### 3.1547 $$\int \sqrt{a^2+2 a b x+b^2 x^2} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.005164, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, 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*}

Mathematica [A]  time = 0.008513, 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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Maple [A]  time = 0.041, size = 27, normalized size = 0.8 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45479, size = 23, normalized size = 0.72 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

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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Sympy [A]  time = 0.111854, size = 8, normalized size = 0.25 \begin{align*} a x + \frac{b x^{2}}{2} \end{align*}

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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Giac [A]  time = 1.19896, size = 45, normalized size = 1.41 \begin{align*} \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} \end{align*}

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