Integrand size = 28, antiderivative size = 156 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {3896 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {164 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{847 \sqrt {33}} \]
[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 = {106, 157, 164, 114, 120} \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=-\frac {164 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {3896 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}+\frac {19480 \sqrt {1-2 x} \sqrt {3 x+2}}{27951 \sqrt {5 x+3}}-\frac {410 \sqrt {1-2 x} \sqrt {3 x+2}}{2541 (5 x+3)^{3/2}}+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} (5 x+3)^{3/2}} \]
[In]
[Out]
Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {95}{2}-45 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {4 \int \frac {-\frac {605}{2}+\frac {615 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2541} \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {18885}{4}-7305 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27951} \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}+\frac {82}{847} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {3896 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{9317} \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {3896 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {164 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {2+3 x} \left (27691-5230 x-97400 x^2\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}}+2 i \sqrt {33} \left (974 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1015 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{27951} \]
[In]
[Out]
Time = 1.36 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.40
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (9570 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9740 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5742 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5844 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-292200 x^{3}-210490 x^{2}+72613 x +55382\right )}{27951 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(219\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {5036 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{195657 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {7792 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{195657 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {5600}{1331} x^{2}-\frac {2800}{3993} x +\frac {5600}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {8 \left (-30 x^{2}-38 x -12\right )}{9317 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
[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 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (97400 \, x^{2} + 5230 \, x - 27691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 34253 \, \sqrt {-30} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 87660 \, \sqrt {-30} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{1257795 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
[In]
[Out]