Integrand size = 28, antiderivative size = 129 \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {4}{11} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2}{11} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 157, 164, 114, 120} \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2}{11} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {4}{11} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {20 \sqrt {1-2 x} \sqrt {3 x+2}}{121 \sqrt {5 x+3}}+\frac {2 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}} \]
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Rule 101
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{11} \int \frac {-\frac {11}{2}-\frac {15 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {4}{121} \int \frac {-\frac {3}{4}-15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}-\frac {12}{121} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {3}{11} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {4}{11} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2}{11} \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 = 5.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {2+3 x} \sqrt {3+5 x} (1+20 x)-4 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+2 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{121 \sqrt {1-2 x} (3+5 x)} \]
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Time = 1.43 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {2 \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-70 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2100 x^{2}-1505 x -70\right )}{4235 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(135\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {1}{1210}+\frac {2 x}{121}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{4235 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(195\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {90 \, {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 37 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 180 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{5445 \, {\left (10 \, x^{2} + x - 3\right )}} \]
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\[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {3 x + 2}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {3\,x+2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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