3.142 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x} \, dx\)

Optimal. Leaf size=62 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.0628911, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.92435, size = 49, normalized size = 0.79 \[ \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + sqrt(a**2 + 2*a*b*x + b**2
*x**2)

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Mathematica [A]  time = 0.0166625, size = 27, normalized size = 0.44 \[ \frac{\sqrt{(a+b x)^2} (a \log (x)+b x)}{a+b x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x,x]

[Out]

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

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Maple [C]  time = 0.039, size = 19, normalized size = 0.3 \[{\it csgn} \left ( bx+a \right ) \left ( bx+a+a\ln \left ( bx \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/x,x)

[Out]

csgn(b*x+a)*(b*x+a+a*ln(b*x))

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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)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230485, size = 11, normalized size = 0.18 \[ b x + a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x + a*log(x)

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Sympy [A]  time = 0.191528, size = 7, normalized size = 0.11 \[ a \log{\left (x \right )} + b x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*log(x) + b*x

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GIAC/XCAS [A]  time = 0.208518, size = 28, normalized size = 0.45 \[ b x{\rm sign}\left (b x + a\right ) + a{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*sign(b*x + a) + a*ln(abs(x))*sign(b*x + a)