Integrand size = 15, antiderivative size = 144 \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221} \[ \int x^{5/2} (2+b x)^{5/2} \, dx=-\frac {5 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{16 b^3}-\frac {5 x^{3/2} \sqrt {b x+2}}{48 b^2}+\frac {1}{6} x^{7/2} (b x+2)^{5/2}+\frac {1}{6} x^{7/2} (b x+2)^{3/2}+\frac {1}{8} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{24 b} \]
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Rule 52
Rule 56
Rule 221
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {5}{6} \int x^{5/2} (2+b x)^{3/2} \, dx \\ & = \frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {1}{2} \int x^{5/2} \sqrt {2+b x} \, dx \\ & = \frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx \\ & = \frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx}{24 b} \\ & = -\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{16 b^2} \\ & = \frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{16 b^3} \\ & = \frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3} \\ & = \frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\frac {\sqrt {x} \sqrt {2+b x} \left (15-5 b x+2 b^2 x^2+54 b^3 x^3+40 b^4 x^4+8 b^5 x^5\right )}{48 b^3}+\frac {5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2+b x}}\right )}{4 b^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.60
method | result | size |
meijerg | \(-\frac {120 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (56 b^{5} x^{5}+280 b^{4} x^{4}+378 b^{3} x^{3}+14 b^{2} x^{2}-35 b x +105\right ) \sqrt {\frac {b x}{2}+1}}{40320}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{192}\right )}{b^{\frac {7}{2}} \sqrt {\pi }}\) | \(87\) |
risch | \(\frac {\left (8 b^{5} x^{5}+40 b^{4} x^{4}+54 b^{3} x^{3}+2 b^{2} x^{2}-5 b x +15\right ) \sqrt {x}\, \sqrt {b x +2}}{48 b^{3}}-\frac {5 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{16 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +2}}\) | \(101\) |
default | \(\frac {x^{\frac {5}{2}} \left (b x +2\right )^{\frac {7}{2}}}{6 b}-\frac {5 \left (\frac {x^{\frac {3}{2}} \left (b x +2\right )^{\frac {7}{2}}}{5 b}-\frac {3 \left (\frac {\sqrt {x}\, \left (b x +2\right )^{\frac {7}{2}}}{4 b}-\frac {\frac {\left (b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {x}\, \sqrt {b x +2}}{2}+\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}}{4 b}\right )}{5 b}\right )}{6 b}\) | \(147\) |
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Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\left [\frac {{\left (8 \, b^{6} x^{5} + 40 \, b^{5} x^{4} + 54 \, b^{4} x^{3} + 2 \, b^{3} x^{2} - 5 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 15 \, \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{48 \, b^{4}}, \frac {{\left (8 \, b^{6} x^{5} + 40 \, b^{5} x^{4} + 54 \, b^{4} x^{3} + 2 \, b^{3} x^{2} - 5 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 30 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{48 \, b^{4}}\right ] \]
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Timed out. \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (99) = 198\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.55 \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\frac {\frac {15 \, \sqrt {b x + 2} b^{5}}{\sqrt {x}} - \frac {85 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}} + \frac {198 \, {\left (b x + 2\right )}^{\frac {7}{2}} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (b x + 2\right )}^{\frac {9}{2}} b}{x^{\frac {9}{2}}} + \frac {15 \, {\left (b x + 2\right )}^{\frac {11}{2}}}{x^{\frac {11}{2}}}}{24 \, {\left (b^{9} - \frac {6 \, {\left (b x + 2\right )} b^{8}}{x} + \frac {15 \, {\left (b x + 2\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x + 2\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x + 2\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x + 2\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x + 2\right )}^{6} b^{3}}{x^{6}}\right )}} + \frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{16 \, b^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (99) = 198\).
Time = 23.79 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.03 \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\frac {{\left ({\left ({\left (2 \, {\left ({\left (b x + 2\right )} {\left (4 \, {\left (b x + 2\right )} {\left (\frac {5 \, {\left (b x + 2\right )}}{b^{5}} - \frac {61}{b^{5}}\right )} + \frac {1251}{b^{5}}\right )} - \frac {3481}{b^{5}}\right )} {\left (b x + 2\right )} + \frac {11395}{b^{5}}\right )} {\left (b x + 2\right )} - \frac {11895}{b^{5}}\right )} \sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} - \frac {6930 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{b^{\frac {9}{2}}}\right )} b {\left | b \right |} + 36 \, {\left ({\left ({\left (2 \, {\left (b x + 2\right )} {\left ({\left (b x + 2\right )} {\left (\frac {4 \, {\left (b x + 2\right )}}{b^{4}} - \frac {41}{b^{4}}\right )} + \frac {171}{b^{4}}\right )} - \frac {745}{b^{4}}\right )} {\left (b x + 2\right )} + \frac {965}{b^{4}}\right )} \sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} + \frac {630 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{b^{\frac {7}{2}}}\right )} {\left | b \right |} + \frac {120 \, {\left ({\left ({\left (b x + 2\right )} {\left (2 \, {\left (b x + 2\right )} {\left (\frac {3 \, {\left (b x + 2\right )}}{b^{3}} - \frac {25}{b^{3}}\right )} + \frac {163}{b^{3}}\right )} - \frac {279}{b^{3}}\right )} \sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} - \frac {210 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{b^{\frac {5}{2}}}\right )} {\left | b \right |}}{b} + \frac {320 \, {\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^{2}} - \frac {13}{b^{2}}\right )} + \frac {33}{b^{2}}\right )} + \frac {30 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{b^{\frac {3}{2}}}\right )} {\left | b \right |}}{b^{2}}}{240 \, b} \]
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Timed out. \[ \int x^{5/2} (2+b x)^{5/2} \, dx=\int x^{5/2}\,{\left (b\,x+2\right )}^{5/2} \,d x \]
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