\(\int \sqrt {x} (1+x^2) \, dx\) [223]

Optimal result

Integrand size = 11, antiderivative size = 19 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2 x^{3/2}}{3}+\frac {2 x^{7/2}}{7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2 x^{7/2}}{7}+\frac {2 x^{3/2}}{3} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2}{21} x^{3/2} \left (7+3 x^2\right ) \]

[In]

Integrate[Sqrt[x]*(1 + x^2),x]

[Out]

(2*x^(3/2)*(7 + 3*x^2))/21

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 x^{\frac {7}{2}}}{7}\)	\(12\)
default	\(\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 x^{\frac {7}{2}}}{7}\)	\(12\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (3 x^{2}+7\right )}{21}\)	\(13\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (3 x^{2}+7\right )}{21}\)	\(13\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (3 x^{2}+7\right )}{21}\)	\(13\)

[In]

int(x^(1/2)*(x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2}{21} \, {\left (3 \, x^{3} + 7 \, x\right )} \sqrt {x} \]

[In]

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

[Out]

2/21*(3*x^3 + 7*x)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2 x^{\frac {7}{2}}}{7} + \frac {2 x^{\frac {3}{2}}}{3} \]

[In]

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

[Out]

2*x**(7/2)/7 + 2*x**(3/2)/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2}{7} \, x^{\frac {7}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \]

[In]

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

[Out]

2/7*x^(7/2) + 2/3*x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2}{7} \, x^{\frac {7}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \]

[In]

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

[Out]

2/7*x^(7/2) + 2/3*x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \sqrt {x} \left (1+x^2\right ) \, dx=\frac {2\,x^{3/2}\,\left (3\,x^2+7\right )}{21} \]

[In]

int(x^(1/2)*(x^2 + 1),x)

[Out]

(2*x^(3/2)*(3*x^2 + 7))/21