3.2019 \(\int \frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=24 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b} \]

[Out]

Sqrt[a^2 + 2*a*b*x + b^2*x^2]/b

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Rubi [A]  time = 0.013966, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a^2 + 2*a*b*x + b^2*x^2]/b

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Rubi in Sympy [A]  time = 2.49171, size = 10, normalized size = 0.42 \[ \frac{\sqrt{\left (a + b x\right )^{2}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((a + b*x)**2)/b

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Mathematica [A]  time = 0.00816341, size = 18, normalized size = 0.75 \[ \frac{x (a+b x)}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x))/Sqrt[(a + b*x)^2]

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Maple [A]  time = 0.002, size = 17, normalized size = 0.7 \[{ \left ( bx+a \right ) x{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.687061, size = 18, normalized size = 0.75 \[ \frac{\sqrt{{\left (b x + a\right )}^{2}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((b*x + a)^2)/b

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Fricas [A]  time = 0.274946, size = 1, normalized size = 0.04 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x

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Sympy [A]  time = 0.136042, size = 0, normalized size = 0. \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x

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GIAC/XCAS [A]  time = 0.275021, size = 27, normalized size = 1.12 \[ x{\rm sign}\left (b x + a\right ) + \frac{a{\rm sign}\left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sign(b*x + a) + a*sign(b*x + a)/b