Integrand size = 28, antiderivative size = 156 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {148831 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6050 \sqrt {33}}-\frac {2252 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3025 \sqrt {33}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 156, 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 = {100, 155, 164, 114, 120} \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2252 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3025 \sqrt {33}}-\frac {148831 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6050 \sqrt {33}}+\frac {7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {434 (3 x+2)^{3/2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {2129 \sqrt {1-2 x} \sqrt {3 x+2}}{19965 \sqrt {5 x+3}} \]
[In]
[Out]
Rule 100
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^{3/2} \left (\frac {233}{2}+201 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx \\ & = -\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-3730-\frac {13143 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx \\ & = \frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {-\frac {282759}{4}-\frac {446493 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{19965} \\ & = \frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {1126 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3025}+\frac {148831 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{66550} \\ & = \frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {148831 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6050 \sqrt {33}}-\frac {2252 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {\frac {10 \sqrt {2+3 x} \left (-28671+66174 x+189851 x^2\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}}+i \sqrt {33} \left (148831 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-153335 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{199650} \]
[In]
[Out]
Time = 1.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.46
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (289146 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-297662 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-144573 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+148831 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5695530 x^{3}-5782240 x^{2}-463350 x +573420\right )}{199650 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\) | \(228\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-5 x +10\right )}{33275 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {94253 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{698775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {148831 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{698775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2904 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {37975}{2662} x^{2}-\frac {144305}{7986} x -\frac {7595}{1331}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {900 \, {\left (189851 \, x^{2} + 66174 \, x - 28671\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5059657 \, \sqrt {-30} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13394790 \, \sqrt {-30} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{17968500 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
[In]
[Out]