3.2802 \(\int \sqrt{(6+10 x)^2} \, dx\)

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

[Out]

((3 + 5*x)*Sqrt[(3 + 5*x)^2])/5

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Rubi [A]  time = 0.0134281, 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 \[ \frac{1}{5} (5 x+3) \sqrt{(5 x+3)^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(6 + 10*x)^2],x]

[Out]

((3 + 5*x)*Sqrt[(3 + 5*x)^2])/5

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Rubi in Sympy [A]  time = 1.4266, size = 19, normalized size = 0.95 \[ \frac{\left (200 x + 120\right ) \sqrt{100 x^{2} + 120 x + 36}}{400} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((6+10*x)**2)**(1/2),x)

[Out]

(200*x + 120)*sqrt(100*x**2 + 120*x + 36)/400

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Mathematica [A]  time = 0.00919343, size = 25, normalized size = 1.25 \[ \frac{x \sqrt{(5 x+3)^2} (5 x+6)}{5 x+3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(6 + 10*x)^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 1.2 \[{\frac{x \left ( 5\,x+6 \right ) }{3+5\,x}\sqrt{ \left ( 3+5\,x \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((6+10*x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49382, size = 39, normalized size = 1.95 \[ \sqrt{25 \, x^{2} + 30 \, x + 9} x + \frac{3}{5} \, \sqrt{25 \, x^{2} + 30 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(25*x^2 + 30*x + 9)*x + 3/5*sqrt(25*x^2 + 30*x + 9)

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Fricas [A]  time = 0.209063, size = 12, normalized size = 0.6 \[ 5 \, x^{2} + 6 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x^2 + 6*x

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Sympy [A]  time = 0.098585, size = 7, normalized size = 0.35 \[ 5 x^{2} + 6 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((6+10*x)**2)**(1/2),x)

[Out]

5*x**2 + 6*x

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GIAC/XCAS [A]  time = 0.216737, size = 34, normalized size = 1.7 \[{\left (5 \, x^{2} + 6 \, x\right )}{\rm sign}\left (5 \, x + 3\right ) + \frac{9}{5} \,{\rm sign}\left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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