3.4.73 \(\int x \sqrt {1+2 x} \, dx\) [373]

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

[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 = {45} \begin {gather*} \frac {1}{10} (2 x+1)^{5/2}-\frac {1}{6} (2 x+1)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 + 2*x],x]

[Out]

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

Rule 45

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 \begin {gather*} \frac {1}{15} (1+2 x)^{3/2} (-1+3 x) \end {gather*}

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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Mathics [A]
time = 2.29, size = 17, normalized size = 0.63 \begin {gather*} \frac {\left (-1+x+6 x^2\right ) \sqrt {1+2 x}}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*Sqrt[1 + 2*x],x]')

[Out]

(-1 + x + 6 x ^ 2) Sqrt[1 + 2 x] / 15

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Maple [A]
time = 0.04, size = 20, normalized size = 0.74

method	result	size
gosper	\(\frac {\left (1+2 x \right )^{\frac {3}{2}} \left (3 x -1\right )}{15}\)	\(15\)
risch	\(\frac {\left (6 x^{2}+x -1\right ) \sqrt {1+2 x}}{15}\)	\(18\)
trager	\(\left (\frac {2}{5} x^{2}+\frac {1}{15} x -\frac {1}{15}\right ) \sqrt {1+2 x}\)	\(19\)
derivativedivides	\(-\frac {\left (1+2 x \right )^{\frac {3}{2}}}{6}+\frac {\left (1+2 x \right )^{\frac {5}{2}}}{10}\)	\(20\)
default	\(-\frac {\left (1+2 x \right )^{\frac {3}{2}}}{6}+\frac {\left (1+2 x \right )^{\frac {5}{2}}}{10}\)	\(20\)
meijerg	\(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+2 x \right )^{\frac {3}{2}} \left (-6 x +2\right )}{15}}{8 \sqrt {\pi }}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1+2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

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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Fricas [A]
time = 0.33, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{15} \, {\left (6 \, x^{2} + x - 1\right )} \sqrt {2 \, x + 1} \end {gather*}

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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Sympy [A]
time = 0.52, size = 36, normalized size = 1.33 \begin {gather*} \frac {2 x^{2} \sqrt {2 x + 1}}{5} + \frac {x \sqrt {2 x + 1}}{15} - \frac {\sqrt {2 x + 1}}{15} \end {gather*}

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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Giac [A]
time = 0.00, size = 84, normalized size = 3.11 \begin {gather*} \frac {2}{4} \left (\frac {1}{5} \sqrt {2 x+1} \left (2 x+1\right )^{2}-\frac {2}{3} \sqrt {2 x+1} \left (2 x+1\right )+\sqrt {2 x+1}\right )+\frac {\frac {1}{3} \sqrt {2 x+1} \left (2 x+1\right )-\sqrt {2 x+1}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \frac {{\left (2\,x+1\right )}^{3/2}\,\left (6\,x-2\right )}{30} \end {gather*}

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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