Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=-\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {17830424 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}-\frac {1717916 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8751645} \]
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Time = 0.06 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=-\frac {1717916 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8751645}-\frac {17830424 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac {1864 \sqrt {1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{5/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {5 x+3}}{8751645 \sqrt {3 x+2}}-\frac {558524 \sqrt {1-2 x} \sqrt {5 x+3}}{1250235 (3 x+2)^{3/2}} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {(3+5 x)^{3/2} \left (-584+\frac {345 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx \\ & = -\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {8 \int \frac {\sqrt {3+5 x} \left (-\frac {127341}{4}+\frac {18435 x}{4}\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{59535} \\ & = -\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {16 \int \frac {-744972-\frac {2253255 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{3750705} \\ & = -\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {32 \int \frac {-\frac {94409715}{16}-\frac {33432045 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26254935} \\ & = -\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {9448538 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8751645}+\frac {17830424 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{8751645} \\ & = -\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {17830424 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}-\frac {1717916 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (159578303+955601637 x+2115318249 x^2+2043155529 x^3+722132172 x^4\right )}{2 (2+3 x)^{9/2}}+i \sqrt {33} \left (4457606 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4887085 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{26254935} \]
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Time = 1.33 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.36
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {1370 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{413343 \left (\frac {2}{3}+x \right )^{4}}-\frac {205474 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4822335 \left (\frac {2}{3}+x \right )^{3}}+\frac {1243066 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11252115 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {35660848}{1750329} x^{2}-\frac {17830424}{8751645} x +\frac {17830424}{2917215}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {25175924 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{183784545 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {35660848 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{183784545 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{177147 \left (\frac {2}{3}+x \right )^{5}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(302\) |
default | \(-\frac {2 \left (746467326 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-722132172 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1990579536 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1925685792 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1990579536 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1925685792 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+884702016 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-855860352 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+147450336 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-142643392 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-21663965160 x^{6}-63461062386 x^{5}-63089824509 x^{4}-16625604096 x^{3}+11383710240 x^{2}+8121679824 x +1436204727\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{26254935 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(504\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (722132172 \, x^{4} + 2043155529 \, x^{3} + 2115318249 \, x^{2} + 955601637 \, x + 159578303\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 180704207 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 401184540 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{1181472075 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]
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