Integrand size = 17, antiderivative size = 80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}}{2 x^{3/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{4 b \sqrt {x}}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^{3/2}} \] Output:
-1/2*(a+b/x)^(1/2)/x^(3/2)-1/4*a*(a+b/x)^(1/2)/b/x^(1/2)+1/4*a^2*arctanh(b ^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(3/2)
Time = 4.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {(-2 b-a x) \sqrt {b+a x}}{4 b x^2}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{4 b^{3/2}}\right )}{\sqrt {b+a x}} \] Input:
Integrate[Sqrt[a + b/x]/x^(5/2),x]
Output:
(Sqrt[a + b/x]*Sqrt[x]*(((-2*b - a*x)*Sqrt[b + a*x])/(4*b*x^2) + (a^2*ArcT anh[Sqrt[b + a*x]/Sqrt[b]])/(4*b^(3/2))))/Sqrt[b + a*x]
Time = 0.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {860, 248, 262, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \frac {\sqrt {a+\frac {b}{x}}}{x}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 248 |
\(\displaystyle -2 \left (\frac {1}{4} a \int \frac {1}{\sqrt {a+\frac {b}{x}} x}d\frac {1}{\sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{4 x^{3/2}}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -2 \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x}}}{2 b \sqrt {x}}-\frac {a \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}}{2 b}\right )+\frac {\sqrt {a+\frac {b}{x}}}{4 x^{3/2}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x}}}{2 b \sqrt {x}}-\frac {a \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}}{2 b}\right )+\frac {\sqrt {a+\frac {b}{x}}}{4 x^{3/2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x}}}{2 b \sqrt {x}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{2 b^{3/2}}\right )+\frac {\sqrt {a+\frac {b}{x}}}{4 x^{3/2}}\right )\) |
Input:
Int[Sqrt[a + b/x]/x^(5/2),x]
Output:
-2*(Sqrt[a + b/x]/(4*x^(3/2)) + (a*(Sqrt[a + b/x]/(2*b*Sqrt[x]) - (a*ArcTa nh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(2*b^(3/2))))/4)
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[-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]
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-\frac {\left (a x +2 b \right ) \sqrt {\frac {a x +b}{x}}}{4 x^{\frac {3}{2}} b}+\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{4 b^{\frac {3}{2}} \sqrt {a x +b}}\) | \(69\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{2} x^{2}+2 \sqrt {a x +b}\, b^{\frac {3}{2}}+a x \sqrt {a x +b}\, \sqrt {b}\right )}{4 x^{\frac {3}{2}} b^{\frac {3}{2}} \sqrt {a x +b}}\) | \(73\) |
Input:
int((a+b/x)^(1/2)/x^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(a*x+2*b)/x^(3/2)/b*((a*x+b)/x)^(1/2)+1/4/b^(3/2)*a^2*arctanh((a*x+b) ^(1/2)/b^(1/2))*((a*x+b)/x)^(1/2)/(a*x+b)^(1/2)*x^(1/2)
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=\left [\frac {a^{2} \sqrt {b} x^{2} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (a b x + 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{8 \, b^{2} x^{2}}, -\frac {a^{2} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (a b x + 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{4 \, b^{2} x^{2}}\right ] \] Input:
integrate((a+b/x)^(1/2)/x^(5/2),x, algorithm="fricas")
Output:
[1/8*(a^2*sqrt(b)*x^2*log((a*x + 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b )/x) - 2*(a*b*x + 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(b^2*x^2), -1/4*(a^2*s qrt(-b)*x^2*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/(a*x + b)) + (a*b*x + 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(b^2*x^2)]
Time = 4.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=- \frac {a^{\frac {3}{2}}}{4 b \sqrt {x} \sqrt {1 + \frac {b}{a x}}} - \frac {3 \sqrt {a}}{4 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} + \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{4 b^{\frac {3}{2}}} - \frac {b}{2 \sqrt {a} x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} \] Input:
integrate((a+b/x)**(1/2)/x**(5/2),x)
Output:
-a**(3/2)/(4*b*sqrt(x)*sqrt(1 + b/(a*x))) - 3*sqrt(a)/(4*x**(3/2)*sqrt(1 + b/(a*x))) + a**2*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(4*b**(3/2)) - b/(2*sqr t(a)*x**(5/2)*sqrt(1 + b/(a*x)))
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (58) = 116\).
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=-\frac {a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{8 \, b^{\frac {3}{2}}} - \frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} x^{\frac {3}{2}} + \sqrt {a + \frac {b}{x}} a^{2} b \sqrt {x}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} b x^{2} - 2 \, {\left (a + \frac {b}{x}\right )} b^{2} x + b^{3}\right )}} \] Input:
integrate((a+b/x)^(1/2)/x^(5/2),x, algorithm="maxima")
Output:
-1/8*a^2*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sq rt(b)))/b^(3/2) - 1/4*((a + b/x)^(3/2)*a^2*x^(3/2) + sqrt(a + b/x)*a^2*b*s qrt(x))/((a + b/x)^2*b*x^2 - 2*(a + b/x)*b^2*x + b^3)
Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=-\frac {\frac {a^{3} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {{\left (a x + b\right )}^{\frac {3}{2}} a^{3} \mathrm {sgn}\left (x\right ) + \sqrt {a x + b} a^{3} b \mathrm {sgn}\left (x\right )}{a^{2} b x^{2}}}{4 \, a} \] Input:
integrate((a+b/x)^(1/2)/x^(5/2),x, algorithm="giac")
Output:
-1/4*(a^3*arctan(sqrt(a*x + b)/sqrt(-b))*sgn(x)/(sqrt(-b)*b) + ((a*x + b)^ (3/2)*a^3*sgn(x) + sqrt(a*x + b)*a^3*b*sgn(x))/(a^2*b*x^2))/a
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \,d x \] Input:
int((a + b/x)^(1/2)/x^(5/2),x)
Output:
int((a + b/x)^(1/2)/x^(5/2), x)
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{5/2}} \, dx=\frac {-2 \sqrt {a x +b}\, a b x -4 \sqrt {a x +b}\, b^{2}-\sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) a^{2} x^{2}+\sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) a^{2} x^{2}}{8 b^{2} x^{2}} \] Input:
int((a+b/x)^(1/2)/x^(5/2),x)
Output:
( - 2*sqrt(a*x + b)*a*b*x - 4*sqrt(a*x + b)*b**2 - sqrt(b)*log(sqrt(a*x + b) - sqrt(b))*a**2*x**2 + sqrt(b)*log(sqrt(a*x + b) + sqrt(b))*a**2*x**2)/ (8*b**2*x**2)