Integrand size = 28, antiderivative size = 160 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {192}{49} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {31940 \sqrt {1-2 x} \sqrt {3 x+2}}{539 \sqrt {5 x+3}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}} \]
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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} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {42-45 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{147} \int \frac {\frac {3495}{2}-1080 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {45495}{2}+\frac {71865 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {288}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {19164}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2}{539} \left (-\frac {\sqrt {1-2 x} \left (60635+186888 x+143730 x^2\right )}{(2+3 x)^{3/2} \sqrt {3+5 x}}-2 i \sqrt {33} \left (1597 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1645 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]
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Time = 1.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.37
method | result | size |
default | \(\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (9306 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9582 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+6204 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-6388 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-287460 x^{3}-230046 x^{2}+65618 x +60635\right )}{539 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(219\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21 \left (\frac {2}{3}+x \right )^{2}}-\frac {358 \left (-30 x^{2}-3 x +9\right )}{49 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {8088 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {12776 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {50 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (143730 \, x^{2} + 186888 \, x + 60635\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 54259 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 143730 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{24255 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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