3.223 \(\int \sqrt{x} (1+x^2) \, dx\)

Optimal. Leaf size=19 \[ \frac{2 x^{7/2}}{7}+\frac{2 x^{3/2}}{3} \]

[Out]

(2*x^(3/2))/3 + (2*x^(7/2))/7

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Rubi [A]  time = 0.0025534, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{2 x^{7/2}}{7}+\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(1 + x^2),x]

[Out]

(2*x^(3/2))/3 + (2*x^(7/2))/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sqrt{x} \left (1+x^2\right ) \, dx &=\int \left (\sqrt{x}+x^{5/2}\right ) \, dx\\ &=\frac{2 x^{3/2}}{3}+\frac{2 x^{7/2}}{7}\\ \end{align*}

Mathematica [A]  time = 0.0027283, size = 16, normalized size = 0.84 \[ \frac{2}{21} x^{3/2} \left (3 x^2+7\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(1 + x^2),x]

[Out]

(2*x^(3/2)*(7 + 3*x^2))/21

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Maple [A]  time = 0.002, size = 13, normalized size = 0.7 \begin{align*}{\frac{6\,{x}^{2}+14}{21}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*x^(3/2)*(3*x^2+7)

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Maxima [A]  time = 0.930852, size = 15, normalized size = 0.79 \begin{align*} \frac{2}{7} \, x^{\frac{7}{2}} + \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*x^(7/2) + 2/3*x^(3/2)

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Fricas [A]  time = 1.58067, size = 38, normalized size = 2. \begin{align*} \frac{2}{21} \,{\left (3 \, x^{3} + 7 \, x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*x^3 + 7*x)*sqrt(x)

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Sympy [A]  time = 0.962552, size = 15, normalized size = 0.79 \begin{align*} \frac{2 x^{\frac{7}{2}}}{7} + \frac{2 x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(7/2)/7 + 2*x**(3/2)/3

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Giac [A]  time = 1.07425, size = 15, normalized size = 0.79 \begin{align*} \frac{2}{7} \, x^{\frac{7}{2}} + \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*x^(7/2) + 2/3*x^(3/2)