3.1600 \(\int \frac {1}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac {1}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 34, 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 = {607} \[ -\frac {1}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(-3/2),x]

[Out]

-1/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 607

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.68 \[ -\frac {a+b x}{2 b \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-3/2),x]

[Out]

-1/2*(a + b*x)/(b*((a + b*x)^2)^(3/2))

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fricas [A]  time = 0.68, size = 24, normalized size = 0.71 \[ -\frac {1}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.04, size = 20, normalized size = 0.59 \[ -\frac {b x +a}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

-1/2*(b*x+a)/b/((b*x+a)^2)^(3/2)

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maxima [A]  time = 1.16, size = 14, normalized size = 0.41 \[ -\frac {1}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2/(b^3*(x + a/b)^2)

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mupad [B]  time = 0.59, size = 30, normalized size = 0.88 \[ -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b\,{\left (a+b\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

-(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)/(2*b*(a + b*x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a**2 + 2*a*b*x + b**2*x**2)**(-3/2), x)

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