3.441 \(\int x^5 \sqrt{9+4 x^2} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{448} \left (4 x^2+9\right )^{7/2}-\frac{9}{160} \left (4 x^2+9\right )^{5/2}+\frac{27}{64} \left (4 x^2+9\right )^{3/2} \]

[Out]

(27*(9 + 4*x^2)^(3/2))/64 - (9*(9 + 4*x^2)^(5/2))/160 + (9 + 4*x^2)^(7/2)/448

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Rubi [A]  time = 0.0195838, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{448} \left (4 x^2+9\right )^{7/2}-\frac{9}{160} \left (4 x^2+9\right )^{5/2}+\frac{27}{64} \left (4 x^2+9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[9 + 4*x^2],x]

[Out]

(27*(9 + 4*x^2)^(3/2))/64 - (9*(9 + 4*x^2)^(5/2))/160 + (9 + 4*x^2)^(7/2)/448

Rule 266

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]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 \sqrt{9+4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt{9+4 x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16} \sqrt{9+4 x}-\frac{9}{8} (9+4 x)^{3/2}+\frac{1}{16} (9+4 x)^{5/2}\right ) \, dx,x,x^2\right )\\ &=\frac{27}{64} \left (9+4 x^2\right )^{3/2}-\frac{9}{160} \left (9+4 x^2\right )^{5/2}+\frac{1}{448} \left (9+4 x^2\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0100116, size = 27, normalized size = 0.59 \[ \frac{1}{280} \left (4 x^2+9\right )^{3/2} \left (10 x^4-18 x^2+27\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[9 + 4*x^2],x]

[Out]

((9 + 4*x^2)^(3/2)*(27 - 18*x^2 + 10*x^4))/280

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Maple [A]  time = 0.003, size = 24, normalized size = 0.5 \begin{align*}{\frac{10\,{x}^{4}-18\,{x}^{2}+27}{280} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(4*x^2+9)^(1/2),x)

[Out]

1/280*(4*x^2+9)^(3/2)*(10*x^4-18*x^2+27)

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Maxima [A]  time = 1.7444, size = 54, normalized size = 1.17 \begin{align*} \frac{1}{28} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{4} - \frac{9}{140} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{2} + \frac{27}{280} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

1/28*(4*x^2 + 9)^(3/2)*x^4 - 9/140*(4*x^2 + 9)^(3/2)*x^2 + 27/280*(4*x^2 + 9)^(3/2)

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Fricas [A]  time = 1.64842, size = 76, normalized size = 1.65 \begin{align*} \frac{1}{280} \,{\left (40 \, x^{6} + 18 \, x^{4} - 54 \, x^{2} + 243\right )} \sqrt{4 \, x^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/280*(40*x^6 + 18*x^4 - 54*x^2 + 243)*sqrt(4*x^2 + 9)

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Sympy [A]  time = 1.88655, size = 61, normalized size = 1.33 \begin{align*} \frac{x^{6} \sqrt{4 x^{2} + 9}}{7} + \frac{9 x^{4} \sqrt{4 x^{2} + 9}}{140} - \frac{27 x^{2} \sqrt{4 x^{2} + 9}}{140} + \frac{243 \sqrt{4 x^{2} + 9}}{280} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(4*x**2+9)**(1/2),x)

[Out]

x**6*sqrt(4*x**2 + 9)/7 + 9*x**4*sqrt(4*x**2 + 9)/140 - 27*x**2*sqrt(4*x**2 + 9)/140 + 243*sqrt(4*x**2 + 9)/28
0

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Giac [A]  time = 1.41474, size = 46, normalized size = 1. \begin{align*} \frac{1}{448} \,{\left (4 \, x^{2} + 9\right )}^{\frac{7}{2}} - \frac{9}{160} \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} + \frac{27}{64} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/448*(4*x^2 + 9)^(7/2) - 9/160*(4*x^2 + 9)^(5/2) + 27/64*(4*x^2 + 9)^(3/2)