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]

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

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.87, size = 1, normalized size = 0.04 \[ x \]

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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giac [A]  time = 0.15, size = 20, normalized size = 0.83 \[ x \mathrm {sgn}\left (b x + a\right ) + \frac {a \mathrm {sgn}\left (b x + a\right )}{b} \]

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

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maple [A]  time = 0.05, size = 17, normalized size = 0.71 \[ \frac {\left (b x +a \right ) x}{\sqrt {\left (b x +a \right )^{2}}} \]

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 = 0.55, size = 13, normalized size = 0.54 \[ \frac {\sqrt {{\left (b x + a\right )}^{2}}}{b} \]

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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mupad [B]  time = 2.38, size = 76, normalized size = 3.17 \[ \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b}+\frac {a\,\ln \left (a+b\,x+\sqrt {{\left (a+b\,x\right )}^2}\right )}{b}-\frac {a\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{\sqrt {b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/((a + b*x)^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.08, size = 0, normalized size = 0.00 \[ x \]

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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