Integrand size = 26, antiderivative size = 115 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}+\frac {1815}{4} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {1815}{4} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}+\frac {55 (1-2 x)^{3/2}}{4 (3 x+2) \sqrt {5 x+3}}-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {5 x+3}} \]
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Rule 95
Rule 96
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55}{4} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx \\ & = \frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}+\frac {1815}{8} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx \\ & = -\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}-\frac {12705}{8} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}-\frac {12705}{4} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}+\frac {1815}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {\sqrt {1-2 x} \left (7148+21843 x+16657 x^2\right )}{4 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1815}{4} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(88)=176\).
Time = 3.87 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.76
method | result | size |
default | \(-\frac {\left (81675 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+157905 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+101640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +33314 x^{2} \sqrt {-10 x^{2}-x +3}+21780 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+43686 x \sqrt {-10 x^{2}-x +3}+14296 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{8 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(202\) |
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Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {1815 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2 \, {\left (16657 \, x^{2} + 21843 \, x + 7148\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{8 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {1815}{8} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {16657 \, x}{18 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {52169}{108 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{54 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {833}{12 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (88) = 176\).
Time = 0.43 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.70 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {363}{16} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121}{10} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {847 \, \sqrt {10} {\left (9 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1960 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {7840 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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