3.2797 \(\int \sqrt{c (a+b x)^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b} \]

[Out]

((a + b*x)*Sqrt[c*(a + b*x)^2])/(2*b)

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Rubi [A]  time = 0.0257906, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*(a + b*x)^2],x]

[Out]

((a + b*x)*Sqrt[c*(a + b*x)^2])/(2*b)

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Rubi in Sympy [A]  time = 2.74969, size = 36, normalized size = 1.44 \[ \frac{\left (2 a + 2 b x\right ) \sqrt{a^{2} c + 2 a b c x + b^{2} c x^{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*(b*x+a)**2)**(1/2),x)

[Out]

(2*a + 2*b*x)*sqrt(a**2*c + 2*a*b*c*x + b**2*c*x**2)/(4*b)

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Mathematica [A]  time = 0.0223195, size = 31, normalized size = 1.24 \[ \frac{c x (a+b x) (2 a+b x)}{2 \sqrt{c (a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*(a + b*x)^2],x]

[Out]

(c*x*(a + b*x)*(2*a + b*x))/(2*Sqrt[c*(a + b*x)^2])

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Maple [A]  time = 0.004, size = 29, normalized size = 1.2 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}\sqrt{c \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^2)^(1/2),x)

[Out]

1/2*x*(b*x+2*a)*(c*(b*x+a)^2)^(1/2)/(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2*c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.211785, size = 55, normalized size = 2.2 \[ \frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{\left (b x^{2} + 2 \, a x\right )}}{2 \,{\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2*c),x, algorithm="fricas")

[Out]

1/2*sqrt(b^2*c*x^2 + 2*a*b*c*x + a^2*c)*(b*x^2 + 2*a*x)/(b*x + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*(b*x+a)**2)**(1/2),x)

[Out]

Integral(sqrt(c*(a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.217353, size = 49, normalized size = 1.96 \[ \frac{1}{2} \,{\left ({\left (b x^{2} + 2 \, a x\right )}{\rm sign}\left (b x + a\right ) + \frac{a^{2}{\rm sign}\left (b x + a\right )}{b}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2*c),x, algorithm="giac")

[Out]

1/2*((b*x^2 + 2*a*x)*sign(b*x + a) + a^2*sign(b*x + a)/b)*sqrt(c)