3.2029 \(\int \frac{a+b x}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0080638, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ -\frac{1}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0083225, size = 16, normalized size = 0.64 \[ -\frac{1}{b \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*Sqrt[(a + b*x)^2]))

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Maple [A]  time = 0.005, size = 22, normalized size = 0.9 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{2}}{b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965342, size = 31, normalized size = 1.24 \begin{align*} -\frac{1}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b)

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Fricas [A]  time = 1.49737, size = 24, normalized size = 0.96 \begin{align*} -\frac{1}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x + a*b)

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Sympy [A]  time = 0.780296, size = 34, normalized size = 1.36 \begin{align*} \begin{cases} - \frac{1}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} & \text{for}\: b \neq 0 \\\frac{a x}{\left (a^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(b*sqrt(a**2 + 2*a*b*x + b**2*x**2)), Ne(b, 0)), (a*x/(a**2)**(3/2), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef