3.18.91 \(\int \frac {1}{(a+\frac {b}{x})^{3/2} x^{5/2}} \, dx\) [1791]

Optimal. Leaf size=52 \[ \frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {344, 294, 223, 212} \begin {gather*} \frac {2}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^(5/2)),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 294

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

Rule 344

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[-k/c, Subst[
Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n,
0] && FractionQ[m]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{5/2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 4.62, size = 64, normalized size = 1.23 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {2}{b \sqrt {b+a x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{b^{3/2}}\right )}{\sqrt {b+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*x^(5/2)),x]

[Out]

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

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Maple [A]
time = 0.03, size = 53, normalized size = 1.02

method	result	size
default	\(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (\arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}-\sqrt {b}\right )}{b^{\frac {3}{2}} \left (a x +b \right )}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x*b)^(3/2)/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 62, normalized size = 1.19 \begin {gather*} \frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} b \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 138, normalized size = 2.65 \begin {gather*} \left [\frac {{\left (a x + b\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a b^{2} x + b^{3}}, \frac {2 \, {\left ({\left (a x + b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + b \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{a b^{2} x + b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

[((a*x + b)*sqrt(b)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2*b*sqrt(x)*sqrt((a*x + b)/x))/
(a*b^2*x + b^3), 2*((a*x + b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + b*sqrt(x)*sqrt((a*x + b)
/x))/(a*b^2*x + b^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (42) = 84\).
time = 6.55, size = 146, normalized size = 2.81 \begin {gather*} \frac {a b^{2} x \log {\left (\frac {a x}{b} \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} - \frac {2 a b^{2} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} + \frac {2 b^{3} \sqrt {\frac {a x}{b} + 1}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} + \frac {b^{3} \log {\left (\frac {a x}{b} \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} - \frac {2 b^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2)/x**(5/2),x)

[Out]

a*b**2*x*log(a*x/b)/(a*b**(7/2)*x + b**(9/2)) - 2*a*b**2*x*log(sqrt(a*x/b + 1) + 1)/(a*b**(7/2)*x + b**(9/2))
+ 2*b**3*sqrt(a*x/b + 1)/(a*b**(7/2)*x + b**(9/2)) + b**3*log(a*x/b)/(a*b**(7/2)*x + b**(9/2)) - 2*b**3*log(sq
rt(a*x/b + 1) + 1)/(a*b**(7/2)*x + b**(9/2))

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Giac [A]
time = 1.27, size = 37, normalized size = 0.71 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {2}{\sqrt {a x + b} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^{5/2}\,{\left (a+\frac {b}{x}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b/x)^(3/2)),x)

[Out]

int(1/(x^(5/2)*(a + b/x)^(3/2)), x)

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