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 = 0.0145727, 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 \[ \frac{2}{45} (3 x+1)^{5/2}-\frac{2}{27} (3 x+1)^{3/2} \]

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

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Rubi in Sympy [A]  time = 1.47911, size = 22, normalized size = 0.81 \[ \frac{2 \left (3 x + 1\right )^{\frac{5}{2}}}{45} - \frac{2 \left (3 x + 1\right )^{\frac{3}{2}}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(1+3*x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00604768, 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 = 0.004, size = 15, normalized size = 0.6 \[{\frac{18\,x-4}{135} \left ( 1+3\,x \right ) ^{{\frac{3}{2}}}} \]

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 = 1.35391, size = 26, normalized size = 0.96 \[ \frac{2}{45} \,{\left (3 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (3 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x + 1)*x,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 = 0.218311, size = 26, normalized size = 0.96 \[ \frac{2}{135} \,{\left (27 \, x^{2} + 3 \, x - 2\right )} \sqrt{3 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x + 1)*x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.48857, 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

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GIAC/XCAS [A]  time = 0.213925, size = 26, normalized size = 0.96 \[ \frac{2}{45} \,{\left (3 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (3 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x + 1)*x,x, algorithm="giac")

[Out]

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