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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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)

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 9, normalized size = 0.45 \[ \frac {5}{2} \, x^{2} + 3 \, x \]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 26, normalized size = 1.30 \[ \frac {1}{2} \, {\left (5 \, x^{2} + 6 \, x\right )} \mathrm {sgn}\left (5 \, x + 3\right ) + \frac {9}{10} \, \mathrm {sgn}\left (5 \, x + 3\right ) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 25, normalized size = 1.25 \[ \frac {\left (5 x +6\right ) \sqrt {\left (5 x +3\right )^{2}}\, x}{6+10 x} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 30, normalized size = 1.50 \[ \frac {1}{2} \, \sqrt {25 \, x^{2} + 30 \, x + 9} x + \frac {3}{10} \, \sqrt {25 \, x^{2} + 30 \, x + 9} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 13, normalized size = 0.65 \[ \frac {\left |5\,x+3\right |\,\left (5\,x+3\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x + 3)^2)^(1/2),x)

[Out]

(abs(5*x + 3)*(5*x + 3))/10

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 8, normalized size = 0.40 \[ \frac {5 x^{2}}{2} + 3 x \]

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

________________________________________________________________________________________