Integrand size = 17, antiderivative size = 100 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{8 \sqrt {x}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}}-\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 \sqrt {b}} \] Output:
-5/8*a^2*(a+b/x)^(1/2)/x^(1/2)-5/12*a*(a+b/x)^(3/2)/x^(1/2)-1/3*(a+b/x)^(5 /2)/x^(1/2)-5/8*a^3*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(1/2)
Time = 5.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {\sqrt {b+a x} \left (-8 b^2-26 a b x-33 a^2 x^2\right )}{24 x^3}-\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{8 \sqrt {b}}\right )}{\sqrt {b+a x}} \] Input:
Integrate[(a + b/x)^(5/2)/x^(3/2),x]
Output:
(Sqrt[a + b/x]*Sqrt[x]*((Sqrt[b + a*x]*(-8*b^2 - 26*a*b*x - 33*a^2*x^2))/( 24*x^3) - (5*a^3*ArcTanh[Sqrt[b + a*x]/Sqrt[b]])/(8*Sqrt[b])))/Sqrt[b + a* x]
Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {860, 211, 211, 211, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \left (a+\frac {b}{x}\right )^{5/2}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -2 \left (\frac {5}{6} a \int \left (a+\frac {b}{x}\right )^{3/2}d\frac {1}{\sqrt {x}}+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{6 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -2 \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {a+\frac {b}{x}}d\frac {1}{\sqrt {x}}+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{6 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -2 \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{6 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{6 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{2 \sqrt {b}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{6 \sqrt {x}}\right )\) |
Input:
Int[(a + b/x)^(5/2)/x^(3/2),x]
Output:
-2*((a + b/x)^(5/2)/(6*Sqrt[x]) + (5*a*((a + b/x)^(3/2)/(4*Sqrt[x]) + (3*a *(Sqrt[a + b/x]/(2*Sqrt[x]) + (a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]) /(2*Sqrt[b])))/4))/6)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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_)^(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 = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\left (33 a^{2} x^{2}+26 a b x +8 b^{2}\right ) \sqrt {\frac {a x +b}{x}}}{24 x^{\frac {5}{2}}}-\frac {5 a^{3} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{8 \sqrt {b}\, \sqrt {a x +b}}\) | \(78\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (15 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{3} x^{3}+33 a^{2} x^{2} \sqrt {a x +b}\, \sqrt {b}+26 a \,b^{\frac {3}{2}} x \sqrt {a x +b}+8 \sqrt {a x +b}\, b^{\frac {5}{2}}\right )}{24 x^{\frac {5}{2}} \sqrt {a x +b}\, \sqrt {b}}\) | \(92\) |
Input:
int((a+b/x)^(5/2)/x^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/24*(33*a^2*x^2+26*a*b*x+8*b^2)/x^(5/2)*((a*x+b)/x)^(1/2)-5/8*a^3/b^(1/2 )*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 = 178, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=\left [\frac {15 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (33 \, a^{2} b x^{2} + 26 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b x^{3}}, \frac {15 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (33 \, a^{2} b x^{2} + 26 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b x^{3}}\right ] \] Input:
integrate((a+b/x)^(5/2)/x^(3/2),x, algorithm="fricas")
Output:
[1/48*(15*a^3*sqrt(b)*x^3*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) - 2*(33*a^2*b*x^2 + 26*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x) )/(b*x^3), 1/24*(15*a^3*sqrt(-b)*x^3*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b )/x)/(a*x + b)) - (33*a^2*b*x^2 + 26*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b*x^3)]
Time = 7.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=- \frac {11 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}}{8 \sqrt {x}} - \frac {13 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x}}}{12 x^{\frac {3}{2}}} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x}}}{3 x^{\frac {5}{2}}} - \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 \sqrt {b}} \] Input:
integrate((a+b/x)**(5/2)/x**(3/2),x)
Output:
-11*a**(5/2)*sqrt(1 + b/(a*x))/(8*sqrt(x)) - 13*a**(3/2)*b*sqrt(1 + b/(a*x ))/(12*x**(3/2)) - sqrt(a)*b**2*sqrt(1 + b/(a*x))/(3*x**(5/2)) - 5*a**3*as inh(sqrt(b)/(sqrt(a)*sqrt(x)))/(8*sqrt(b))
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (72) = 144\).
Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=\frac {5 \, a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, \sqrt {b}} - \frac {33 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} - 40 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} + 15 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{2} x - b^{3}\right )}} \] Input:
integrate((a+b/x)^(5/2)/x^(3/2),x, algorithm="maxima")
Output:
5/16*a^3*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sq rt(b)))/sqrt(b) - 1/24*(33*(a + b/x)^(5/2)*a^3*x^(5/2) - 40*(a + b/x)^(3/2 )*a^3*b*x^(3/2) + 15*sqrt(a + b/x)*a^3*b^2*sqrt(x))/((a + b/x)^3*x^3 - 3*( a + b/x)^2*b*x^2 + 3*(a + b/x)*b^2*x - b^3)
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=\frac {1}{24} \, {\left (\frac {15 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} - \frac {33 \, {\left (a x + b\right )}^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - 40 \, {\left (a x + b\right )}^{\frac {3}{2}} b \mathrm {sgn}\left (x\right ) + 15 \, \sqrt {a x + b} b^{2} \mathrm {sgn}\left (x\right )}{a^{3} x^{3}}\right )} a^{3} \] Input:
integrate((a+b/x)^(5/2)/x^(3/2),x, algorithm="giac")
Output:
1/24*(15*arctan(sqrt(a*x + b)/sqrt(-b))*sgn(x)/sqrt(-b) - (33*(a*x + b)^(5 /2)*sgn(x) - 40*(a*x + b)^(3/2)*b*sgn(x) + 15*sqrt(a*x + b)*b^2*sgn(x))/(a ^3*x^3))*a^3
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^{5/2}}{x^{3/2}} \,d x \] Input:
int((a + b/x)^(5/2)/x^(3/2),x)
Output:
int((a + b/x)^(5/2)/x^(3/2), x)
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{3/2}} \, dx=\frac {-66 \sqrt {a x +b}\, a^{2} b \,x^{2}-52 \sqrt {a x +b}\, a \,b^{2} x -16 \sqrt {a x +b}\, b^{3}+15 \sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) a^{3} x^{3}-15 \sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) a^{3} x^{3}}{48 b \,x^{3}} \] Input:
int((a+b/x)^(5/2)/x^(3/2),x)
Output:
( - 66*sqrt(a*x + b)*a**2*b*x**2 - 52*sqrt(a*x + b)*a*b**2*x - 16*sqrt(a*x + b)*b**3 + 15*sqrt(b)*log(sqrt(a*x + b) - sqrt(b))*a**3*x**3 - 15*sqrt(b )*log(sqrt(a*x + b) + sqrt(b))*a**3*x**3)/(48*b*x**3)