Integrand size = 17, antiderivative size = 77 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Output:
-(a+b*x^(1/2))^(1/2)/x-1/2*b*(a+b*x^(1/2))^(1/2)/a/x^(1/2)+1/2*b^2*arctanh ((a+b*x^(1/2))^(1/2)/a^(1/2))/a^(3/2)
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=\frac {\left (-2 a-b \sqrt {x}\right ) \sqrt {a+b \sqrt {x}}}{2 a x}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Input:
Integrate[Sqrt[a + b*Sqrt[x]]/x^2,x]
Output:
((-2*a - b*Sqrt[x])*Sqrt[a + b*Sqrt[x]])/(2*a*x) + (b^2*ArcTanh[Sqrt[a + b *Sqrt[x]]/Sqrt[a]])/(2*a^(3/2))
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {798, 51, 52, 73, 221}
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.
\(\displaystyle \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\sqrt {a+b \sqrt {x}}}{x^{3/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle 2 \left (\frac {1}{4} b \int \frac {1}{\sqrt {a+b \sqrt {x}} x}d\sqrt {x}-\frac {\sqrt {a+b \sqrt {x}}}{2 x}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle 2 \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{\sqrt {a+b \sqrt {x}} \sqrt {x}}d\sqrt {x}}{2 a}-\frac {\sqrt {a+b \sqrt {x}}}{a \sqrt {x}}\right )-\frac {\sqrt {a+b \sqrt {x}}}{2 x}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 \left (\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {x}{b}-\frac {a}{b}}d\sqrt {a+b \sqrt {x}}}{a}-\frac {\sqrt {a+b \sqrt {x}}}{a \sqrt {x}}\right )-\frac {\sqrt {a+b \sqrt {x}}}{2 x}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 2 \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b \sqrt {x}}}{a \sqrt {x}}\right )-\frac {\sqrt {a+b \sqrt {x}}}{2 x}\right )\) |
Input:
Int[Sqrt[a + b*Sqrt[x]]/x^2,x]
Output:
2*(-1/2*Sqrt[a + b*Sqrt[x]]/x + (b*(-(Sqrt[a + b*Sqrt[x]]/(a*Sqrt[x])) + ( b*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/a^(3/2)))/4)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.54 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(4 b^{2} \left (-\frac {\frac {\left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{8 a}+\frac {\sqrt {a +b \sqrt {x}}}{8}}{b^{2} x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {x}}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )\) | \(60\) |
default | \(4 b^{2} \left (-\frac {\frac {\left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{8 a}+\frac {\sqrt {a +b \sqrt {x}}}{8}}{b^{2} x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {x}}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )\) | \(60\) |
Input:
int((a+b*x^(1/2))^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
4*b^2*(-(1/8/a*(a+b*x^(1/2))^(3/2)+1/8*(a+b*x^(1/2))^(1/2))/b^2/x+1/8/a^(3 /2)*arctanh((a+b*x^(1/2))^(1/2)/a^(1/2)))
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=\left [\frac {\sqrt {a} b^{2} x \log \left (\frac {b x + 2 \, \sqrt {b \sqrt {x} + a} \sqrt {a} \sqrt {x} + 2 \, a \sqrt {x}}{x}\right ) - 2 \, {\left (a b \sqrt {x} + 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{4 \, a^{2} x}, -\frac {\sqrt {-a} b^{2} x \arctan \left (\frac {{\left (\sqrt {-a} b \sqrt {x} - \sqrt {-a} a\right )} \sqrt {b \sqrt {x} + a}}{b^{2} x - a^{2}}\right ) + {\left (a b \sqrt {x} + 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{2 \, a^{2} x}\right ] \] Input:
