Integrand size = 28, antiderivative size = 158 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=-\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {438 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {146}{343} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{343} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=-\frac {17}{343} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {146}{343} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}+\frac {438 \sqrt {1-2 x} \sqrt {5 x+3}}{343 \sqrt {3 x+2}}-\frac {143 \sqrt {5 x+3}}{49 \sqrt {1-2 x} \sqrt {3 x+2}} \]
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {1}{21} \int \frac {\sqrt {3+5 x} \left (\frac {249}{2}+180 x\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx \\ & = -\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {1}{147} \int \frac {-174+\frac {135 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {438 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {2 \int \frac {-\frac {8445}{4}-3285 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1029} \\ & = -\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {438 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {187}{686} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {438}{343} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {438 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {146}{343} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{343} \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 = 8.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.59 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\frac {\frac {\sqrt {3+5 x} \left (-72+3445 x+5256 x^2\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}}+i \sqrt {33} \left (438 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-455 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{1029} \]
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Time = 1.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.44
method | result | size |
default | \(-\frac {\left (858 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-876 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-429 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+438 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-26280 x^{3}-32993 x^{2}-9975 x +216\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{1029 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\) | \(228\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-3 x +9\right )}{1029 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {563 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {292 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2401 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {4400}{343} x^{2}-\frac {16720}{1029} x -\frac {1760}{343}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{588 \left (x -\frac {1}{2}\right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
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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 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\frac {30 \, {\left (5256 \, x^{2} + 3445 \, x - 72\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5087 \, \sqrt {-30} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13140 \, \sqrt {-30} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{30870 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}} \,d x \]
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