Integrand size = 28, antiderivative size = 218 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {1473539 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750 \sqrt {33}}-\frac {31288 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{109375 \sqrt {33}} \]
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Time = 0.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159, 164, 114, 120} \[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=-\frac {31288 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{109375 \sqrt {33}}-\frac {1473539 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac {118 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 \sqrt {5 x+3}}+\frac {958 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{1925}+\frac {5153 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{48125}-\frac {12601 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{240625} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {23}{2}-30 x\right ) (2+3 x)^{7/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (\frac {4875}{4}-\frac {7185 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {4 \int \frac {(2+3 x)^{3/2} \left (-\frac {32295}{4}+\frac {77295 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{28875} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}+\frac {4 \int \frac {\sqrt {2+3 x} \left (\frac {1297125}{8}+\frac {567045 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{721875} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {4 \int \frac {-\frac {42883065}{8}-\frac {66309255 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10828125} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}+\frac {15644 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{109375}+\frac {1473539 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2406250} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {1473539 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750 \sqrt {33}}-\frac {31288 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=\frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (54083+1854575 x+6882975 x^2+8575875 x^3+3341250 x^4\right )}{(3+5 x)^{3/2}}+1473539 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1536115 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{7218750} \]
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Time = 1.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {\left (7241685 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-7367695 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+4345011 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4420617 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-200475000 x^{6}-547965000 x^{5}-431912250 x^{4}-8586750 x^{3}+115868770 x^{2}+36550670 x +1081660\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{7218750 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(234\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {1107 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {243 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {952957 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{25265625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1473539 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{25265625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {162 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{234375 \left (x +\frac {3}{5}\right )^{2}}-\frac {854 \left (-30 x^{2}-5 x +10\right )}{515625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(280\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=\frac {900 \, {\left (3341250 \, x^{4} + 8575875 \, x^{3} + 6882975 \, x^{2} + 1854575 \, x + 54083\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 51874733 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 132618510 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{649687500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{9/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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