Integrand size = 17, antiderivative size = 70 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=-\frac {b \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+2 a \sqrt {a+\frac {b}{x}} \sqrt {x}-3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \] Output:
-b*(a+b/x)^(1/2)/x^(1/2)+2*a*(a+b/x)^(1/2)*x^(1/2)-3*a*b^(1/2)*arctanh(b^( 1/2)/(a+b/x)^(1/2)/x^(1/2))
Time = 4.82 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=-\frac {\sqrt {a+\frac {b}{x}} \left ((b-2 a x) \sqrt {b+a x}+3 a \sqrt {b} x \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )\right )}{\sqrt {x} \sqrt {b+a x}} \] Input:
Integrate[(a + b/x)^(3/2)/Sqrt[x],x]
Output:
-((Sqrt[a + b/x]*((b - 2*a*x)*Sqrt[b + a*x] + 3*a*Sqrt[b]*x*ArcTanh[Sqrt[b + a*x]/Sqrt[b]]))/(Sqrt[x]*Sqrt[b + a*x]))
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {860, 247, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \left (a+\frac {b}{x}\right )^{3/2} xd\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -2 \left (3 b \int \sqrt {a+\frac {b}{x}}d\frac {1}{\sqrt {x}}-\sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -2 \left (3 b \left (\frac {1}{2} a \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )-\sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (3 b \left (\frac {1}{2} a \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )-\sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (3 b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{2 \sqrt {b}}+\frac {\sqrt {a+\frac {b}{x}}}{2 \sqrt {x}}\right )-\sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}\right )\) |
Input:
Int[(a + b/x)^(3/2)/Sqrt[x],x]
Output:
-2*(-((a + b/x)^(3/2)*Sqrt[x]) + 3*b*(Sqrt[a + b/x]/(2*Sqrt[x]) + (a*ArcTa nh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(2*Sqrt[b])))
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_)^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 = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (\sqrt {a x +b}\, b^{\frac {3}{2}}-2 a x \sqrt {a x +b}\, \sqrt {b}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a b x \right )}{\sqrt {x}\, \sqrt {a x +b}\, \sqrt {b}}\) | \(70\) |
risch | \(-\frac {b \sqrt {\frac {a x +b}{x}}}{\sqrt {x}}+\frac {a \left (4 \sqrt {a x +b}-6 \sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{2 \sqrt {a x +b}}\) | \(70\) |
Input:
int((a+b/x)^(3/2)/x^(1/2),x,method=_RETURNVERBOSE)
Output:
-((a*x+b)/x)^(1/2)*((a*x+b)^(1/2)*b^(3/2)-2*a*x*(a*x+b)^(1/2)*b^(1/2)+3*ar ctanh((a*x+b)^(1/2)/b^(1/2))*a*b*x)/x^(1/2)/(a*x+b)^(1/2)/b^(1/2)
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.91 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=\left [\frac {3 \, a \sqrt {b} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (2 \, a x - b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, x}, \frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (2 \, a x - b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{x}\right ] \] Input:
integrate((a+b/x)^(3/2)/x^(1/2),x, algorithm="fricas")
Output:
[1/2*(3*a*sqrt(b)*x*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/ x) + 2*(2*a*x - b)*sqrt(x)*sqrt((a*x + b)/x))/x, (3*a*sqrt(-b)*x*arctan(sq rt(-b)*sqrt(x)*sqrt((a*x + b)/x)/(a*x + b)) + (2*a*x - b)*sqrt(x)*sqrt((a* x + b)/x))/x]
Time = 2.87 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=\frac {2 a^{\frac {3}{2}} \sqrt {x}}{\sqrt {1 + \frac {b}{a x}}} + \frac {\sqrt {a} b}{\sqrt {x} \sqrt {1 + \frac {b}{a x}}} - 3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )} - \frac {b^{2}}{\sqrt {a} x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} \] Input:
integrate((a+b/x)**(3/2)/x**(1/2),x)
Output:
2*a**(3/2)*sqrt(x)/sqrt(1 + b/(a*x)) + sqrt(a)*b/(sqrt(x)*sqrt(1 + b/(a*x) )) - 3*a*sqrt(b)*asinh(sqrt(b)/(sqrt(a)*sqrt(x))) - b**2/(sqrt(a)*x**(3/2) *sqrt(1 + b/(a*x)))
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=\frac {3}{2} \, a \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right ) + 2 \, \sqrt {a + \frac {b}{x}} a \sqrt {x} - \frac {\sqrt {a + \frac {b}{x}} a b \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \] Input:
integrate((a+b/x)^(3/2)/x^(1/2),x, algorithm="maxima")
Output:
3/2*a*sqrt(b)*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b))) + 2*sqrt(a + b/x)*a*sqrt(x) - sqrt(a + b/x)*a*b*sqrt(x)/((a + b/x)*x - b)
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx={\left (\frac {3 \, b \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + 2 \, \sqrt {a x + b} \mathrm {sgn}\left (x\right ) - \frac {\sqrt {a x + b} b \mathrm {sgn}\left (x\right )}{a x}\right )} a \] Input:
integrate((a+b/x)^(3/2)/x^(1/2),x, algorithm="giac")
Output:
(3*b*arctan(sqrt(a*x + b)/sqrt(-b))*sgn(x)/sqrt(-b) + 2*sqrt(a*x + b)*sgn( x) - sqrt(a*x + b)*b*sgn(x)/(a*x))*a
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=\int \frac {{\left (a+\frac {b}{x}\right )}^{3/2}}{\sqrt {x}} \,d x \] Input:
int((a + b/x)^(3/2)/x^(1/2),x)
Output:
int((a + b/x)^(3/2)/x^(1/2), x)
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx=\frac {4 \sqrt {a x +b}\, a x -2 \sqrt {a x +b}\, b +3 \sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) a x -3 \sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) a x}{2 x} \] Input:
int((a+b/x)^(3/2)/x^(1/2),x)
Output:
(4*sqrt(a*x + b)*a*x - 2*sqrt(a*x + b)*b + 3*sqrt(b)*log(sqrt(a*x + b) - s qrt(b))*a*x - 3*sqrt(b)*log(sqrt(a*x + b) + sqrt(b))*a*x)/(2*x)