### 3.172 $$\int \frac{x^4}{(9+x^2)^3} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0082532, antiderivative size = 37, normalized size of antiderivative = 1., 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 = {288, 203} $-\frac{x^3}{4 \left (x^2+9\right )^2}-\frac{3 x}{8 \left (x^2+9\right )}+\frac{1}{8} \tan ^{-1}\left (\frac{x}{3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[x^4/(9 + x^2)^3,x]

[Out]

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

Rule 288

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]

Rule 203

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

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*}

Mathematica [A]  time = 0.0127404, size = 28, normalized size = 0.76 $\frac{1}{8} \left (\tan ^{-1}\left (\frac{x}{3}\right )-\frac{x \left (5 x^2+27\right )}{\left (x^2+9\right )^2}\right )$

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 = 0.007, size = 25, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ({x}^{2}+9 \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8}}-{\frac{27\,x}{8}} \right ) }+{\frac{1}{8}\arctan \left ({\frac{x}{3}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.41383, size = 41, normalized size = 1.11 \begin{align*} -\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{align*}

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 = 1.92872, size = 104, normalized size = 2.81 \begin{align*} -\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{align*}

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 = 0.116243, size = 26, normalized size = 0.7 \begin{align*} - \frac{5 x^{3} + 27 x}{8 x^{4} + 144 x^{2} + 648} + \frac{\operatorname{atan}{\left (\frac{x}{3} \right )}}{8} \end{align*}

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 = 1.0556, size = 34, normalized size = 0.92 \begin{align*} -\frac{5 \, x^{3} + 27 \, x}{8 \,{\left (x^{2} + 9\right )}^{2}} + \frac{1}{8} \, \arctan \left (\frac{1}{3} \, x\right ) \end{align*}

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)