Integrand size = 20, antiderivative size = 96 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {3}{4} a^{3/2} c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {38, 65, 223, 209} \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {3}{4} a^{3/2} c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )+\frac {3}{8} a c x \sqrt {a x+a} \sqrt {c-c x}+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2} \]
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Rule 38
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{4} (3 a c) \int \sqrt {a+a x} \sqrt {c-c x} \, dx \\ & = \frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{8} \left (3 a^2 c^2\right ) \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx \\ & = \frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{4} \left (3 a c^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+a x}\right ) \\ & = \frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{4} \left (3 a c^2\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+a x}}{\sqrt {c-c x}}\right ) \\ & = \frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {3}{4} a^{3/2} c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {\sqrt {c} (a (1+x))^{3/2} \sqrt {c-c x} \left (\sqrt {c} x \sqrt {1+x} \left (-5+5 x+2 x^2-2 x^3\right )+6 \sqrt {c-c x} \arcsin \left (\frac {\sqrt {c-c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{8 (-1+x) (1+x)^{3/2}} \]
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Time = 0.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {x \left (2 x^{2}-5\right ) \left (-1+x \right ) \left (1+x \right ) a^{2} c^{2}}{8 \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}+\frac {3 \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right ) a^{2} c^{2} \sqrt {-a \left (1+x \right ) c \left (-1+x \right )}}{8 \sqrt {a c}\, \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}\) | \(100\) |
default | \(-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {5}{2}}}{4 c}+\frac {3 a \left (-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {5}{2}}}{3 c}+\frac {a \left (\frac {\left (-c x +c \right )^{\frac {3}{2}} \sqrt {a x +a}}{2 a}+\frac {3 c \left (\frac {\sqrt {-c x +c}\, \sqrt {a x +a}}{a}+\frac {c \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}\right )}{2}\right )}{3}\right )}{4}\) | \(150\) |
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Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.61 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\left [\frac {3}{16} \, \sqrt {-a c} a c \log \left (2 \, a c x^{2} + 2 \, \sqrt {-a c} \sqrt {a x + a} \sqrt {-c x + c} x - a c\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}, -\frac {3}{8} \, \sqrt {a c} a c \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}\right ] \]
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\[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\int \left (a \left (x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (x - 1\right )\right )^{\frac {3}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.52 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {3 \, a^{2} c^{2} \arcsin \left (x\right )}{8 \, \sqrt {a c}} + \frac {3}{8} \, \sqrt {-a c x^{2} + a c} a c x + \frac {1}{4} \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (70) = 140\).
Time = 0.45 (sec) , antiderivative size = 403, normalized size of antiderivative = 4.20 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=-\frac {{\left (\frac {18 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (a x + a\right )} {\left (2 \, {\left (a x + a\right )} {\left (\frac {3 \, {\left (a x + a\right )}}{a^{3}} - \frac {13}{a^{2}}\right )} + \frac {43}{a}\right )} - 39\right )} \sqrt {a x + a}\right )} c {\left | a \right |}}{24 \, a} + \frac {{\left (\frac {6 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left ({\left (a x + a\right )} {\left (\frac {2 \, {\left (a x + a\right )}}{a^{2}} - \frac {7}{a}\right )} + 9\right )}\right )} c {\left | a \right |}}{6 \, a} - \frac {{\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a}\right )} c {\left | a \right |}}{a} + \frac {{\left (\frac {2 \, a^{3} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left (a x - 2 \, a\right )}\right )} c {\left | a \right |}}{2 \, a^{2}} \]
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Timed out. \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\int {\left (a+a\,x\right )}^{3/2}\,{\left (c-c\,x\right )}^{3/2} \,d x \]
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