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]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.40, size = 14, normalized size = 0.74 \[ \frac {2}{21} \, {\left (3 \, x^{3} + 7 \, x\right )} \sqrt {x} \]

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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giac [A]  time = 1.11, size = 11, normalized size = 0.58 \[ \frac {2}{7} \, x^{\frac {7}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.68 \[ \frac {2 \left (3 x^{2}+7\right ) x^{\frac {3}{2}}}{21} \]

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 = 0.42, size = 11, normalized size = 0.58 \[ \frac {2}{7} \, x^{\frac {7}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \]

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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mupad [B]  time = 0.02, size = 12, normalized size = 0.63 \[ \frac {2\,x^{3/2}\,\left (3\,x^2+7\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(x^2 + 1),x)

[Out]

(2*x^(3/2)*(3*x^2 + 7))/21

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sympy [A]  time = 1.09, size = 15, normalized size = 0.79 \[ \frac {2 x^{\frac {7}{2}}}{7} + \frac {2 x^{\frac {3}{2}}}{3} \]

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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