3.1.34 \(\int x \sqrt {1+x} \, dx\) [34]

Optimal. Leaf size=23 \[ -\frac {2}{3} (1+x)^{3/2}+\frac {2}{5} (1+x)^{5/2} \]

[Out]

-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 + x],x]

[Out]

(-2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5

Rule 45

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

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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.78 \begin {gather*} \frac {2}{15} (1+x)^{3/2} (-5+3 (1+x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2)*(-5 + 3*(1 + x)))/15

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Maple [A]
time = 0.03, size = 16, normalized size = 0.70

method	result	size
gosper	\(\frac {2 \left (1+x \right )^{\frac {3}{2}} \left (-2+3 x \right )}{15}\)	\(13\)
derivativedivides	\(-\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}+\frac {2 \left (1+x \right )^{\frac {5}{2}}}{5}\)	\(16\)
default	\(-\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}+\frac {2 \left (1+x \right )^{\frac {5}{2}}}{5}\)	\(16\)
risch	\(\frac {2 \left (3 x^{2}+x -2\right ) \sqrt {1+x}}{15}\)	\(16\)
trager	\(\left (\frac {2}{5} x^{2}+\frac {2}{15} x -\frac {4}{15}\right ) \sqrt {1+x}\)	\(17\)
meijerg	\(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+x \right )^{\frac {3}{2}} \left (2-3 x \right )}{15}}{2 \sqrt {\pi }}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)

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Maxima [A]
time = 1.22, size = 15, normalized size = 0.65 \begin {gather*} \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)

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Fricas [A]
time = 0.52, size = 15, normalized size = 0.65 \begin {gather*} \frac {2}{15} \, {\left (3 \, x^{2} + x - 2\right )} \sqrt {x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^2 + x - 2)*sqrt(x + 1)

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Sympy [A]
time = 0.46, size = 34, normalized size = 1.48 \begin {gather*} \frac {2 x^{2} \sqrt {x + 1}}{5} + \frac {2 x \sqrt {x + 1}}{15} - \frac {4 \sqrt {x + 1}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**2*sqrt(x + 1)/5 + 2*x*sqrt(x + 1)/15 - 4*sqrt(x + 1)/15

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Giac [A]
time = 0.45, size = 15, normalized size = 0.65 \begin {gather*} \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)

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Mupad [B]
time = 0.00, size = 12, normalized size = 0.52 \begin {gather*} \frac {2\,\left (3\,x-2\right )\,{\left (x+1\right )}^{3/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x + 1)^(1/2),x)

[Out]

(2*(3*x - 2)*(x + 1)^(3/2))/15

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