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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 27, 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 = {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 = 0.01, 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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 19, normalized size = 0.70 \[ \frac {2}{135} \, {\left (27 \, x^{2} + 3 \, x - 2\right )} \sqrt {3 \, x + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.01, size = 19, normalized size = 0.70 \[ \frac {2}{45} \, {\left (3 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (3 \, x + 1\right )}^{\frac {3}{2}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 15, normalized size = 0.56 \[ \frac {2 \left (3 x +1\right )^{\frac {3}{2}} \left (9 x -2\right )}{135} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 19, normalized size = 0.70 \[ \frac {2}{45} \, {\left (3 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (3 \, x + 1\right )}^{\frac {3}{2}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 14, normalized size = 0.52 \[ \frac {2\,{\left (3\,x+1\right )}^{3/2}\,\left (9\,x-2\right )}{135} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(3*x + 1)^(1/2),x)

[Out]

(2*(3*x + 1)^(3/2)*(9*x - 2))/135

________________________________________________________________________________________

sympy [A]  time = 1.04, size = 39, normalized size = 1.44 \[ \frac {2 x^{2} \sqrt {3 x + 1}}{5} + \frac {2 x \sqrt {3 x + 1}}{45} - \frac {4 \sqrt {3 x + 1}}{135} \]

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

________________________________________________________________________________________