3.34 \(\int x \sqrt{1+x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0039135, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {43} \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2} \]

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 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 \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*}

Mathematica [A]  time = 0.0031991, size = 16, normalized size = 0.7 \[ \frac{2}{15} (x+1)^{3/2} (3 x-2) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2)*(-2 + 3*x))/15

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Maple [A]  time = 0., size = 13, normalized size = 0.6 \begin{align*}{\frac{-4+6\,x}{15} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(1+x)^(3/2)*(-2+3*x)

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Maxima [A]  time = 0.926539, size = 20, normalized size = 0.87 \begin{align*} \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}

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 = 1.79048, size = 46, normalized size = 2. \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + x - 2\right )} \sqrt{x + 1} \end{align*}

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 = 0.839955, size = 34, normalized size = 1.48 \begin{align*} \frac{2 x^{2} \sqrt{x + 1}}{5} + \frac{2 x \sqrt{x + 1}}{15} - \frac{4 \sqrt{x + 1}}{15} \end{align*}

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 = 1.09204, size = 20, normalized size = 0.87 \begin{align*} \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}

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)