Integrand size = 17, antiderivative size = 73 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\frac {8}{3} b \sqrt {a+\frac {b}{x}} \sqrt {x}+\frac {2}{3} a \sqrt {a+\frac {b}{x}} x^{3/2}-2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \] Output:
8/3*b*(a+b/x)^(1/2)*x^(1/2)+2/3*a*(a+b/x)^(1/2)*x^(3/2)-2*b^(3/2)*arctanh( b^(1/2)/(a+b/x)^(1/2)/x^(1/2))
Time = 4.70 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {2}{3} \sqrt {b+a x} (4 b+a x)-2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )\right )}{\sqrt {b+a x}} \] Input:
Integrate[(a + b/x)^(3/2)*Sqrt[x],x]
Output:
(Sqrt[a + b/x]*Sqrt[x]*((2*Sqrt[b + a*x]*(4*b + a*x))/3 - 2*b^(3/2)*ArcTan h[Sqrt[b + a*x]/Sqrt[b]]))/Sqrt[b + a*x]
Time = 0.31 (sec) , antiderivative size = 73, 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 = {860, 247, 247, 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.
\(\displaystyle \int \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \left (a+\frac {b}{x}\right )^{3/2} x^2d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -2 \left (b \int \sqrt {a+\frac {b}{x}} xd\frac {1}{\sqrt {x}}-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -2 \left (b \left (b \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}-\sqrt {x} \sqrt {a+\frac {b}{x}}\right )-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (b \left (b \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}-\sqrt {x} \sqrt {a+\frac {b}{x}}\right )-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (b \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )-\sqrt {x} \sqrt {a+\frac {b}{x}}\right )-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
Input:
Int[(a + b/x)^(3/2)*Sqrt[x],x]
Output:
-2*(-1/3*((a + b/x)^(3/2)*x^(3/2)) + b*(-(Sqrt[a + b/x]*Sqrt[x]) + Sqrt[b] *ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]))
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 + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 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 = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (-3 b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+a x \sqrt {a x +b}+4 b \sqrt {a x +b}\right )}{3 \sqrt {a x +b}}\) | \(62\) |
Input:
int((a+b/x)^(3/2)*x^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*((a*x+b)/x)^(1/2)*x^(1/2)*(-3*b^(3/2)*arctanh((a*x+b)^(1/2)/b^(1/2))+a *x*(a*x+b)^(1/2)+4*b*(a*x+b)^(1/2))/(a*x+b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\left [b^{\frac {3}{2}} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + \frac {2}{3} \, {\left (a x + 4 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}, 2 \, \sqrt {-b} b \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + \frac {2}{3} \, {\left (a x + 4 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}\right ] \] Input:
integrate((a+b/x)^(3/2)*x^(1/2),x, algorithm="fricas")
Output:
[b^(3/2)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2/3*(a *x + 4*b)*sqrt(x)*sqrt((a*x + b)/x), 2*sqrt(-b)*b*arctan(sqrt(-b)*sqrt(x)* sqrt((a*x + b)/x)/(a*x + b)) + 2/3*(a*x + 4*b)*sqrt(x)*sqrt((a*x + b)/x)]
Time = 3.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\frac {2 a \sqrt {b} x \sqrt {\frac {a x}{b} + 1}}{3} + \frac {8 b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{3} + b^{\frac {3}{2}} \log {\left (\frac {a x}{b} \right )} - 2 b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )} \] Input:
integrate((a+b/x)**(3/2)*x**(1/2),x)
Output:
2*a*sqrt(b)*x*sqrt(a*x/b + 1)/3 + 8*b**(3/2)*sqrt(a*x/b + 1)/3 + b**(3/2)* log(a*x/b) - 2*b**(3/2)*log(sqrt(a*x/b + 1) + 1)
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\frac {2}{3} \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}} + b^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right ) + 2 \, \sqrt {a + \frac {b}{x}} b \sqrt {x} \] Input:
integrate((a+b/x)^(3/2)*x^(1/2),x, algorithm="maxima")
Output:
2/3*(a + b/x)^(3/2)*x^(3/2) + b^(3/2)*log((sqrt(a + b/x)*sqrt(x) - sqrt(b) )/(sqrt(a + b/x)*sqrt(x) + sqrt(b))) + 2*sqrt(a + b/x)*b*sqrt(x)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {2}{3} \, {\left (a x + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 2 \, \sqrt {a x + b} b \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b}} \] Input:
integrate((a+b/x)^(3/2)*x^(1/2),x, algorithm="giac")
Output:
2*b^2*arctan(sqrt(a*x + b)/sqrt(-b))*sgn(x)/sqrt(-b) + 2/3*(a*x + b)^(3/2) *sgn(x) + 2*sqrt(a*x + b)*b*sgn(x) - 2/3*(3*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b)*b^(3/2))*sgn(x)/sqrt(-b)
Timed out. \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\int \sqrt {x}\,{\left (a+\frac {b}{x}\right )}^{3/2} \,d x \] Input:
int(x^(1/2)*(a + b/x)^(3/2),x)
Output:
int(x^(1/2)*(a + b/x)^(3/2), x)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x} \, dx=\frac {2 \sqrt {a x +b}\, a x}{3}+\frac {8 \sqrt {a x +b}\, b}{3}+\sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) b -\sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) b \] Input:
int((a+b/x)^(3/2)*x^(1/2),x)
Output:
(2*sqrt(a*x + b)*a*x + 8*sqrt(a*x + b)*b + 3*sqrt(b)*log(sqrt(a*x + b) - s qrt(b))*b - 3*sqrt(b)*log(sqrt(a*x + b) + sqrt(b))*b)/3