Optimal. Leaf size=53 \[ \frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}}-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}}-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx &=\int \frac {x^{3/2}}{b+a x} \, dx\\ &=\frac {2 x^{3/2}}{3 a}-\frac {b \int \frac {\sqrt {x}}{b+a x} \, dx}{a}\\ &=-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a}+\frac {b^2 \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{a^2}\\ &=-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a}+\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 0.92 \[ \frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}}+\frac {2 \sqrt {x} (a x-3 b)}{3 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 103, normalized size = 1.94 \[ \left [\frac {3 \, b \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (a x - 3 \, b\right )} \sqrt {x}}{3 \, a^{2}}, \frac {2 \, {\left (3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (a x - 3 \, b\right )} \sqrt {x}\right )}}{3 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 45, normalized size = 0.85 \[ \frac {2 \, b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (a^{2} x^{\frac {3}{2}} - 3 \, a b \sqrt {x}\right )}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 43, normalized size = 0.81 \[ \frac {2 b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {2 x^{\frac {3}{2}}}{3 a}-\frac {2 b \sqrt {x}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.33, size = 41, normalized size = 0.77 \[ \frac {2 \, {\left (a - \frac {3 \, b}{x}\right )} x^{\frac {3}{2}}}{3 \, a^{2}} - \frac {2 \, b^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 37, normalized size = 0.70 \[ \frac {2\,x^{3/2}}{3\,a}-\frac {2\,b\,\sqrt {x}}{a^2}+\frac {2\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.37, size = 105, normalized size = 1.98 \[ \begin {cases} \frac {2 x^{\frac {3}{2}}}{3 a} - \frac {2 b \sqrt {x}}{a^{2}} - \frac {i b^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a^{3} \sqrt {\frac {1}{a}}} + \frac {i b^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a^{3} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\\frac {2 x^{\frac {5}{2}}}{5 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________