3.2808 \(\int \sqrt{(3+5 x)^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{1}{10} (5 x+3) \sqrt{(5 x+3)^2} \]

[Out]

((3 + 5*x)*Sqrt[(3 + 5*x)^2])/10

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Rubi [A]  time = 0.0048126, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 15, 30} \[ \frac{1}{10} (5 x+3) \sqrt{(5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(3 + 5*x)^2],x]

[Out]

((3 + 5*x)*Sqrt[(3 + 5*x)^2])/10

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0081942, size = 25, normalized size = 1.25 \[ \frac{x \sqrt{(5 x+3)^2} (5 x+6)}{10 x+6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(3 + 5*x)^2],x]

[Out]

(x*Sqrt[(3 + 5*x)^2]*(6 + 5*x))/(6 + 10*x)

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Maple [A]  time = 0.003, size = 25, normalized size = 1.3 \begin{align*}{\frac{x \left ( 5\,x+6 \right ) }{6+10\,x}\sqrt{ \left ( 3+5\,x \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+5*x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.45134, size = 41, normalized size = 2.05 \begin{align*} \frac{1}{2} \, \sqrt{25 \, x^{2} + 30 \, x + 9} x + \frac{3}{10} \, \sqrt{25 \, x^{2} + 30 \, x + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+5*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(25*x^2 + 30*x + 9)*x + 3/10*sqrt(25*x^2 + 30*x + 9)

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Fricas [A]  time = 1.24234, size = 20, normalized size = 1. \begin{align*} \frac{5}{2} \, x^{2} + 3 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+5*x)^2)^(1/2),x, algorithm="fricas")

[Out]

5/2*x^2 + 3*x

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Sympy [A]  time = 0.065068, size = 8, normalized size = 0.4 \begin{align*} \frac{5 x^{2}}{2} + 3 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+5*x)**2)**(1/2),x)

[Out]

5*x**2/2 + 3*x

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Giac [A]  time = 1.10181, size = 35, normalized size = 1.75 \begin{align*} \frac{1}{2} \,{\left (5 \, x^{2} + 6 \, x\right )} \mathrm{sgn}\left (5 \, x + 3\right ) + \frac{9}{10} \, \mathrm{sgn}\left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+5*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/2*(5*x^2 + 6*x)*sgn(5*x + 3) + 9/10*sgn(5*x + 3)