3.2801 \(\int \sqrt{(3+5 x)^2} \, dx\)

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 = 0.0143624, 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 \[ \frac{1}{10} (5 x+3) \sqrt{(5 x+3)^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(3 + 5*x)^2],x]

[Out]

((3 + 5*x)*Sqrt[(3 + 5*x)^2])/10

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Rubi in Sympy [A]  time = 1.38744, size = 19, normalized size = 0.95 \[ \frac{\left (50 x + 30\right ) \sqrt{25 x^{2} + 30 x + 9}}{100} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((3+5*x)**2)**(1/2),x)

[Out]

(50*x + 30)*sqrt(25*x**2 + 30*x + 9)/100

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Mathematica [A]  time = 0.0145698, 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 = 0.004, size = 25, normalized size = 1.3 \[{\frac{x \left ( 5\,x+6 \right ) }{6+10\,x}\sqrt{ \left ( 3+5\,x \right ) ^{2}}} \]

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 = 1.51976, size = 41, normalized size = 2.05 \[ \frac{1}{2} \, \sqrt{25 \, x^{2} + 30 \, x + 9} x + \frac{3}{10} \, \sqrt{25 \, x^{2} + 30 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((5*x + 3)^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 = 0.208223, size = 12, normalized size = 0.6 \[ \frac{5}{2} \, x^{2} + 3 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((5*x + 3)^2),x, algorithm="fricas")

[Out]

5/2*x^2 + 3*x

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Sympy [A]  time = 0.095481, size = 8, normalized size = 0.4 \[ \frac{5 x^{2}}{2} + 3 x \]

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

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GIAC/XCAS [A]  time = 0.217768, size = 35, normalized size = 1.75 \[ \frac{1}{2} \,{\left (5 \, x^{2} + 6 \, x\right )}{\rm sign}\left (5 \, x + 3\right ) + \frac{9}{10} \,{\rm sign}\left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((5*x + 3)^2),x, algorithm="giac")

[Out]

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