3.160 \(\int x^{3/2} \tan ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=36 \[ -\frac{x^2}{10}+\frac{2}{5} x^{5/2} \tan ^{-1}\left (\sqrt{x}\right )+\frac{x}{5}-\frac{1}{5} \log (x+1) \]

[Out]

x/5 - x^2/10 + (2*x^(5/2)*ArcTan[Sqrt[x]])/5 - Log[1 + x]/5

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Rubi [A]  time = 0.0150146, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5033, 43} \[ -\frac{x^2}{10}+\frac{2}{5} x^{5/2} \tan ^{-1}\left (\sqrt{x}\right )+\frac{x}{5}-\frac{1}{5} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*ArcTan[Sqrt[x]],x]

[Out]

x/5 - x^2/10 + (2*x^(5/2)*ArcTan[Sqrt[x]])/5 - Log[1 + x]/5

Rule 5033

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTan
[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

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^{3/2} \tan ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{2}{5} x^{5/2} \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{5} \int \frac{x^2}{1+x} \, dx\\ &=\frac{2}{5} x^{5/2} \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{5} \int \left (-1+x+\frac{1}{1+x}\right ) \, dx\\ &=\frac{x}{5}-\frac{x^2}{10}+\frac{2}{5} x^{5/2} \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{5} \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0152966, size = 30, normalized size = 0.83 \[ \frac{1}{10} \left (4 x^{5/2} \tan ^{-1}\left (\sqrt{x}\right )-(x-2) x-2 \log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*ArcTan[Sqrt[x]],x]

[Out]

(-((-2 + x)*x) + 4*x^(5/2)*ArcTan[Sqrt[x]] - 2*Log[1 + x])/10

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Maple [A]  time = 0.023, size = 25, normalized size = 0.7 \begin{align*}{\frac{x}{5}}-{\frac{{x}^{2}}{10}}+{\frac{2}{5}{x}^{{\frac{5}{2}}}\arctan \left ( \sqrt{x} \right ) }-{\frac{\ln \left ( x+1 \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*arctan(x^(1/2)),x)

[Out]

1/5*x-1/10*x^2+2/5*x^(5/2)*arctan(x^(1/2))-1/5*ln(x+1)

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Maxima [A]  time = 0.977442, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \arctan \left (\sqrt{x}\right ) - \frac{1}{10} \, x^{2} + \frac{1}{5} \, x - \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2)*arctan(sqrt(x)) - 1/10*x^2 + 1/5*x - 1/5*log(x + 1)

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Fricas [A]  time = 2.2254, size = 88, normalized size = 2.44 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \arctan \left (\sqrt{x}\right ) - \frac{1}{10} \, x^{2} + \frac{1}{5} \, x - \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2)*arctan(sqrt(x)) - 1/10*x^2 + 1/5*x - 1/5*log(x + 1)

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Sympy [B]  time = 7.67426, size = 92, normalized size = 2.56 \begin{align*} \frac{4 x^{\frac{7}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{10 x + 10} + \frac{4 x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{10 x + 10} - \frac{x^{3}}{10 x + 10} + \frac{x^{2}}{10 x + 10} - \frac{2 x \log{\left (x + 1 \right )}}{10 x + 10} + \frac{x}{10 x + 10} - \frac{2 \log{\left (x + 1 \right )}}{10 x + 10} - \frac{1}{10 x + 10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x**(7/2)*atan(sqrt(x))/(10*x + 10) + 4*x**(5/2)*atan(sqrt(x))/(10*x + 10) - x**3/(10*x + 10) + x**2/(10*x +
10) - 2*x*log(x + 1)/(10*x + 10) + x/(10*x + 10) - 2*log(x + 1)/(10*x + 10) - 1/(10*x + 10)

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Giac [A]  time = 1.10913, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \arctan \left (\sqrt{x}\right ) - \frac{1}{10} \, x^{2} + \frac{1}{5} \, x - \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2)*arctan(sqrt(x)) - 1/10*x^2 + 1/5*x - 1/5*log(x + 1)