3.3.55 \(\int \frac {1}{(a+\frac {b}{x})^{3/2}} \, dx\) [255]

Optimal. Leaf size=60 \[ \frac {3 b}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \sqrt {a+\frac {b}{x}}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

-3*b*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(5/2)+3*b/a^2/(a+b/x)^(1/2)+x/a/(a+b/x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {248, 44, 53, 65, 214} \begin {gather*} -\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {3 b}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \sqrt {a+\frac {b}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(-3/2),x]

[Out]

(3*b)/(a^2*Sqrt[a + b/x]) + x/(a*Sqrt[a + b/x]) - (3*b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)

Rule 44

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] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 248

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 57, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}} x (3 b+a x)}{a^2 (b+a x)}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(-3/2),x]

[Out]

(Sqrt[a + b/x]*x*(3*b + a*x))/(a^2*(b + a*x)) - (3*b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(50)=100\).
time = 0.03, size = 198, normalized size = 3.30

method	result	size
risch	\(\frac {a x +b}{a^{2} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {3 b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{2 a^{\frac {5}{2}}}+\frac {2 b \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a^{3} \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\)	\(116\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (-6 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} x^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b \,x^{2}+4 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-12 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b x +6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} x -6 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3}\right )}{2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{2}}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a*x+b)/x)^(1/2)*x/a^(5/2)*(-6*(x*(a*x+b))^(1/2)*a^(5/2)*x^2+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x
+b)/a^(1/2))*a^2*b*x^2+4*a^(3/2)*(x*(a*x+b))^(3/2)-12*(x*(a*x+b))^(1/2)*a^(3/2)*b*x+6*ln(1/2*(2*(x*(a*x+b))^(1
/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^2*x-6*(x*(a*x+b))^(1/2)*a^(1/2)*b^2+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*b^3)/(x*(a*x+b))^(1/2)/(a*x+b)^2

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Maxima [A]
time = 0.54, size = 85, normalized size = 1.42 \begin {gather*} \frac {3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, a^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*(a + b/x)*b - 2*a*b)/((a + b/x)^(3/2)*a^2 - sqrt(a + b/x)*a^3) + 3/2*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(
a + b/x) + sqrt(a)))/a^(5/2)

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Fricas [A]
time = 2.10, size = 156, normalized size = 2.60 \begin {gather*} \left [\frac {3 \, {\left (a b x + b^{2}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {3 \, {\left (a b x + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*(a*b*x + b^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(a^2*x^2 + 3*a*b*x)*sqrt((a*x
+ b)/x))/(a^4*x + a^3*b), (3*(a*b*x + b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (a^2*x^2 + 3*a*b*x
)*sqrt((a*x + b)/x))/(a^4*x + a^3*b)]

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Sympy [A]
time = 1.73, size = 71, normalized size = 1.18 \begin {gather*} \frac {x^{\frac {3}{2}}}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} \sqrt {x}}{a^{2} \sqrt {\frac {a x}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(3/2)/(a*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*sqrt(x)/(a**2*sqrt(a*x/b + 1)) - 3*b*asinh(sqrt(a)*sqrt(x)/sq
rt(b))/a**(5/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (50) = 100\).
time = 1.72, size = 118, normalized size = 1.97 \begin {gather*} -\frac {{\left (3 \, b \log \left ({\left | b \right |}\right ) + 4 \, b\right )} \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {5}{2}}} + \frac {3 \, b \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, b^{2}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a + \sqrt {a} b\right )} a^{2} \mathrm {sgn}\left (x\right )} + \frac {\sqrt {a x^{2} + b x}}{a^{2} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*b*log(abs(b)) + 4*b)*sgn(x)/a^(5/2) + 3/2*b*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))/(
a^(5/2)*sgn(x)) + 2*b^2/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*a + sqrt(a)*b)*a^2*sgn(x)) + sqrt(a*x^2 + b*x)/(a^2*
sgn(x))

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Mupad [B]
time = 1.87, size = 34, normalized size = 0.57 \begin {gather*} \frac {2\,x\,{\left (\frac {a\,x}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,x}{b}\right )}{5\,{\left (a+\frac {b}{x}\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x)^(3/2),x)

[Out]

(2*x*((a*x)/b + 1)^(3/2)*hypergeom([3/2, 5/2], 7/2, -(a*x)/b))/(5*(a + b/x)^(3/2))

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