Integrand size = 16, antiderivative size = 124 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {3 a^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\frac {3 a^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}}-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2} \]
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {1}{8} (3 a) \int x^{3/2} \sqrt {a-b x} \, dx \\ & = \frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {1}{16} a^2 \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx \\ & = -\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^3\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{64 b} \\ & = -\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{128 b^2} \\ & = -\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^2} \\ & = -\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^2} \\ & = -\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {3 a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\frac {-\sqrt {b} \sqrt {x} \sqrt {a-b x} \left (3 a^3+2 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )+6 a^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{64 b^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\left (16 b^{3} x^{3}-24 a \,b^{2} x^{2}+2 a^{2} b x +3 a^{3}\right ) \sqrt {x}\, \sqrt {-b x +a}}{64 b^{2}}+\frac {3 a^{4} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{128 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(102\) |
default | \(-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {5}{2}}}{4 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {5}{2}}}{3 b}+\frac {a \left (\frac {\left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\) | \(129\) |
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Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.31 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\left [-\frac {3 \, a^{4} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (16 \, b^{4} x^{3} - 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x + 3 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{128 \, b^{3}}, -\frac {3 \, a^{4} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (16 \, b^{4} x^{3} - 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x + 3 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{64 \, b^{3}}\right ] \]
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Result contains complex when optimal does not.
Time = 17.43 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.60 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\begin {cases} \frac {3 i a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {-1 + \frac {b x}{a}}} - \frac {13 i a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {-1 + \frac {b x}{a}}} + \frac {5 i \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{4} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} - \frac {i b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {3 a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {1 - \frac {b x}{a}}} + \frac {13 a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {1 - \frac {b x}{a}}} - \frac {5 \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {1 - \frac {b x}{a}}} + \frac {3 a^{4} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} + \frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.37 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=-\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{64 \, b^{\frac {5}{2}}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{4} b^{3}}{\sqrt {x}} + \frac {11 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{64 \, {\left (b^{6} - \frac {4 \, {\left (b x - a\right )} b^{5}}{x} + \frac {6 \, {\left (b x - a\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x - a\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x - a\right )}^{4} b^{2}}{x^{4}}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (90) = 180\).
Time = 230.84 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.75 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\frac {{\left (\frac {105 \, a^{4} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b^{2}} - {\left (2 \, {\left (b x - a\right )} {\left (4 \, {\left (b x - a\right )} {\left (\frac {6 \, {\left (b x - a\right )}}{b^{3}} + \frac {25 \, a}{b^{3}}\right )} + \frac {163 \, a^{2}}{b^{3}}\right )} + \frac {279 \, a^{3}}{b^{3}}\right )} \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}\right )} {\left | b \right |} - \frac {16 \, {\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (2 \, {\left (b x - a\right )} {\left (\frac {4 \, {\left (b x - a\right )}}{b^{2}} + \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} a {\left | b \right |}}{b} + \frac {48 \, {\left (\frac {3 \, a^{2} b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} {\left (2 \, b x + 3 \, a\right )} \sqrt {-b x + a}\right )} a^{2} {\left | b \right |}}{b^{3}}}{192 \, b} \]
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Timed out. \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\int x^{3/2}\,{\left (a-b\,x\right )}^{3/2} \,d x \]
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