Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {28174 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{28125} \]
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Time = 0.04 (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 = {99, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=-\frac {28174 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{28125}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {2 \sqrt {3 x+2} (1-2 x)^{5/2}}{5 \sqrt {5 x+3}}-\frac {24}{125} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {3028 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{5625} \]
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (-\frac {17}{2}-18 x\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{375} \int \frac {\left (-\frac {1887}{4}-\frac {2271 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {8 \int \frac {-\frac {53121}{8}-\frac {60873 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16875} \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {81164 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{28125}+\frac {154957 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{28125} \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {28174 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \left (-7287-2530 x+900 x^2\right )}{\sqrt {3+5 x}}-40582 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+26495 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{84375} \]
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Time = 1.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (24981 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-40582 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+81000 x^{4}-214200 x^{3}-720780 x^{2}-33405 x +218610\right )}{84375 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}-\frac {1228 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5625}-\frac {35414 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{590625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {162328 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{590625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\frac {1350 \, {\left (900 \, x^{2} - 2530 \, x - 7287\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 136571 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 3652380 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{3796875 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {3 x + 2}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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