3.16.47 \(\int \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [1547]

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 = 0.00, 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 = {623} \begin {gather*} \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b} \end {gather*}

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 623

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 \begin {gather*} \frac {x \sqrt {(a+b x)^2} (2 a+b x)}{2 (a+b x)} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 2.
time = 0.15, size = 19, normalized size = 0.59

method	result	size
default	\(\frac {\mathrm {csgn}\left (b x +a \right ) \left (b x +a \right )^{2}}{2 b}\)	\(19\)
gosper	\(\frac {x \left (b x +2 a \right ) \sqrt {\left (b x +a \right )^{2}}}{2 b x +2 a}\)	\(27\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, a x}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b \,x^{2}}{2 b x +2 a}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
time = 0.29, size = 46, normalized size = 1.44 \begin {gather*} \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} \end {gather*}

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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Fricas [A]
time = 4.28, size = 10, normalized size = 0.31 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

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 = 0.01, size = 8, normalized size = 0.25 \begin {gather*} a x + \frac {b x^{2}}{2} \end {gather*}

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 = 1.72, size = 33, normalized size = 1.03 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 0.56, size = 19, normalized size = 0.59 \begin {gather*} \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \end {gather*}

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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