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 = 0.0071298, 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, Rules used = {629} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b} \]

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

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0045291, 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 = 0.003, size = 17, normalized size = 0.7 \begin{align*}{ \left ( bx+a \right ) x{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

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 = 0.978684, size = 18, normalized size = 0.75 \begin{align*} \frac{\sqrt{{\left (b x + a\right )}^{2}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

sqrt((b*x + a)^2)/b

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Fricas [A]  time = 1.47548, size = 4, normalized size = 0.17 \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

x

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Sympy [A]  time = 0.079097, size = 0, normalized size = 0. \begin{align*} x \end{align*}

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 [A]  time = 1.12489, size = 27, normalized size = 1.12 \begin{align*} x \mathrm{sgn}\left (b x + a\right ) + \frac{a \mathrm{sgn}\left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

x*sgn(b*x + a) + a*sgn(b*x + a)/b