Integrand size = 28, antiderivative size = 191 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {496 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {582 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2401} \]
[Out]
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 = {100, 157, 164, 114, 120} \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=-\frac {582 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2401}+\frac {496 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {496 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 \sqrt {3 x+2}}-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{343 (3 x+2)^{3/2}}+\frac {58 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}} \]
[In]
[Out]
Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {1}{21} \int \frac {-\frac {169}{2}-150 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {2 \int \frac {\frac {16203}{4}+\frac {14355 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1617} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {10593}{2}+\frac {44055 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{33957} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {8 \int \frac {-\frac {60885}{8}-30690 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{237699} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}-\frac {496 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2401}+\frac {873 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2401} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {496 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {582 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\frac {2 \left (-\frac {11 \sqrt {3+5 x} \left (-885-4616 x+762 x^2+8928 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}-i \sqrt {33} \left (2728 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1855 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{79233} \]
[In]
[Out]
Time = 1.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.19
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {23}{2646}+\frac {31 x}{2646}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (-\frac {121}{43218}+\frac {248 x}{7203}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {82 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{16807 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {992 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
default | \(\frac {2 \sqrt {1-2 x}\, \left (954 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1488 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+159 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-248 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-318 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+496 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-44640 x^{4}-30594 x^{3}+20794 x^{2}+18273 x +2655\right )}{7203 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(311\) |
[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 = 0.67 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=-\frac {90 \, {\left (8928 \, x^{3} + 762 \, x^{2} - 4616 \, x - 885\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 169 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 22320 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{324135 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}} \,d x \]
[In]
[Out]