Optimal. Leaf size=42 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b x-a}}{x} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {47, 63, 205} \begin {gather*} \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b x-a}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {\sqrt {-a+b x}}{x^2} \, dx &=-\frac {\sqrt {-a+b x}}{x}+\frac {1}{2} b \int \frac {1}{x \sqrt {-a+b x}} \, dx\\ &=-\frac {\sqrt {-a+b x}}{x}+\operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )\\ &=-\frac {\sqrt {-a+b x}}{x}+\frac {b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 52, normalized size = 1.24 \begin {gather*} \frac {-b x \sqrt {1-\frac {b x}{a}} \tanh ^{-1}\left (\sqrt {1-\frac {b x}{a}}\right )+a-b x}{x \sqrt {b x-a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.04, size = 42, normalized size = 1.00 \begin {gather*} \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b x-a}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.00, size = 98, normalized size = 2.33 \begin {gather*} \left [-\frac {\sqrt {-a} b x \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a} a}{2 \, a x}, \frac {\sqrt {a} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) - \sqrt {b x - a} a}{a x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.08, size = 41, normalized size = 0.98 \begin {gather*} \frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {\sqrt {b x - a} b}{x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 35, normalized size = 0.83 \begin {gather*} \frac {b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b x -a}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.94, size = 34, normalized size = 0.81 \begin {gather*} \frac {b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {\sqrt {b x - a}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 34, normalized size = 0.81 \begin {gather*} \frac {b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b\,x-a}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.14, size = 121, normalized size = 2.88 \begin {gather*} \begin {cases} - \frac {i a}{\sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {i \sqrt {b}}{\sqrt {x} \sqrt {\frac {a}{b x} - 1}} + \frac {i b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {\sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{\sqrt {x}} - \frac {b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________