Integrand size = 28, antiderivative size = 246 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {269045681 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{207900}+\frac {4066493 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23100}+\frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {17888580643 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{189000 \sqrt {33}}+\frac {269045681 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{94500 \sqrt {33}} \]
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Time = 0.06 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {269045681 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{94500 \sqrt {33}}+\frac {17888580643 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{189000 \sqrt {33}}+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}+\frac {18}{11} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}+\frac {9741}{385} \sqrt {1-2 x} (5 x+3)^{5/2} \sqrt {3 x+2}+\frac {4066493 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{23100}+\frac {269045681 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{207900} \]
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac {113}{2}+90 x\right )}{\sqrt {1-2 x}} \, dx \\ & = \frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{55} \int \frac {\left (-9950-\frac {31425 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx \\ & = \frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {\int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {5624625}{4}+2191725 x\right )}{\sqrt {1-2 x}} \, dx}{2475} \\ & = \frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\int \frac {\left (-149936475-\frac {914960925 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{86625} \\ & = \frac {4066493 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23100}+\frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {\int \frac {\sqrt {3+5 x} \left (\frac {78681075975}{8}+\frac {60535278225 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1299375} \\ & = \frac {269045681 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{207900}+\frac {4066493 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23100}+\frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\int \frac {-\frac {637033999725}{2}-\frac {4024930644675 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11694375} \\ & = \frac {269045681 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{207900}+\frac {4066493 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23100}+\frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {269045681 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{189000}-\frac {17888580643 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2079000} \\ & = \frac {269045681 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{207900}+\frac {4066493 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23100}+\frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {17888580643 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{189000 \sqrt {33}}+\frac {269045681 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{94500 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-477155552+273928969 x+198895770 x^2+133330950 x^3+60196500 x^4+12757500 x^5\right )-17888580643 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+18426672005 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{6237000 \sqrt {1-2 x}} \]
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Time = 4.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (5740875000 x^{7}+17373719319 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-17888580643 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+34360200000 x^{6}+96607282500 x^{5}+176337108000 x^{4}+260638195950 x^{3}-22779247470 x^{2}-222671450220 x -85887999360\right )}{187110000 x^{3}+143451000 x^{2}-43659000 x -37422000}\) | \(160\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {12542447 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18480}+\frac {1660125991 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1663200}-\frac {2831262221 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5457375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {17888580643 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{21829500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {740527 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1848}+\frac {675 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{22}+\frac {7045 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{44}-\frac {41503 \left (-30 x^{2}-38 x -12\right )}{64 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(312\) |
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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 = 0.38 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {2700 \, {\left (12757500 \, x^{5} + 60196500 \, x^{4} + 133330950 \, x^{3} + 198895770 \, x^{2} + 273928969 \, x - 477155552\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 607817044771 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 1609972257870 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{561330000 \, {\left (2 \, x - 1\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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