3.18.88 \(\int \frac {\sqrt {x}}{(a+\frac {b}{x})^{3/2}} \, dx\) [1788]

Optimal. Leaf size=74 \[ -\frac {16 b^2}{3 a^3 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}} \]

[Out]

2/3*x^(3/2)/a/(a+b/x)^(1/2)-16/3*b^2/a^3/(a+b/x)^(1/2)/x^(1/2)-8/3*b*x^(1/2)/a^2/(a+b/x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 b^2}{3 a^3 \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b/x)^(3/2),x]

[Out]

(-16*b^2)/(3*a^3*Sqrt[a + b/x]*Sqrt[x]) - (8*b*Sqrt[x])/(3*a^2*Sqrt[a + b/x]) + (2*x^(3/2))/(3*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 {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx &=\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}}-\frac {(4 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx}{3 a}\\ &=-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}}+\frac {\left (8 b^2\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{3 a^2}\\ &=-\frac {16 b^2}{3 a^3 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}}\\ \end {align*}

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Mathematica [A]
time = 4.57, size = 48, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} \left (-8 b^2-4 a b x+a^2 x^2\right )}{3 a^3 (b+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b/x)^(3/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-8*b^2 - 4*a*b*x + a^2*x^2))/(3*a^3*(b + a*x))

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Maple [A]
time = 0.03, size = 45, normalized size = 0.61

method	result	size
gosper	\(\frac {2 \left (a x +b \right ) \left (a^{2} x^{2}-4 a b x -8 b^{2}\right )}{3 a^{3} x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\)	\(43\)
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a^{2} x^{2}-4 a b x -8 b^{2}\right )}{3 \left (a x +b \right ) a^{3}}\)	\(45\)
risch	\(\frac {2 \left (a x -5 b \right ) \left (a x +b \right )}{3 a^{3} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}-\frac {2 b^{2}}{a^{3} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((a*x+b)/x)^(1/2)*x^(1/2)*(a^2*x^2-4*a*b*x-8*b^2)/(a*x+b)/a^3

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Maxima [A]
time = 0.29, size = 55, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}} - 6 \, \sqrt {a + \frac {b}{x}} b \sqrt {x}\right )}}{3 \, a^{3}} - \frac {2 \, b^{2}}{\sqrt {a + \frac {b}{x}} a^{3} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((a + b/x)^(3/2)*x^(3/2) - 6*sqrt(a + b/x)*b*sqrt(x))/a^3 - 2*b^2/(sqrt(a + b/x)*a^3*sqrt(x))

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Fricas [A]
time = 0.35, size = 47, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (a^{2} x^{2} - 4 \, a b x - 8 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{4} x + a^{3} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a^2*x^2 - 4*a*b*x - 8*b^2)*sqrt(x)*sqrt((a*x + b)/x)/(a^4*x + a^3*b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (63) = 126\).
time = 1.11, size = 206, normalized size = 2.78 \begin {gather*} \frac {2 a^{3} b^{\frac {9}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac {6 a^{2} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac {24 a b^{\frac {13}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac {16 b^{\frac {15}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*b**(9/2)*x**3*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a**3*b**6) - 6*a**2*b**(11/2)*x**2*
sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a**3*b**6) - 24*a*b**(13/2)*x*sqrt(a*x/b + 1)/(3*a**5*b*
*4*x**2 + 6*a**4*b**5*x + 3*a**3*b**6) - 16*b**(15/2)*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a*
*3*b**6)

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Giac [A]
time = 0.57, size = 46, normalized size = 0.62 \begin {gather*} -\frac {2 \, b^{2}}{\sqrt {a x + b} a^{3}} + \frac {2 \, {\left ({\left (a x + b\right )}^{\frac {3}{2}} a^{6} - 6 \, \sqrt {a x + b} a^{6} b\right )}}{3 \, a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^2/(sqrt(a*x + b)*a^3) + 2/3*((a*x + b)^(3/2)*a^6 - 6*sqrt(a*x + b)*a^6*b)/a^9

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Mupad [B]
time = 1.49, size = 49, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {8\,b\,x^{3/2}}{3\,a^3}-\frac {2\,x^{5/2}}{3\,a^2}+\frac {16\,b^2\,\sqrt {x}}{3\,a^4}\right )}{x+\frac {b}{a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a + b/x)^(3/2),x)

[Out]

-((a + b/x)^(1/2)*((8*b*x^(3/2))/(3*a^3) - (2*x^(5/2))/(3*a^2) + (16*b^2*x^(1/2))/(3*a^4)))/(x + b/a)

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