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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

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 = 0.01, size = 16, normalized size = 0.64 \begin {gather*} -\frac {1}{b \sqrt {(a+b x)^2}} \end {gather*}

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]))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 16, normalized size = 0.64 \begin {gather*} -\frac {1}{b \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*Sqrt[(a + b*x)^2]))

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 13, normalized size = 0.52 \begin {gather*} -\frac {1}{b^{2} x + a b} \end {gather*}

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 0.88 \begin {gather*} -\frac {\left (b x +a \right )^{2}}{\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b} \end {gather*}

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)

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 23, normalized size = 0.92 \begin {gather*} -\frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 2.15, size = 23, normalized size = 0.92 \begin {gather*} -\frac {1}{b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.85, size = 34, normalized size = 1.36 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________