Integrand size = 28, antiderivative size = 126 \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {10}{3} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {133}{6} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2}{3} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 126, 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 = {99, 159, 164, 114, 120} \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {2}{3} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {133}{6} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {\sqrt {3 x+2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}+\frac {10}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {\sqrt {3+5 x} \left (\frac {39}{2}+30 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {10}{3} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{9} \int \frac {-\frac {1263}{2}-\frac {1995 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {10}{3} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {11}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {133}{6} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {10}{3} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {133}{6} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2}{3} \sqrt {\frac {11}{3}} 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 = 5.43 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {6 (19-5 x) \sqrt {2+3 x} \sqrt {3+5 x}-133 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+137 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{18 \sqrt {1-2 x}} \]
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Time = 1.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (4521 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4655 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+15750 x^{3}-39900 x^{2}-69510 x -23940\right )}{18900 x^{3}+14490 x^{2}-4410 x -3780}\) | \(140\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {5 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6}-\frac {421 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{315 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {19 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {11 \left (-30 x^{2}-38 x -12\right )}{4 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {540 \, \sqrt {5 \, x + 3} {\left (5 \, x - 19\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 4519 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 11970 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{1620 \, {\left (2 \, x - 1\right )}} \]
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\[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {\sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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