\(\int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx\) [580]

Optimal result

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

[Out]

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

[In]

Int[1/(Sqrt[x]*(a + b*x)^(3/2)),x]

[Out]

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

Mathematica [A] (verified)

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

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a + b*x])

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} \sqrt {x}}{a b x + a^{2}} \]

[In]

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

[Out]

2*sqrt(b*x + a)*sqrt(x)/(a*b*x + a^2)

Sympy [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx=\frac {2}{a \sqrt {b} \sqrt {\frac {a}{b x} + 1}} \]

[In]

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

[Out]

2/(a*sqrt(b)*sqrt(a/(b*x) + 1))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx=\frac {2 \, \sqrt {x}}{\sqrt {b x + a} a} \]

[In]

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

[Out]

2*sqrt(x)/(sqrt(b*x + a)*a)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (15) = 30\).

Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.37 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx=\frac {4 \, b^{\frac {3}{2}}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} {\left | b \right |}} \]

[In]

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

[Out]

4*b^(3/2)/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*abs(b))

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx=\frac {2\,\sqrt {x}\,\sqrt {a+b\,x}}{a^2+b\,x\,a} \]

[In]

int(1/(x^(1/2)*(a + b*x)^(3/2)),x)

[Out]

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