Optimal. Leaf size=39 \[ -\frac {\sqrt {a+b x}}{x}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {43, 65, 214}
\begin {gather*} -\frac {\sqrt {a+b x}}{x}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 65
Rule 214
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}+\text {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 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 39, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b x}}{x}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 37, normalized size = 0.95
method | result | size |
risch | \(-\frac {b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b x +a}}{x}\) | \(32\) |
derivativedivides | \(2 b \left (-\frac {\sqrt {b x +a}}{2 b x}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\) | \(37\) |
default | \(2 b \left (-\frac {\sqrt {b x +a}}{2 b x}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.24, size = 47, normalized size = 1.21 \begin {gather*} \frac {b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2 \, \sqrt {a}} - \frac {\sqrt {b x + a}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.49, size = 93, normalized size = 2.38 \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}}{a}\right ) - \sqrt {b x + a} a}{a x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.88, size = 44, normalized size = 1.13 \begin {gather*} - \frac {\sqrt {b} \sqrt {\frac {a}{b x} + 1}}{\sqrt {x}} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.47, size = 41, normalized size = 1.05 \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]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 31, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {a+b\,x}}{x}-\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________