Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{539 \sqrt {33}} \]
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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 = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {37768 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {1255552 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}+\frac {6277760 \sqrt {1-2 x} \sqrt {3 x+2}}{17787 \sqrt {5 x+3}}-\frac {94420 \sqrt {1-2 x} \sqrt {3 x+2}}{1617 (5 x+3)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}} \]
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Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{21} \int \frac {57-75 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4}{147} \int \frac {\frac {8385}{2}-4815 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {343365}{2}-\frac {212445 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{4851} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {8942355}{4}+3531240 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{53361} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}+\frac {18884}{539} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {1255552 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5929} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.81 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {1-2 x} \left (35747225+169778606 x+268408770 x^2+141249600 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (156944 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-161665 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{17787} \]
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Time = 1.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {1294}{51975}-\frac {136 x}{3465}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {5966174}{266805}-\frac {627776 x}{17787}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {1589752 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2511104 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (9145620 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9416640 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+11584452 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-11927744 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3658248 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-3766656 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-282499200 x^{4}-395567940 x^{3}-71148442 x^{2}+98284156 x +35747225\right )}{17787 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(311\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (141249600 \, x^{3} + 268408770 \, x^{2} + 169778606 \, x + 35747225\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 10665286 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 28249920 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{800415 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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