Optimal. Leaf size=67 \[ -\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {3 \sqrt {x} \sqrt {b x+2}}{2 b^2}+\frac {x^{3/2} \sqrt {b x+2}}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx &=\frac {x^{3/2} \sqrt {2+b x}}{2 b}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{2 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 57, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x} (-3+b x) \sqrt {2+b x}}{2 b^2}-\frac {3 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 4.25, size = 73, normalized size = 1.09 \begin {gather*} \frac {3 \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{b^{\frac {5}{2}}}-\frac {6 \sqrt {x}}{b^2 \left (2+b x\right )^{\frac {3}{2}}}-\frac {4 x^{\frac {3}{2}}}{b \left (2+b x\right )^{\frac {3}{2}}}+\frac {x^{\frac {5}{2}}}{2 \left (2+b x\right )^{\frac {3}{2}}}+\frac {b x^{\frac {7}{2}}}{2 \left (2+b x\right )^{\frac {3}{2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 83, normalized size = 1.24
method | result | size |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (-5 b x +15\right ) \sqrt {\frac {b x}{2}+1}}{10}+3 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(55\) |
risch | \(\frac {\left (b x -3\right ) \sqrt {x}\, \sqrt {b x +2}}{2 b^{2}}+\frac {3 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right ) \sqrt {x \left (b x +2\right )}}{2 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +2}}\) | \(68\) |
default | \(\frac {x^{\frac {3}{2}} \sqrt {b x +2}}{2 b}-\frac {3 \left (\frac {\sqrt {x}\, \sqrt {b x +2}}{b}-\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{b^{\frac {3}{2}} \sqrt {b x +2}\, \sqrt {x}}\right )}{2 b}\) | \(83\) |
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. 102 vs.
\(2 (48) = 96\).
time = 0.35, size = 102, normalized size = 1.52 \begin {gather*} \frac {\frac {5 \, \sqrt {b x + 2} b}{\sqrt {x}} - \frac {3 \, {\left (b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{4} - \frac {2 \, {\left (b x + 2\right )} b^{3}}{x} + \frac {{\left (b x + 2\right )}^{2} b^{2}}{x^{2}}} - \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 105, normalized size = 1.57 \begin {gather*} \left [\frac {{\left (b^{2} x - 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 3 \, \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{2 \, b^{3}}, \frac {{\left (b^{2} x - 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} - 6 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{2 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.52, size = 75, normalized size = 1.12 \begin {gather*} \frac {x^{\frac {5}{2}}}{2 \sqrt {b x + 2}} - \frac {x^{\frac {3}{2}}}{2 b \sqrt {b x + 2}} - \frac {3 \sqrt {x}}{b^{2} \sqrt {b x + 2}} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 90, normalized size = 1.34 \begin {gather*} 2 \left (2 \left (\frac {\frac {1}{8} b^{2} \sqrt {x} \sqrt {x}}{b^{3}}-\frac {\frac {1}{8}\cdot 3 b}{b^{3}}\right ) \sqrt {x} \sqrt {b x+2}-\frac {3 \ln \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{2 b^{2} \sqrt {b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {b\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________