Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {481339 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}+\frac {124724 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{70875} \]
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Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\frac {124724 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{70875}-\frac {481339 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{3 \sqrt {3 x+2}}-\frac {32}{63} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {2108 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{1575}+\frac {124724 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{14175} \]
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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} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {4}{315} \int \frac {\left (-\frac {1335}{4}-\frac {7905 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {8 \int \frac {\left (\frac {326745}{8}-\frac {467715 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{23625} \\ & = \frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {8 \int \frac {-\frac {2274105}{8}-\frac {7220085 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{212625} \\ & = \frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {481339 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{70875}-\frac {685982 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{70875} \\ & = \frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {481339 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}+\frac {124724 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.48 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (32033+14727 x-21690 x^2+13500 x^3\right )}{\sqrt {2+3 x}}+481339 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-356615 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{212625} \]
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Time = 1.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (336237 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-481339 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4050000 x^{5}+6102000 x^{4}-2552400 x^{3}-12003810 x^{2}+364440 x +2882970\right )}{212625 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(150\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {1364 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945}+\frac {23458 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14175}+\frac {303214 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1488375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {962678 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1488375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {40 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}+\frac {-\frac {980}{81} x^{2}-\frac {98}{81} x +\frac {98}{27}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\frac {2700 \, {\left (13500 \, x^{3} - 21690 \, x^{2} + 14727 \, x + 32033\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 2573833 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 43320510 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{19136250 \, {\left (3 \, x + 2\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
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