### 3.129 $$\int \frac{x^3}{\sqrt{4+x^2}} \, dx$$

Optimal. Leaf size=25 $\frac{1}{3} \left (x^2+4\right )^{3/2}-4 \sqrt{x^2+4}$

[Out]

-4*Sqrt[4 + x^2] + (4 + x^2)^(3/2)/3

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Rubi [A]  time = 0.0111117, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {266, 43} $\frac{1}{3} \left (x^2+4\right )^{3/2}-4 \sqrt{x^2+4}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3/Sqrt[4 + x^2],x]

[Out]

-4*Sqrt[4 + x^2] + (4 + x^2)^(3/2)/3

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

Mathematica [A]  time = 0.0044458, size = 18, normalized size = 0.72 $\frac{1}{3} \left (x^2-8\right ) \sqrt{x^2+4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3/Sqrt[4 + x^2],x]

[Out]

((-8 + x^2)*Sqrt[4 + x^2])/3

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Maple [A]  time = 0.005, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{2}-8}{3}\sqrt{{x}^{2}+4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^2+4)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.40537, size = 30, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \sqrt{x^{2} + 4} x^{2} - \frac{8}{3} \, \sqrt{x^{2} + 4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^2+4)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73206, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{3} \, \sqrt{x^{2} + 4}{\left (x^{2} - 8\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^2+4)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(x^2 + 4)*(x^2 - 8)

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Sympy [A]  time = 0.341551, size = 24, normalized size = 0.96 \begin{align*} \frac{x^{2} \sqrt{x^{2} + 4}}{3} - \frac{8 \sqrt{x^{2} + 4}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**2+4)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.04427, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 4\right )}^{\frac{3}{2}} - 4 \, \sqrt{x^{2} + 4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^2+4)^(1/2),x, algorithm="giac")

[Out]

1/3*(x^2 + 4)^(3/2) - 4*sqrt(x^2 + 4)