\(\int \frac {1}{(a+b x^2)^{3/2}} \, dx\) [85]

Optimal result

Integrand size = 11, antiderivative size = 16 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{a \sqrt {a+b x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197} \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{a \sqrt {a+b x^2}} \]

[In]

Int[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{a \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{a \sqrt {a+b x^2}} \]

[In]

Integrate[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
gosper	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)
default	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)
trager	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)
pseudoelliptic	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b x^{2} + a} x}{a b x^{2} + a^{2}} \]

[In]

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

[Out]

sqrt(b*x^2 + a)*x/(a*b*x^2 + a^2)

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \]

[In]

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

[Out]

x/(a**(3/2)*sqrt(1 + b*x**2/a))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {b x^{2} + a} a} \]

[In]

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

[Out]

x/(sqrt(b*x^2 + a)*a)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {b x^{2} + a} a} \]

[In]

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

[Out]

x/(sqrt(b*x^2 + a)*a)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x}{a\,\sqrt {b\,x^2+a}} \]

[In]

int(1/(a + b*x^2)^(3/2),x)

[Out]

x/(a*(a + b*x^2)^(1/2))