Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}-\frac {184636 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {9124 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{26411 \sqrt {33}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 187, 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 = {106, 157, 164, 114, 120} \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {9124 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {184636 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}+\frac {184636 \sqrt {1-2 x} \sqrt {5 x+3}}{290521 \sqrt {3 x+2}}+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{41503 (3 x+2)^{3/2}}+\frac {1072 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}} \]
[In]
[Out]
Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {193}{2}-75 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {17541}{4}+6030 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{17787} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {123867}{4}-\frac {21915 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{373527} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {2718405}{8}+\frac {2077155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2614689} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}+\frac {4562 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26411}+\frac {184636 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{290521} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}-\frac {184636 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {9124 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (597945-1478206 x-1066908 x^2+3323448 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}+2 i \sqrt {33} \left (46159 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-48440 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{871563} \]
[In]
[Out]
Time = 1.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {107}{29106}-\frac {31 x}{4851}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (\frac {135068}{2614689}-\frac {92318 x}{871563}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}+\frac {241636 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6100941 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {369272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{6100941 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (548064 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-553908 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+91344 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-92318 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-182688 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+184636 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16617240 x^{4}-4635804 x^{3}+10591754 x^{2}+1444893 x -1793835\right )}{871563 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(311\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (45 \, {\left (3323448 \, x^{3} - 1066908 \, x^{2} - 1478206 \, x + 597945\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1656748 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4154310 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{39220335 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
[In]
[Out]