\(\int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\) [612]

Optimal result

Integrand size = 15, antiderivative size = 16 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {2+b x}}{\sqrt {x}} \]

[Out]

-(b*x+2)^(1/2)/x^(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {b x+2}}{\sqrt {x}} \]

[In]

Int[1/(x^(3/2)*Sqrt[2 + b*x]),x]

[Out]

-(Sqrt[2 + b*x]/Sqrt[x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {2+b x}}{\sqrt {x}} \]

[In]

Integrate[1/(x^(3/2)*Sqrt[2 + b*x]),x]

[Out]

-(Sqrt[2 + b*x]/Sqrt[x])

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(-\frac {\sqrt {b x +2}}{\sqrt {x}}\)	\(13\)
default	\(-\frac {\sqrt {b x +2}}{\sqrt {x}}\)	\(13\)
risch	\(-\frac {\sqrt {b x +2}}{\sqrt {x}}\)	\(13\)
meijerg	\(-\frac {\sqrt {2}\, \sqrt {\frac {b x}{2}+1}}{\sqrt {x}}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {b x + 2}}{\sqrt {x}} \]

[In]

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

[Out]

-sqrt(b*x + 2)/sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=- \sqrt {b} \sqrt {1 + \frac {2}{b x}} \]

[In]

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

[Out]

-sqrt(b)*sqrt(1 + 2/(b*x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {b x + 2}}{\sqrt {x}} \]

[In]

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

[Out]

-sqrt(b*x + 2)/sqrt(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {b x + 2} b^{2}}{\sqrt {{\left (b x + 2\right )} b - 2 \, b} {\left | b \right |}} \]

[In]

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

[Out]

-sqrt(b*x + 2)*b^2/(sqrt((b*x + 2)*b - 2*b)*abs(b))

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {b\,x+2}}{\sqrt {x}} \]

[In]

int(1/(x^(3/2)*(b*x + 2)^(1/2)),x)

[Out]

-(b*x + 2)^(1/2)/x^(1/2)