3.373 \(\int x \sqrt {1+2 x} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{10} (2 x+1)^{5/2}-\frac {1}{6} (2 x+1)^{3/2} \]

[Out]

-1/6*(1+2*x)^(3/2)+1/10*(1+2*x)^(5/2)

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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} \[ \frac {1}{10} (2 x+1)^{5/2}-\frac {1}{6} (2 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 + 2*x],x]

[Out]

-(1 + 2*x)^(3/2)/6 + (1 + 2*x)^(5/2)/10

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+2 x} \, dx &=\int \left (-\frac {1}{2} \sqrt {1+2 x}+\frac {1}{2} (1+2 x)^{3/2}\right ) \, dx\\ &=-\frac {1}{6} (1+2 x)^{3/2}+\frac {1}{10} (1+2 x)^{5/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.67 \[ \frac {1}{15} (2 x+1)^{3/2} (3 x-1) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + 2*x],x]

[Out]

((1 + 2*x)^(3/2)*(-1 + 3*x))/15

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fricas [A]  time = 0.40, size = 17, normalized size = 0.63 \[ \frac {1}{15} \, {\left (6 \, x^{2} + x - 1\right )} \sqrt {2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(6*x^2 + x - 1)*sqrt(2*x + 1)

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giac [A]  time = 0.94, size = 19, normalized size = 0.70 \[ \frac {1}{10} \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*x + 1)^(5/2) - 1/6*(2*x + 1)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 19, normalized size = 0.70 \[ \frac {1}{10} \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*x + 1)^(5/2) - 1/6*(2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 14, normalized size = 0.52 \[ \frac {{\left (2\,x+1\right )}^{3/2}\,\left (6\,x-2\right )}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*x + 1)^(1/2),x)

[Out]

((2*x + 1)^(3/2)*(6*x - 2))/30

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sympy [A]  time = 1.02, size = 36, normalized size = 1.33 \[ \frac {2 x^{2} \sqrt {2 x + 1}}{5} + \frac {x \sqrt {2 x + 1}}{15} - \frac {\sqrt {2 x + 1}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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