\(\int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx\) [573]

Optimal result

Integrand size = 15, antiderivative size = 19 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{a \sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, 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} \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{a \sqrt {x}} \]

[In]

Int[1/(x^(3/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(a*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 {2 \sqrt {a+b x}}{a \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{a \sqrt {x}} \]

[In]

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

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[x])

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {x}}\)	\(16\)
default	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {x}}\)	\(16\)
risch	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {x}}\)	\(16\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x + a}}{a \sqrt {x}} \]

[In]

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a} \]

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/a

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x + a}}{a \sqrt {x}} \]

[In]

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {{\left (b x + a\right )} b - a b} a {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx=-\frac {2\,\sqrt {a+b\,x}}{a\,\sqrt {x}} \]

[In]

int(1/(x^(3/2)*(a + b*x)^(1/2)),x)

[Out]

-(2*(a + b*x)^(1/2))/(a*x^(1/2))