\(\int \frac {a+b x}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\) [2029]

Optimal result

Integrand size = 26, antiderivative size = 25 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {643} \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \]

[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 643

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*} \text {integral}& = -\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b \sqrt {(a+b x)^2}} \]

[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]))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(-\frac {\operatorname {csgn}\left (b x +a \right )}{b \left (b x +a \right )}\)	\(19\)
gosper	\(-\frac {\left (b x +a \right )^{2}}{b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\)	\(22\)
default	\(-\frac {\left (b x +a \right )^{2}}{b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\)	\(22\)
risch	\(-\frac {\sqrt {\left (b x +a \right )^{2}}}{\left (b x +a \right )^{2} b}\)	\(22\)

[In]

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

[Out]

-1/b*csgn(b*x+a)/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b^{2} x + a b} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {1}{b \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}} & \text {for}\: b \neq 0 \\\frac {a x}{\left (a^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{{\left (b x + a\right )} b \mathrm {sgn}\left (b x + a\right )} \]

[In]

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

[Out]

-1/((b*x + a)*b*sgn(b*x + a))

Mupad [B] (verification not implemented)

Time = 10.85 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}} \]

[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))