Optimal. Leaf size=41 \[ -\frac {2 \sqrt {2+b x}}{\sqrt {x}}+2 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 41, 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 = {49, 56, 221}
\begin {gather*} 2 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 \sqrt {b x+2}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {\sqrt {2+b x}}{x^{3/2}} \, dx &=-\frac {2 \sqrt {2+b x}}{\sqrt {x}}+b \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=-\frac {2 \sqrt {2+b x}}{\sqrt {x}}+(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \sqrt {2+b x}}{\sqrt {x}}+2 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 47, normalized size = 1.15 \begin {gather*} -\frac {2 \sqrt {2+b x}}{\sqrt {x}}-2 \sqrt {b} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 49, normalized size = 1.20
method | result | size |
meijerg | \(-\frac {\sqrt {b}\, \left (\frac {4 \sqrt {\pi }\, \sqrt {2}\, \sqrt {\frac {b x}{2}+1}}{\sqrt {x}\, \sqrt {b}}-4 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )\right )}{2 \sqrt {\pi }}\) | \(49\) |
risch | \(-\frac {2 \sqrt {b x +2}}{\sqrt {x}}+\frac {\sqrt {b}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right ) \sqrt {x \left (b x +2\right )}}{\sqrt {x}\, \sqrt {b x +2}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 54, normalized size = 1.32 \begin {gather*} -\sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + 2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.49, size = 87, normalized size = 2.12 \begin {gather*} \left [\frac {\sqrt {b} x \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) - 2 \, \sqrt {b x + 2} \sqrt {x}}{x}, -\frac {2 \, {\left (\sqrt {-b} x \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + 2} \sqrt {x}\right )}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.82, size = 48, normalized size = 1.17 \begin {gather*} - 2 \sqrt {b} \sqrt {1 + \frac {2}{b x}} - \sqrt {b} \log {\left (\frac {1}{b x} \right )} + 2 \sqrt {b} \log {\left (\sqrt {1 + \frac {2}{b x}} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {b\,x+2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________