Integrand size = 26, antiderivative size = 96 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{19965 (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {8182 \sqrt {1-2 x}}{219615 \sqrt {3+5 x}} \]
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Time = 0.01 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 91, 79, 37} \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {8182 \sqrt {1-2 x}}{219615 \sqrt {5 x+3}}-\frac {3679 \sqrt {1-2 x}}{19965 (5 x+3)^{3/2}}+\frac {49}{121 \sqrt {1-2 x} (5 x+3)^{3/2}} \]
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Rule 37
Rule 79
Rule 91
Rule 96
Rubi steps \begin{align*} \text {integral}& = \frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{11} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx \\ & = \frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{121} \int \frac {-\frac {617}{2}+99 x}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{19965 (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {4091 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{19965} \\ & = \frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{19965 (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {8182 \sqrt {1-2 x}}{219615 \sqrt {3+5 x}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.39 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (13040+52044 x+62232 x^2+19573 x^3\right )}{43923 (1-2 x)^{3/2} (3+5 x)^{3/2}} \]
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Time = 4.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.33
method | result | size |
gosper | \(\frac {\frac {39146}{43923} x^{3}+\frac {41488}{14641} x^{2}+\frac {34696}{14641} x +\frac {26080}{43923}}{\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}\) | \(32\) |
default | \(\frac {2 \sqrt {1-2 x}\, \left (19573 x^{3}+62232 x^{2}+52044 x +13040\right )}{43923 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2}}\) | \(39\) |
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Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (19573 \, x^{3} + 62232 \, x^{2} + 52044 \, x + 13040\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
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\[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {19573 \, x}{219615 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {27 \, x^{2}}{10 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {19573}{4392300 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {95567 \, x}{36300 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {22039}{36300 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (72) = 144\).
Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.68 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{17569200 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {19 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{133100 \, \sqrt {5 \, x + 3}} + \frac {98 \, {\left (17 \, \sqrt {5} {\left (5 \, x + 3\right )} + 99 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1098075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {627 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1098075 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {\sqrt {5\,x+3}\,\left (\frac {19573\,x^3}{1098075}+\frac {20744\,x^2}{366025}+\frac {17348\,x}{366025}+\frac {2608}{219615}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\sqrt {1-2\,x}} \]
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