Integrand size = 28, antiderivative size = 191 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}+\frac {1344984 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {40456 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715} \]
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Time = 0.06 (sec) , antiderivative size = 191, 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} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {40456 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715}+\frac {1344984 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {1344984 \sqrt {1-2 x} \sqrt {3 x+2}}{3773 \sqrt {5 x+3}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {436 \sqrt {1-2 x}}{245 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}} \]
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Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2}{35} \int \frac {59-75 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {4}{735} \int \frac {\frac {8631}{2}-4905 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {8 \int \frac {183915-\frac {227565 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{5145} \\ & = \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}-\frac {16 \int \frac {\frac {9579285}{4}+\frac {7565535 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{56595} \\ & = \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}-\frac {60684 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}-\frac {4034952 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{18865} \\ & = \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}+\frac {1344984 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {40456 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.68 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {\sqrt {1-2 x} \left (25529443+116993058 x+178568982 x^2+90786420 x^3\right )}{(2+3 x)^{5/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (168123 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-173180 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{18865} \]
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Time = 1.39 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.42
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105 \left (\frac {2}{3}+x \right )^{3}}-\frac {646 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{735 \left (\frac {2}{3}+x \right )^{2}}-\frac {83294 \left (-30 x^{2}-3 x +9\right )}{1715 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1702984 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{132055 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2689968 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{132055 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {250 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(271\) |
default | \(\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (5878224 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-6052428 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+7837632 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8069904 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+2612544 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2689968 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-181572840 x^{4}-266351544 x^{3}-55417134 x^{2}+65934172 x +25529443\right )}{18865 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(314\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (90786420 \, x^{3} + 178568982 \, x^{2} + 116993058 \, x + 25529443\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 3808304 \, \sqrt {-30} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 10087380 \, \sqrt {-30} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{282975 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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