Integrand size = 28, antiderivative size = 156 \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {133}{22} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {100 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {4621 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10 \sqrt {33}}-\frac {139 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10 \sqrt {33}} \]
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Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {139 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10 \sqrt {33}}-\frac {4621 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10 \sqrt {33}}+\frac {\sqrt {5 x+3} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {100 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 \sqrt {1-2 x}}-\frac {133}{22} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^{3/2} \left (\frac {55}{2}+45 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {100 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-1845-\frac {5985 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {133}{22} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {100 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {1}{495} \int \frac {\frac {263295}{4}+\frac {207945 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {133}{22} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {100 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {139}{20} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {4621}{110} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {133}{22} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {100 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {4621 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10 \sqrt {33}}-\frac {139 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.86 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {5 \sqrt {2+3 x} \sqrt {3+5 x} \left (711-2060 x+198 x^2\right )+4621 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4760 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{330 (1-2 x)^{3/2}} \]
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Time = 1.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.49
method | result | size |
default | \(-\frac {\left (8976 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9242 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4488 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+4621 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+14850 x^{4}-135690 x^{3}-136435 x^{2}+5745 x +21330\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{330 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(233\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {3 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4}+\frac {5851 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2310 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4621 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1155 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {4655}{22} x^{2}-\frac {17689}{66} x -\frac {931}{11}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {49 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{48 \left (x -\frac {1}{2}\right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(238\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {225 \, {\left (198 \, x^{2} - 2060 \, x + 711\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 78506 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 207945 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{14850 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
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