Integrand size = 16, antiderivative size = 149 \[ \int x^{5/2} (a-b x)^{3/2} \, dx=-\frac {3 a^4 \sqrt {x} \sqrt {a-b x}}{128 b^3}-\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {3 a^5 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{128 b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int x^{5/2} (a-b x)^{3/2} \, dx=\frac {3 a^5 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{128 b^{7/2}}-\frac {3 a^4 \sqrt {x} \sqrt {a-b x}}{128 b^3}-\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2} \]
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {1}{10} (3 a) \int x^{5/2} \sqrt {a-b x} \, dx \\ & = \frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {1}{80} \left (3 a^2\right ) \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx \\ & = -\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {a^3 \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{32 b} \\ & = -\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{128 b^2} \\ & = -\frac {3 a^4 \sqrt {x} \sqrt {a-b x}}{128 b^3}-\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{256 b^3} \\ & = -\frac {3 a^4 \sqrt {x} \sqrt {a-b x}}{128 b^3}-\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {\left (3 a^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^3} \\ & = -\frac {3 a^4 \sqrt {x} \sqrt {a-b x}}{128 b^3}-\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {\left (3 a^5\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{128 b^3} \\ & = -\frac {3 a^4 \sqrt {x} \sqrt {a-b x}}{128 b^3}-\frac {a^3 x^{3/2} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{5/2} \sqrt {a-b x}}{80 b}+\frac {3}{40} a x^{7/2} \sqrt {a-b x}+\frac {1}{5} x^{7/2} (a-b x)^{3/2}+\frac {3 a^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{128 b^{7/2}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72 \[ \int x^{5/2} (a-b x)^{3/2} \, dx=-\frac {\sqrt {x} \sqrt {a-b x} \left (15 a^4+10 a^3 b x+8 a^2 b^2 x^2-176 a b^3 x^3+128 b^4 x^4\right )}{640 b^3}+\frac {3 a^5 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{64 b^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {\left (128 b^{4} x^{4}-176 a \,b^{3} x^{3}+8 a^{2} b^{2} x^{2}+10 a^{3} b x +15 a^{4}\right ) \sqrt {x}\, \sqrt {-b x +a}}{640 b^{3}}+\frac {3 a^{5} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{256 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(113\) |
default | \(-\frac {x^{\frac {5}{2}} \left (-b x +a \right )^{\frac {5}{2}}}{5 b}+\frac {a \left (-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {5}{2}}}{4 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {5}{2}}}{3 b}+\frac {a \left (\frac {\left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\) | \(152\) |
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Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.24 \[ \int x^{5/2} (a-b x)^{3/2} \, dx=\left [-\frac {15 \, a^{5} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (128 \, b^{5} x^{4} - 176 \, a b^{4} x^{3} + 8 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 15 \, a^{4} b\right )} \sqrt {-b x + a} \sqrt {x}}{1280 \, b^{4}}, -\frac {15 \, a^{5} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (128 \, b^{5} x^{4} - 176 \, a b^{4} x^{3} + 8 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 15 \, a^{4} b\right )} \sqrt {-b x + a} \sqrt {x}}{640 \, b^{4}}\right ] \]
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Result contains complex when optimal does not.
Time = 80.52 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.52 \[ \int x^{5/2} (a-b x)^{3/2} \, dx=\begin {cases} \frac {3 i a^{\frac {9}{2}} \sqrt {x}}{128 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 b \sqrt {-1 + \frac {b x}{a}}} - \frac {23 i a^{\frac {3}{2}} x^{\frac {7}{2}}}{80 \sqrt {-1 + \frac {b x}{a}}} + \frac {19 i \sqrt {a} b x^{\frac {9}{2}}}{40 \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{5} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {7}{2}}} - \frac {i b^{2} x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {3 a^{\frac {9}{2}} \sqrt {x}}{128 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 b \sqrt {1 - \frac {b x}{a}}} + \frac {23 a^{\frac {3}{2}} x^{\frac {7}{2}}}{80 \sqrt {1 - \frac {b x}{a}}} - \frac {19 \sqrt {a} b x^{\frac {9}{2}}}{40 \sqrt {1 - \frac {b x}{a}}} + \frac {3 a^{5} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {7}{2}}} + \frac {b^{2} x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.39 \[ \int x^{5/2} (a-b x)^{3/2} \, dx=-\frac {3 \, a^{5} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{128 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{5} b^{4}}{\sqrt {x}} + \frac {70 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{5} b^{3}}{x^{\frac {3}{2}}} - \frac {128 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{5} b^{2}}{x^{\frac {5}{2}}} - \frac {70 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{5} b}{x^{\frac {7}{2}}} - \frac {15 \, {\left (-b x + a\right )}^{\frac {9}{2}} a^{5}}{x^{\frac {9}{2}}}}{640 \, {\left (b^{8} - \frac {5 \, {\left (b x - a\right )} b^{7}}{x} + \frac {10 \, {\left (b x - a\right )}^{2} b^{6}}{x^{2}} - \frac {10 \, {\left (b x - a\right )}^{3} b^{5}}{x^{3}} + \frac {5 \, {\left (b x - a\right )}^{4} b^{4}}{x^{4}} - \frac {{\left (b x - a\right )}^{5} b^{3}}{x^{5}}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (109) = 218\).
Time = 229.43 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.74 \[ \int x^{5/2} (a-b x)^{3/2} \, dx=\frac {3 \, {\left (\frac {315 \, a^{5} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b^{3}} - {\left (2 \, {\left (4 \, {\left (b x - a\right )} {\left (2 \, {\left (b x - a\right )} {\left (\frac {8 \, {\left (b x - a\right )}}{b^{4}} + \frac {41 \, a}{b^{4}}\right )} + \frac {171 \, a^{2}}{b^{4}}\right )} + \frac {745 \, a^{3}}{b^{4}}\right )} {\left (b x - a\right )} + \frac {965 \, a^{4}}{b^{4}}\right )} \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}\right )} {\left | b \right |} + \frac {80 \, {\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (2 \, {\left (b x - a\right )} {\left (\frac {4 \, {\left (b x - a\right )}}{b^{2}} + \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} a^{2} {\left | b \right |}}{b^{2}} - \frac {20 \, {\left (\frac {105 \, a^{4} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b^{2}} - {\left (2 \, {\left (b x - a\right )} {\left (4 \, {\left (b x - a\right )} {\left (\frac {6 \, {\left (b x - a\right )}}{b^{3}} + \frac {25 \, a}{b^{3}}\right )} + \frac {163 \, a^{2}}{b^{3}}\right )} + \frac {279 \, a^{3}}{b^{3}}\right )} \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}\right )} a {\left | b \right |}}{b}}{1920 \, b} \]
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Timed out. \[ \int x^{5/2} (a-b x)^{3/2} \, dx=\int x^{5/2}\,{\left (a-b\,x\right )}^{3/2} \,d x \]
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