Integrand size = 28, antiderivative size = 187 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {908 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 (3+5 x)^{3/2}}-\frac {63 (2+3 x)^{3/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {29933 \sqrt {1-2 x} \sqrt {2+3 x}}{219615 \sqrt {3+5 x}}-\frac {29933 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{33275 \sqrt {33}}-\frac {1847 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{33275 \sqrt {33}} \]
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Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {1847 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{33275 \sqrt {33}}-\frac {29933 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{33275 \sqrt {33}}+\frac {7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {63 (3 x+2)^{3/2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {29933 \sqrt {1-2 x} \sqrt {3 x+2}}{219615 \sqrt {5 x+3}}+\frac {908 \sqrt {1-2 x} \sqrt {3 x+2}}{19965 (5 x+3)^{3/2}} \]
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^{3/2} \left (\frac {93}{2}+96 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx \\ & = -\frac {63 (2+3 x)^{3/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (\frac {345}{2}-\frac {333 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {908 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 (3+5 x)^{3/2}}-\frac {63 (2+3 x)^{3/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\frac {3993}{2}-\frac {16623 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{59895} \\ & = \frac {908 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 (3+5 x)^{3/2}}-\frac {63 (2+3 x)^{3/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {29933 \sqrt {1-2 x} \sqrt {2+3 x}}{219615 \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {359847}{8}+\frac {269397 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{658845} \\ & = \frac {908 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 (3+5 x)^{3/2}}-\frac {63 (2+3 x)^{3/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {29933 \sqrt {1-2 x} \sqrt {2+3 x}}{219615 \sqrt {3+5 x}}+\frac {1847 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{66550}+\frac {29933 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{366025} \\ & = \frac {908 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 (3+5 x)^{3/2}}-\frac {63 (2+3 x)^{3/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {29933 \sqrt {1-2 x} \sqrt {2+3 x}}{219615 \sqrt {3+5 x}}-\frac {29933 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{33275 \sqrt {33}}-\frac {1847 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{33275 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {\frac {5 \sqrt {2+3 x} \left (57437+423882 x+905823 x^2+598660 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+i \sqrt {33} \left (29933 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-31780 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{1098075} \]
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Time = 1.39 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {25721}{1815000}+\frac {42883 x}{1815000}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {845957}{43923000}-\frac {29933 x}{2196150}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {39983 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{7686525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {59866 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{7686525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
default | \(\frac {\sqrt {1-2 x}\, \left (299330 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-299640 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+29933 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-29964 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-89799 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+89892 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+8979900 x^{4}+19573945 x^{3}+15416460 x^{2}+5100375 x +574370\right )}{1098075 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\) | \(311\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {225 \, {\left (598660 \, x^{3} + 905823 \, x^{2} + 423882 \, x + 57437\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 555388 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 1346985 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{49413375 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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