Optimal. Leaf size=43 \[ \frac {\sqrt {x} \sqrt {2+b x}}{b}-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 43, 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 = {52, 56, 221}
\begin {gather*} \frac {\sqrt {x} \sqrt {b x+2}}{b}-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx &=\frac {\sqrt {x} \sqrt {2+b x}}{b}-\frac {\int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{b}\\ &=\frac {\sqrt {x} \sqrt {2+b x}}{b}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {\sqrt {x} \sqrt {2+b x}}{b}-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 49, normalized size = 1.14 \begin {gather*} \frac {\sqrt {x} \sqrt {2+b x}}{b}+\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 62, normalized size = 1.44
method | result | size |
meijerg | \(\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \sqrt {\frac {b x}{2}+1}-2 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {3}{2}} \sqrt {\pi }}\) | \(49\) |
default | \(\frac {\sqrt {x}\, \sqrt {b x +2}}{b}-\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\) | \(62\) |
risch | \(\frac {\sqrt {x}\, \sqrt {b x +2}}{b}-\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (32) = 64\).
time = 0.51, size = 70, normalized size = 1.63 \begin {gather*} \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{b^{\frac {3}{2}}} - \frac {2 \, \sqrt {b x + 2}}{{\left (b^{2} - \frac {{\left (b x + 2\right )} b}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.54, size = 87, normalized size = 2.02 \begin {gather*} \left [\frac {\sqrt {b x + 2} b \sqrt {x} + \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{b^{2}}, \frac {\sqrt {b x + 2} b \sqrt {x} + 2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.11, size = 54, normalized size = 1.26 \begin {gather*} \frac {x^{\frac {3}{2}}}{\sqrt {b x + 2}} + \frac {2 \sqrt {x}}{b \sqrt {b x + 2}} - \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} \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 [B]
time = 0.59, size = 43, normalized size = 1.00 \begin {gather*} \frac {4\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {2}-\sqrt {b\,x+2}}\right )}{b^{3/2}}+\frac {\sqrt {x}\,\sqrt {b\,x+2}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________