3.931 \(\int \sqrt{x+x^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ \frac{4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt{x}}-\frac{16 \left (x^{3/2}+x\right )^{3/2}}{35 x}+\frac{32 \left (x^{3/2}+x\right )^{3/2}}{105 x^{3/2}} \]

[Out]

(32*(x + x^(3/2))^(3/2))/(105*x^(3/2)) - (16*(x + x^(3/2))^(3/2))/(35*x) + (4*(x + x^(3/2))^(3/2))/(7*Sqrt[x])

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Rubi [A]  time = 0.0515844, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2002, 2016, 2014} \[ \frac{4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt{x}}-\frac{16 \left (x^{3/2}+x\right )^{3/2}}{35 x}+\frac{32 \left (x^{3/2}+x\right )^{3/2}}{105 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + x^(3/2)],x]

[Out]

(32*(x + x^(3/2))^(3/2))/(105*x^(3/2)) - (16*(x + x^(3/2))^(3/2))/(35*x) + (4*(x + x^(3/2))^(3/2))/(7*Sqrt[x])

Rule 2002

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]

Rule 2016

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

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \sqrt{x+x^{3/2}} \, dx &=\frac{4 \left (x+x^{3/2}\right )^{3/2}}{7 \sqrt{x}}-\frac{4}{7} \int \frac{\sqrt{x+x^{3/2}}}{\sqrt{x}} \, dx\\ &=-\frac{16 \left (x+x^{3/2}\right )^{3/2}}{35 x}+\frac{4 \left (x+x^{3/2}\right )^{3/2}}{7 \sqrt{x}}+\frac{8}{35} \int \frac{\sqrt{x+x^{3/2}}}{x} \, dx\\ &=\frac{32 \left (x+x^{3/2}\right )^{3/2}}{105 x^{3/2}}-\frac{16 \left (x+x^{3/2}\right )^{3/2}}{35 x}+\frac{4 \left (x+x^{3/2}\right )^{3/2}}{7 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0194226, size = 39, normalized size = 0.66 \[ \frac{4 \left (\sqrt{x}+1\right ) \left (15 x-12 \sqrt{x}+8\right ) \sqrt{x^{3/2}+x}}{105 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^(3/2)],x]

[Out]

(4*(1 + Sqrt[x])*(8 - 12*Sqrt[x] + 15*x)*Sqrt[x + x^(3/2)])/(105*Sqrt[x])

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Maple [A]  time = 0.007, size = 28, normalized size = 0.5 \begin{align*}{\frac{4}{105}\sqrt{x+{x}^{{\frac{3}{2}}}} \left ( 1+\sqrt{x} \right ) \left ( 15\,x-12\,\sqrt{x}+8 \right ){\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/105*(x+x^(3/2))^(1/2)*(1+x^(1/2))*(15*x-12*x^(1/2)+8)/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{\frac{3}{2}} + x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^(3/2) + x), x)

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Fricas [A]  time = 1.83732, size = 84, normalized size = 1.42 \begin{align*} \frac{4 \,{\left (15 \, x^{2} +{\left (3 \, x + 8\right )} \sqrt{x} - 4 \, x\right )} \sqrt{x^{\frac{3}{2}} + x}}{105 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/105*(15*x^2 + (3*x + 8)*sqrt(x) - 4*x)*sqrt(x^(3/2) + x)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{\frac{3}{2}} + x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**(3/2) + x), x)

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Giac [A]  time = 1.08443, size = 45, normalized size = 0.76 \begin{align*} \frac{4}{105} \,{\left (15 \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - 42 \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + 35 \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - 8\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/105*(15*(sqrt(x) + 1)^(7/2) - 42*(sqrt(x) + 1)^(5/2) + 35*(sqrt(x) + 1)^(3/2) - 8)*sgn(x)