Integrand size = 15, antiderivative size = 79 \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221} \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\frac {5 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}+\frac {1}{3} \sqrt {x} (b x+2)^{5/2}+\frac {5}{6} \sqrt {x} (b x+2)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {b x+2} \]
[In]
[Out]
Rule 52
Rule 56
Rule 221
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5}{3} \int \frac {(2+b x)^{3/2}}{\sqrt {x}} \, dx \\ & = \frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {2+b x}}{\sqrt {x}} \, dx \\ & = \frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx \\ & = \frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+5 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\frac {1}{6} \sqrt {x} \sqrt {2+b x} \left (33+13 b x+2 b^2 x^2\right )-\frac {5 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{\sqrt {b}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80
method | result | size |
meijerg | \(-\frac {15 \left (-\frac {8 \sqrt {\pi }\, \sqrt {b}\, \sqrt {x}\, \sqrt {2}\, \left (\frac {1}{24} b^{2} x^{2}+\frac {13}{48} b x +\frac {11}{16}\right ) \sqrt {\frac {b x}{2}+1}}{15}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{3}\right )}{\sqrt {b}\, \sqrt {\pi }}\) | \(63\) |
risch | \(\frac {\left (2 b^{2} x^{2}+13 b x +33\right ) \sqrt {x}\, \sqrt {b x +2}}{6}+\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(74\) |
default | \(\frac {\left (b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {x}\, \sqrt {b x +2}}{2}+\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(84\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.56 \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\left [\frac {{\left (2 \, b^{3} x^{2} + 13 \, b^{2} x + 33 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 15 \, \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{6 \, b}, \frac {{\left (2 \, b^{3} x^{2} + 13 \, b^{2} x + 33 \, b\right )} \sqrt {b x + 2} \sqrt {x} - 30 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{6 \, b}\right ] \]
[In]
[Out]
Time = 4.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\frac {b^{3} x^{\frac {7}{2}}}{3 \sqrt {b x + 2}} + \frac {17 b^{2} x^{\frac {5}{2}}}{6 \sqrt {b x + 2}} + \frac {59 b x^{\frac {3}{2}}}{6 \sqrt {b x + 2}} + \frac {11 \sqrt {x}}{\sqrt {b x + 2}} + \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (54) = 108\).
Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.63 \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=-\frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, \sqrt {b}} - \frac {\frac {15 \, \sqrt {b x + 2} b^{2}}{\sqrt {x}} - \frac {40 \, {\left (b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {33 \, {\left (b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{3} - \frac {3 \, {\left (b x + 2\right )} b^{2}}{x} + \frac {3 \, {\left (b x + 2\right )}^{2} b}{x^{2}} - \frac {{\left (b x + 2\right )}^{3}}{x^{3}}\right )}} \]
[In]
[Out]
none
Time = 5.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14 \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\frac {{\left (\sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} {\left ({\left (b x + 2\right )} {\left (\frac {2 \, {\left (b x + 2\right )}}{b} + \frac {5}{b}\right )} + \frac {15}{b}\right )} - \frac {30 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{\sqrt {b}}\right )} b}{6 \, {\left | b \right |}} \]
[In]
[Out]
Timed out. \[ \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx=\int \frac {{\left (b\,x+2\right )}^{5/2}}{\sqrt {x}} \,d x \]
[In]
[Out]