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

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Rubi [A]  time = 0.0048999, 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} $\frac{1}{10} (2 x+1)^{5/2}-\frac{1}{6} (2 x+1)^{3/2}$

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*}

Mathematica [A]  time = 0.0057128, 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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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*}{\frac{3\,x-1}{15} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.924912, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{10} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1}{6} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

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 = 2.09458, size = 49, normalized size = 1.81 \begin{align*} \frac{1}{15} \,{\left (6 \, x^{2} + x - 1\right )} \sqrt{2 \, x + 1} \end{align*}

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 = 0.844147, size = 36, normalized size = 1.33 \begin{align*} \frac{2 x^{2} \sqrt{2 x + 1}}{5} + \frac{x \sqrt{2 x + 1}}{15} - \frac{\sqrt{2 x + 1}}{15} \end{align*}

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 = 1.05702, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{10} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1}{6} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

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)