3.2 \(\int x \sqrt{1+3 x} \, dx\)

Optimal. Leaf size=27 \[ \frac{2}{45} (3 x+1)^{5/2}-\frac{2}{27} (3 x+1)^{3/2} \]

[Out]

(-2*(1 + 3*x)^(3/2))/27 + (2*(1 + 3*x)^(5/2))/45

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Rubi [A]  time = 0.0044375, antiderivative size = 27, 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 = {43} \[ \frac{2}{45} (3 x+1)^{5/2}-\frac{2}{27} (3 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 + 3*x],x]

[Out]

(-2*(1 + 3*x)^(3/2))/27 + (2*(1 + 3*x)^(5/2))/45

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+3 x} \, dx &=\int \left (-\frac{1}{3} \sqrt{1+3 x}+\frac{1}{3} (1+3 x)^{3/2}\right ) \, dx\\ &=-\frac{2}{27} (1+3 x)^{3/2}+\frac{2}{45} (1+3 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.005936, size = 18, normalized size = 0.67 \[ \frac{2}{135} (3 x+1)^{3/2} (9 x-2) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + 3*x],x]

[Out]

(2*(1 + 3*x)^(3/2)*(-2 + 9*x))/135

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*}{\frac{18\,x-4}{135} \left ( 1+3\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1+3*x)^(1/2),x)

[Out]

2/135*(1+3*x)^(3/2)*(9*x-2)

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Maxima [A]  time = 0.949593, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{45} \,{\left (3 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (3 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+3*x)^(1/2),x, algorithm="maxima")

[Out]

2/45*(3*x + 1)^(5/2) - 2/27*(3*x + 1)^(3/2)

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Fricas [A]  time = 0.412359, size = 54, normalized size = 2. \begin{align*} \frac{2}{135} \,{\left (27 \, x^{2} + 3 \, x - 2\right )} \sqrt{3 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+3*x)^(1/2),x, algorithm="fricas")

[Out]

2/135*(27*x^2 + 3*x - 2)*sqrt(3*x + 1)

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Sympy [A]  time = 0.854789, size = 39, normalized size = 1.44 \begin{align*} \frac{2 x^{2} \sqrt{3 x + 1}}{5} + \frac{2 x \sqrt{3 x + 1}}{45} - \frac{4 \sqrt{3 x + 1}}{135} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+3*x)**(1/2),x)

[Out]

2*x**2*sqrt(3*x + 1)/5 + 2*x*sqrt(3*x + 1)/45 - 4*sqrt(3*x + 1)/135

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Giac [A]  time = 1.09895, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{45} \,{\left (3 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (3 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+3*x)^(1/2),x, algorithm="giac")

[Out]

2/45*(3*x + 1)^(5/2) - 2/27*(3*x + 1)^(3/2)