Integrand size = 26, antiderivative size = 171 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {575}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {326717 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{13608 \sqrt {7}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {99, 154, 159, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {575}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {326717 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{13608 \sqrt {7}}+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {331 \sqrt {1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac {39745 \sqrt {1-2 x} \sqrt {5 x+3}}{4536} \]
[In]
[Out]
Rule 56
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx \\ & = -\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (\frac {139}{4}-1065 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = \frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {\int \frac {\left (\frac {8757}{8}-\frac {119235 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134} \\ & = -\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}+\frac {\int \frac {-\frac {317283}{4}-241500 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6804} \\ & = -\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {2875}{243} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {326717 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{27216} \\ & = -\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}+\frac {326717 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{13608}-\frac {1}{243} \left (1150 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = -\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {575}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {326717 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{13608 \sqrt {7}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (78416+275022 x+286791 x^2+75600 x^3\right )}{(2+3 x)^3}+225400 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-326717 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{95256} \]
[In]
[Out]
Time = 1.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (75600 x^{3}+286791 x^{2}+275022 x +78416\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4536 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (-\frac {575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{486}+\frac {326717 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{190512}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(142\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (8821359 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-6085800 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+17642718 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-12171600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-3175200 x^{3} \sqrt {-10 x^{2}-x +3}+11761812 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -8114400 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -12045222 x^{2} \sqrt {-10 x^{2}-x +3}+2613736 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1803200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11550924 x \sqrt {-10 x^{2}-x +3}-3293472 \sqrt {-10 x^{2}-x +3}\right )}{190512 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(270\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {326717 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 225400 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (75600 \, x^{3} + 286791 \, x^{2} + 275022 \, x + 78416\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{190512 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
[In]
[Out]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{4}}\, dx \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {865}{2646} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {173 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {34805}{5292} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {575}{486} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {326717}{190512} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {152917}{31752} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2507 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3528 \, {\left (3 \, x + 2\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (129) = 258\).
Time = 0.58 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.36 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {326717}{1905120} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {575}{486} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {10}{81} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, {\left (2463 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1767360 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 377652800 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{756 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \]
[In]
[Out]