3.18.89 \(\int \frac {1}{(a+\frac {b}{x})^{3/2} \sqrt {x}} \, dx\) [1789]

Optimal. Leaf size=44 \[ \frac {4 b}{a^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \begin {gather*} \frac {4 b}{a^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*Sqrt[x]),x]

[Out]

(4*b)/(a^2*Sqrt[a + b/x]*Sqrt[x]) + (2*Sqrt[x])/(a*Sqrt[a + b/x])

Rule 270

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

Rule 277

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx &=\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}}-\frac {(2 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{a}\\ &=\frac {4 b}{a^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}}\\ \end {align*}

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Mathematica [A]
time = 4.51, size = 35, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (2 b+a x)}{a^2 (b+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(2*b + a*x))/(a^2*(b + a*x))

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Maple [A]
time = 0.03, size = 34, normalized size = 0.77

method	result	size
gosper	\(\frac {2 \left (a x +b \right ) \left (a x +2 b \right )}{a^{2} x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\)	\(32\)
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a x +2 b \right )}{\left (a x +b \right ) a^{2}}\)	\(34\)
risch	\(\frac {2 a x +2 b}{a^{2} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}+\frac {2 b}{a^{2} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 36, normalized size = 0.82 \begin {gather*} \frac {2 \, \sqrt {a + \frac {b}{x}} \sqrt {x}}{a^{2}} + \frac {2 \, b}{\sqrt {a + \frac {b}{x}} a^{2} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 36, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (a x + 2 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a^{3} x + a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.40, size = 39, normalized size = 0.89 \begin {gather*} \frac {2 x}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a x}{b} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(a*sqrt(b)*sqrt(a*x/b + 1)) + 4*sqrt(b)/(a**2*sqrt(a*x/b + 1))

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Giac [A]
time = 1.43, size = 29, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {a x + b}}{a} + \frac {b}{\sqrt {a x + b} a}\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(a*x + b)/a + b/(sqrt(a*x + b)*a))/a

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Mupad [B]
time = 1.44, size = 37, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{3/2}}{a^2}+\frac {4\,b\,\sqrt {x}}{a^3}\right )}{x+\frac {b}{a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b/x)^(3/2)),x)

[Out]

((a + b/x)^(1/2)*((2*x^(3/2))/a^2 + (4*b*x^(1/2))/a^3))/(x + b/a)

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