Integrand size = 28, antiderivative size = 125 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {4451 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}-\frac {67 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{55 \sqrt {33}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 155, 164, 114, 120} \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {67 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{55 \sqrt {33}}-\frac {4451 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}-\frac {448 \sqrt {5 x+3} \sqrt {3 x+2}}{363 \sqrt {1-2 x}} \]
[In]
[Out]
Rule 100
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\sqrt {2+3 x} \left (\frac {247}{2}+201 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{363} \int \frac {-4227-\frac {13353 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {67}{110} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {4451 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1210} \\ & = -\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {4451 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}-\frac {67 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{55 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {490 \sqrt {2+3 x} \sqrt {3+5 x} (-6+23 x)-4451 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4585 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{3630 (1-2 x)^{3/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(93)=186\).
Time = 1.39 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {49 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{264 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {5635}{242} x^{2}-\frac {21413}{726} x -\frac {1127}{121}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {2818 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4451 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(219\) |
default | \(-\frac {\left (8646 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8902 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4323 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+4451 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-169050 x^{3}-170030 x^{2}-11760 x +17640\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{3630 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(228\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {44100 \, {\left (23 \, x - 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 151247 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 400590 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{326700 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
[In]
[Out]
\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, dx \]
[In]
[Out]
\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
[In]
[Out]