Integrand size = 26, antiderivative size = 166 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}-\frac {4246733 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {4246733 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {5 x+3}}-\frac {204595 \sqrt {1-2 x}}{168 (5 x+3)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (3 x+2) (5 x+3)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}} \]
[In]
[Out]
Rule 12
Rule 95
Rule 100
Rule 156
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {1}{9} \int \frac {\frac {345}{2}-268 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{126} \int \frac {\frac {87003}{4}-31605 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {1}{882} \int \frac {\frac {16024491}{8}-2569245 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx \\ & = -\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {\frac {1808711289}{16}-\frac {425353005 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{14553} \\ & = -\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {2 \int \frac {97118536977}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{160083} \\ & = -\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {4246733}{112} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {4246733}{56} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}-\frac {4246733 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (43006496+267610802 x+623901861 x^2+645909120 x^3+250551225 x^4\right )}{168 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {4246733 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(127)=254\).
Time = 1.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.80
method | result | size |
default | \(\frac {\left (8599634325 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+27518829840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+35201169837 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+3507717150 x^{4} \sqrt {-10 x^{2}-x +3}+22499191434 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+9042727680 x^{3} \sqrt {-10 x^{2}-x +3}+7185472236 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +8734626054 x^{2} \sqrt {-10 x^{2}-x +3}+917294328 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3746551228 x \sqrt {-10 x^{2}-x +3}+602090944 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{2352 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(298\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {12740199 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (250551225 \, x^{4} + 645909120 \, x^{3} + 623901861 \, x^{2} + 267610802 \, x + 43006496\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2352 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
[In]
[Out]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.45 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {4246733}{784} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {618645 \, x}{28 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1937773}{168 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {199895 \, x}{36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {343}{81 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {4655}{108 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {165739}{216 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1943461}{648 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (127) = 254\).
Time = 0.52 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.59 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {5}{48} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {4246733}{7840} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + 335 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {99 \, \sqrt {10} {\left (21713 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 10391360 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1283172800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5132691200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{28 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
[In]
[Out]