integrate((a+b*x^(1/2))^(1/2)/x^2,x, algorithm="fricas")
Output:
[1/4*(sqrt(a)*b^2*x*log((b*x + 2*sqrt(b*sqrt(x) + a)*sqrt(a)*sqrt(x) + 2*a *sqrt(x))/x) - 2*(a*b*sqrt(x) + 2*a^2)*sqrt(b*sqrt(x) + a))/(a^2*x), -1/2* (sqrt(-a)*b^2*x*arctan((sqrt(-a)*b*sqrt(x) - sqrt(-a)*a)*sqrt(b*sqrt(x) + a)/(b^2*x - a^2)) + (a*b*sqrt(x) + 2*a^2)*sqrt(b*sqrt(x) + a))/(a^2*x)]
Time = 2.52 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=- \frac {a}{\sqrt {b} x^{\frac {5}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {3 \sqrt {b}}{2 x^{\frac {3}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {b^{\frac {3}{2}}}{2 a \sqrt [4]{x} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt [4]{x}} \right )}}{2 a^{\frac {3}{2}}} \] Input:
integrate((a+b*x**(1/2))**(1/2)/x**2,x)
Output:
-a/(sqrt(b)*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) - 3*sqrt(b)/(2*x**(3/4)*sqrt (a/(b*sqrt(x)) + 1)) - b**(3/2)/(2*a*x**(1/4)*sqrt(a/(b*sqrt(x)) + 1)) + b **2*asinh(sqrt(a)/(sqrt(b)*x**(1/4)))/(2*a**(3/2))
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=-\frac {b^{2} \log \left (\frac {\sqrt {b \sqrt {x} + a} - \sqrt {a}}{\sqrt {b \sqrt {x} + a} + \sqrt {a}}\right )}{4 \, a^{\frac {3}{2}}} - \frac {{\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {b \sqrt {x} + a} a b^{2}}{2 \, {\left ({\left (b \sqrt {x} + a\right )}^{2} a - 2 \, {\left (b \sqrt {x} + a\right )} a^{2} + a^{3}\right )}} \] Input:
integrate((a+b*x^(1/2))^(1/2)/x^2,x, algorithm="maxima")
Output:
-1/4*b^2*log((sqrt(b*sqrt(x) + a) - sqrt(a))/(sqrt(b*sqrt(x) + a) + sqrt(a )))/a^(3/2) - 1/2*((b*sqrt(x) + a)^(3/2)*b^2 + sqrt(b*sqrt(x) + a)*a*b^2)/ ((b*sqrt(x) + a)^2*a - 2*(b*sqrt(x) + a)*a^2 + a^3)
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=-\frac {1}{2} \, b^{3} {\left (\frac {\arctan \left (\frac {\sqrt {b \sqrt {x} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a b} + \frac {{\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} + \sqrt {b \sqrt {x} + a} a}{a b^{3} x}\right )} \] Input:
integrate((a+b*x^(1/2))^(1/2)/x^2,x, algorithm="giac")
Output:
-1/2*b^3*(arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a*b) + ((b*sqrt(x ) + a)^(3/2) + sqrt(b*sqrt(x) + a)*a)/(a*b^3*x))
Time = 0.88 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,\sqrt {x}}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {\sqrt {a+b\,\sqrt {x}}}{2\,x}-\frac {{\left (a+b\,\sqrt {x}\right )}^{3/2}}{2\,a\,x} \] Input:
int((a + b*x^(1/2))^(1/2)/x^2,x)
Output:
(b^2*atanh((a + b*x^(1/2))^(1/2)/a^(1/2)))/(2*a^(3/2)) - (a + b*x^(1/2))^( 1/2)/(2*x) - (a + b*x^(1/2))^(3/2)/(2*a*x)
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx=\frac {-2 \sqrt {x}\, \sqrt {\sqrt {x}\, b +a}\, a b -4 \sqrt {\sqrt {x}\, b +a}\, a^{2}-\sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {x}\, b +a}-\sqrt {a}\right ) b^{2} x +\sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {x}\, b +a}+\sqrt {a}\right ) b^{2} x}{4 a^{2} x} \] Input:
int((a+b*x^(1/2))^(1/2)/x^2,x)
Output:
( - 2*sqrt(x)*sqrt(sqrt(x)*b + a)*a*b - 4*sqrt(sqrt(x)*b + a)*a**2 - sqrt( a)*log(sqrt(sqrt(x)*b + a) - sqrt(a))*b**2*x + sqrt(a)*log(sqrt(sqrt(x)*b + a) + sqrt(a))*b**2*x)/(4*a**2*x)