Integrand size = 28, antiderivative size = 253 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{22235661} \]
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Time = 0.08 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=-\frac {1199452 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{22235661}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {5 x+3}}{22235661 \sqrt {3 x+2}}-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{453789 (3 x+2)^{5/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {5 x+3}}{64827 (3 x+2)^{7/2}}+\frac {295 \sqrt {1-2 x} \sqrt {5 x+3}}{1323 (3 x+2)^{9/2}} \]
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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}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {1}{7} \int \frac {\left (-\frac {555}{2}-490 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {2 \int \frac {-\frac {169585}{4}-\frac {144025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx}{1323} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {4 \int \frac {-245765-\frac {1683625 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{64827} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {7378905}{8}-\frac {3135525 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2268945} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {17369985}{8}-\frac {14737425 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{47647845} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {185288025}{16}-\frac {113175525 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{333534915} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}+\frac {6036028 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{22235661}+\frac {6596986 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{22235661} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.44 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {4 \left (-\frac {3 \sqrt {3+5 x} \left (-52688263-243200677 x-227945505 x^2+466728543 x^3+985046292 x^4+488918268 x^5\right )}{2 \sqrt {1-2 x} (2+3 x)^{9/2}}+i \sqrt {33} \left (1509007 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1808870 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{66706983} \]
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Time = 1.37 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.26
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{5}}+\frac {1262 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5250987 \left (\frac {2}{3}+x \right )^{4}}-\frac {30014 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12252303 \left (\frac {2}{3}+x \right )^{3}}-\frac {118570 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28588707 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {65848840}{22235661} x^{2}-\frac {6584884}{22235661} x +\frac {6584884}{7411887}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {9882028 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12072056 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {968 \left (-30 x^{2}-38 x -12\right )}{117649 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(319\) |
default | \(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (276291972 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-244459134 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+736778592 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-651891024 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+736778592 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-651891024 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+327457152 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-289729344 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+54576192 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-48288224 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-7333774020 x^{6}-19175958792 x^{5}-15866344773 x^{4}-781374312 x^{3}+5699519700 x^{2}+2979130038 x +474194367\right )}{66706983 \left (2+3 x \right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )}\) | \(504\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (488918268 \, x^{5} + 985046292 \, x^{4} + 466728543 \, x^{3} - 227945505 \, x^{2} - 243200677 \, x - 52688263\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 76465654 \, \sqrt {-30} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 135810630 \, \sqrt {-30} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{3001814235 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{11/2}} \,d x \]
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