}{8 \, {\left (x^{2} + 9\right )}^{2}} + \frac {1}{8} \, \arctan \left (\frac {1}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^2+9)^3,x, algorithm="giac")

[Out]

-1/8*(5*x^3 + 27*x)/(x^2 + 9)^2 + 1/8*arctan(1/3*x)

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Mupad [B]
time = 0.16, size = 30, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x}{3}\right )}{8}-\frac {\frac {5\,x^3}{8}+\frac {27\,x}{8}}{x^4+18\,x^2+81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^2 + 9)^3,x)

[Out]

atan(x/3)/8 - ((27*x)/8 + (5*x^3)/8)/(18*x^2 + x^4 + 81)

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