Integrand size = 17, antiderivative size = 77 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=-\frac {3 a \sqrt {a+\frac {b}{x}}}{4 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{2 \sqrt {x}}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 \sqrt {b}} \] Output:
-3/4*a*(a+b/x)^(1/2)/x^(1/2)-1/2*(a+b/x)^(3/2)/x^(1/2)-3/4*a^2*arctanh(b^( 1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(1/2)
Time = 4.80 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {(-2 b-5 a x) \sqrt {b+a x}}{4 x^2}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{4 \sqrt {b}}\right )}{\sqrt {b+a x}} \] Input:
Integrate[(a + b/x)^(3/2)/x^(3/2),x]
Output:
(Sqrt[a + b/x]*Sqrt[x]*(((-2*b - 5*a*x)*Sqrt[b + a*x])/(4*x^2) - (3*a^2*Ar cTanh[Sqrt[b + a*x]/Sqrt[b]])/(4*Sqrt[b])))/Sqrt[b + a*x]
Time = 0.30 (sec) , antiderivative size = 82, 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, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \left (a+\frac {b}{x}\right )^{3/2}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -2 \left (\frac {3}{4} a \int \sqrt {a+\frac {b}{x}}d\frac {1}{\sqrt {x}}+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -2 \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{2 \sqrt {b}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{4 \sqrt {x}}\right )\) |
Input:
Int[(a + b/x)^(3/2)/x^(3/2),x]
Output:
-2*((a + b/x)^(3/2)/(4*Sqrt[x]) + (3*a*(Sqrt[a + b/x]/(2*Sqrt[x]) + (a*Arc Tanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(2*Sqrt[b])))/4)
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 = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\left (5 a x +2 b \right ) \sqrt {\frac {a x +b}{x}}}{4 x^{\frac {3}{2}}}-\frac {3 a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{4 \sqrt {b}\, \sqrt {a x +b}}\) | \(67\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{2} x^{2}+5 a x \sqrt {a x +b}\, \sqrt {b}+2 \sqrt {a x +b}\, b^{\frac {3}{2}}\right )}{4 x^{\frac {3}{2}} \sqrt {a x +b}\, \sqrt {b}}\) | \(74\) |
Input:
int((a+b/x)^(3/2)/x^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(5*a*x+2*b)/x^(3/2)*((a*x+b)/x)^(1/2)-3/4*a^2/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 = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} x^{2} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (5 \, a b x + 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{8 \, b x^{2}}, \frac {3 \, a^{2} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (5 \, a b x + 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{4 \, b x^{2}}\right ] \] Input:
integrate((a+b/x)^(3/2)/x^(3/2),x, algorithm="fricas")
Output:
[1/8*(3*a^2*sqrt(b)*x^2*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2 *b)/x) - 2*(5*a*b*x + 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(b*x^2), 1/4*(3*a^ 2*sqrt(-b)*x^2*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/(a*x + b)) - (5*a *b*x + 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(b*x^2)]
Time = 3.00 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=- \frac {5 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}}{4 \sqrt {x}} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x}}}{2 x^{\frac {3}{2}}} - \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{4 \sqrt {b}} \] Input:
integrate((a+b/x)**(3/2)/x**(3/2),x)
Output:
-5*a**(3/2)*sqrt(1 + b/(a*x))/(4*sqrt(x)) - sqrt(a)*b*sqrt(1 + b/(a*x))/(2 *x**(3/2)) - 3*a**2*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(4*sqrt(b))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (55) = 110\).
Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=\frac {3 \, a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{8 \, \sqrt {b}} - \frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} x^{\frac {3}{2}} - 3 \, \sqrt {a + \frac {b}{x}} a^{2} b \sqrt {x}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} x^{2} - 2 \, {\left (a + \frac {b}{x}\right )} b x + b^{2}\right )}} \] Input:
integrate((a+b/x)^(3/2)/x^(3/2),x, algorithm="maxima")
Output:
3/8*a^2*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqr t(b)))/sqrt(b) - 1/4*(5*(a + b/x)^(3/2)*a^2*x^(3/2) - 3*sqrt(a + b/x)*a^2* b*sqrt(x))/((a + b/x)^2*x^2 - 2*(a + b/x)*b*x + b^2)
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=\frac {\frac {3 \, a^{3} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} - \frac {5 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{3} \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {a x + b} a^{3} b \mathrm {sgn}\left (x\right )}{a^{2} x^{2}}}{4 \, a} \] Input:
integrate((a+b/x)^(3/2)/x^(3/2),x, algorithm="giac")
Output:
1/4*(3*a^3*arctan(sqrt(a*x + b)/sqrt(-b))*sgn(x)/sqrt(-b) - (5*(a*x + b)^( 3/2)*a^3*sgn(x) - 3*sqrt(a*x + b)*a^3*b*sgn(x))/(a^2*x^2))/a
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^{3/2}}{x^{3/2}} \,d x \] Input:
int((a + b/x)^(3/2)/x^(3/2),x)
Output:
int((a + b/x)^(3/2)/x^(3/2), x)
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{3/2}} \, dx=\frac {-10 \sqrt {a x +b}\, a b x -4 \sqrt {a x +b}\, b^{2}+3 \sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) a^{2} x^{2}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) a^{2} x^{2}}{8 b \,x^{2}} \] Input:
int((a+b/x)^(3/2)/x^(3/2),x)
Output:
( - 10*sqrt(a*x + b)*a*b*x - 4*sqrt(a*x + b)*b**2 + 3*sqrt(b)*log(sqrt(a*x + b) - sqrt(b))*a**2*x**2 - 3*sqrt(b)*log(sqrt(a*x + b) + sqrt(b))*a**2*x **2)/(8*b*x**2)