\(\int \frac {(a+\frac {b}{x})^{3/2}}{x^{5/2}} \, dx\) [1767]

Optimal result

Integrand size = 17, antiderivative size = 103 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {344, 285, 327, 223, 212} \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}} \]

[In]

Int[(a + b/x)^(3/2)/x^(5/2),x]

[Out]

-1/4*(a*Sqrt[a + b/x])/x^(3/2) - (a + b/x)^(3/2)/(3*x^(3/2)) - (a^2*Sqrt[a + b/x])/(8*b*Sqrt[x]) + (a^3*ArcTan
h[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*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 285

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

Rule 327

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*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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*} \text {integral}& = -\left (2 \text {Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-a \text {Subst}\left (\int x^2 \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {1}{4} a^2 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{8 b} \\ & = -\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b} \\ & = -\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {\sqrt {b+a x} \left (-8 b^2-14 a b x-3 a^2 x^2\right )}{24 b x^3}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{8 b^{3/2}}\right )}{\sqrt {b+a x}} \]

[In]

Integrate[(a + b/x)^(3/2)/x^(5/2),x]

[Out]

(Sqrt[a + b/x]*Sqrt[x]*((Sqrt[b + a*x]*(-8*b^2 - 14*a*b*x - 3*a^2*x^2))/(24*b*x^3) + (a^3*ArcTanh[Sqrt[b + a*x
]/Sqrt[b]])/(8*b^(3/2))))/Sqrt[b + a*x]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {\left (3 a^{2} x^{2}+14 a b x +8 b^{2}\right ) \sqrt {\frac {a x +b}{x}}}{24 x^{\frac {5}{2}} b}+\frac {a^{3} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{8 b^{\frac {3}{2}} \sqrt {a x +b}}\)	\(81\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{3} x^{3}+8 b^{\frac {5}{2}} \sqrt {a x +b}+14 a \,b^{\frac {3}{2}} x \sqrt {a x +b}+3 a^{2} x^{2} \sqrt {b}\, \sqrt {a x +b}\right )}{24 x^{\frac {5}{2}} b^{\frac {3}{2}} \sqrt {a x +b}}\)	\(92\)

[In]

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

[Out]

-1/24*(3*a^2*x^2+14*a*b*x+8*b^2)/x^(5/2)/b*((a*x+b)/x)^(1/2)+1/8/b^(3/2)*a^3*arctanh((a*x+b)^(1/2)/b^(1/2))*((
a*x+b)/x)^(1/2)/(a*x+b)^(1/2)*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=\left [\frac {3 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (3 \, a^{2} b x^{2} + 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b^{2} x^{3}}, -\frac {3 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a^{2} b x^{2} + 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b^{2} x^{3}}\right ] \]

[In]

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

[Out]

[1/48*(3*a^3*sqrt(b)*x^3*log((a*x + 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) - 2*(3*a^2*b*x^2 + 14*a*b^2*
x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b^2*x^3), -1/24*(3*a^3*sqrt(-b)*x^3*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x +
b)/x)/b) + (3*a^2*b*x^2 + 14*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b^2*x^3)]

Sympy [A] (verification not implemented)

Time = 5.52 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=- \frac {a^{\frac {5}{2}}}{8 b \sqrt {x} \sqrt {1 + \frac {b}{a x}}} - \frac {17 a^{\frac {3}{2}}}{24 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {11 \sqrt {a} b}{12 x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} + \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 b^{\frac {3}{2}}} - \frac {b^{2}}{3 \sqrt {a} x^{\frac {7}{2}} \sqrt {1 + \frac {b}{a x}}} \]

[In]

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

[Out]

-a**(5/2)/(8*b*sqrt(x)*sqrt(1 + b/(a*x))) - 17*a**(3/2)/(24*x**(3/2)*sqrt(1 + b/(a*x))) - 11*sqrt(a)*b/(12*x**
(5/2)*sqrt(1 + b/(a*x))) + a**3*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(8*b**(3/2)) - b**2/(3*sqrt(a)*x**(7/2)*sqrt(
1 + b/(a*x)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (75) = 150\).

Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=-\frac {a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} + 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} - 3 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} b x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b^{2} x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{3} x - b^{4}\right )}} \]

[In]

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

[Out]

-1/16*a^3*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/b^(3/2) - 1/24*(3*(a + b/x)
^(5/2)*a^3*x^(5/2) + 8*(a + b/x)^(3/2)*a^3*b*x^(3/2) - 3*sqrt(a + b/x)*a^3*b^2*sqrt(x))/((a + b/x)^3*b*x^3 - 3
*(a + b/x)^2*b^2*x^2 + 3*(a + b/x)*b^3*x - b^4)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=-\frac {\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{4} + 8 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} b - 3 \, \sqrt {a x + b} a^{4} b^{2}}{a^{3} b x^{3}}}{24 \, a} \]

[In]

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

[Out]

-1/24*(3*a^4*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b) + (3*(a*x + b)^(5/2)*a^4 + 8*(a*x + b)^(3/2)*a^4*b -
3*sqrt(a*x + b)*a^4*b^2)/(a^3*b*x^3))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^{3/2}}{x^{5/2}} \,d x \]

[In]

int((a + b/x)^(3/2)/x^(5/2),x)

[Out]

int((a + b/x)^(3/2)/x^(5/2), x)