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

Optimal. Leaf size=32 \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {609} \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.94 \[ \frac {x \sqrt {(a+b x)^2} (2 a+b x)}{2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.88, size = 10, normalized size = 0.31 \[ \frac {1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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giac [A]  time = 0.17, size = 33, normalized size = 1.03 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{2} \mathrm {sgn}\left (b x + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 27, normalized size = 0.84 \[ \frac {\left (b x +2 a \right ) \sqrt {\left (b x +a \right )^{2}}\, x}{2 b x +2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.33, size = 46, normalized size = 1.44 \[ \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 19, normalized size = 0.59 \[ \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2),x)

[Out]

(((a + b*x)^2)^(1/2)*(a + b*x))/(2*b)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.25 \[ a x + \frac {b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x**2/2

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