Optimal. Leaf size=44 \[ \frac {\sqrt {-a+b x}}{a x}+\frac {b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 44, 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 = {44, 65, 211}
\begin {gather*} \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\sqrt {b x-a}}{a x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx &=\frac {\sqrt {-a+b x}}{a x}+\frac {b \int \frac {1}{x \sqrt {-a+b x}} \, dx}{2 a}\\ &=\frac {\sqrt {-a+b x}}{a x}+\frac {\text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{a}\\ &=\frac {\sqrt {-a+b x}}{a x}+\frac {b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 44, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-a+b x}}{a x}+\frac {b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.22, size = 117, normalized size = 2.66 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \sqrt {b} \sqrt {-1+\frac {a}{b x}}}{a \sqrt {x}}+\frac {I b \text {ArcCosh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]}{a^{\frac {3}{2}}},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},-\frac {b \text {ArcSin}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]}{a^{\frac {3}{2}}}-\frac {1}{\sqrt {b} x^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}+\frac {\sqrt {b}}{a \sqrt {x} \sqrt {1-\frac {a}{b x}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 44, normalized size = 1.00
method | result | size |
derivativedivides | \(2 b \left (\frac {\sqrt {b x -a}}{2 a b x}+\frac {\arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )\) | \(44\) |
default | \(2 b \left (\frac {\sqrt {b x -a}}{2 a b x}+\frac {\arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )\) | \(44\) |
risch | \(-\frac {-b x +a}{a x \sqrt {b x -a}}+\frac {b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 46, normalized size = 1.05 \begin {gather*} \frac {\sqrt {b x - a} b}{{\left (b x - a\right )} a + a^{2}} + \frac {b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 97, normalized size = 2.20 \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^{2} x}, \frac {\sqrt {a} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \sqrt {b x - a} a}{a^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.18, size = 121, normalized size = 2.75 \begin {gather*} \begin {cases} \frac {i \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{a \sqrt {x}} + \frac {i b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {3}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {1}{\sqrt {b} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {\sqrt {b}}{a \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} - \frac {b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 63, normalized size = 1.43 \begin {gather*} \frac {2 \left (\frac {\sqrt {-a+b x} b^{2}}{2 a \left (-a+b x+a\right )}+\frac {b^{2} \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a\cdot 2 \sqrt {a}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 36, normalized size = 0.82 \begin {gather*} \frac {\sqrt {b\,x-a}}{a\,x}+\frac {b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________