Integrand size = 26, antiderivative size = 137 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}-\frac {40787 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \]
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Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {40787 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4 \sqrt {7}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {5 x+3}}-\frac {655 \sqrt {1-2 x}}{4 (5 x+3)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (3 x+2) (5 x+3)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{6} \int \frac {\frac {279}{2}-202 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {1}{42} \int \frac {\frac {51303}{4}-16450 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx \\ & = -\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}-\frac {1}{693} \int \frac {\frac {5790477}{8}-\frac {1361745 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx \\ & = -\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}+\frac {2 \int \frac {310919301}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{7623} \\ & = -\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}+\frac {40787}{8} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}+\frac {40787}{4} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}-\frac {40787 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (206524+975325 x+1533090 x^2+802125 x^3\right )}{12 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {40787 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(104)=208\).
Time = 1.14 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82
method | result | size |
default | \(\frac {\left (27531225 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+69745770 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+66197301 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+11229750 x^{3} \sqrt {-10 x^{2}-x +3}+27898308 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +21463260 x^{2} \sqrt {-10 x^{2}-x +3}+4404996 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13654550 x \sqrt {-10 x^{2}-x +3}+2891336 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{168 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(250\) |
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Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {122361 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (802125 \, x^{3} + 1533090 \, x^{2} + 975325 \, x + 206524\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{168 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {40787}{56} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {17825 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {18611}{12 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {13439 \, x}{18 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {343}{54 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {11123}{108 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1613}{4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (104) = 208\).
Time = 0.42 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.72 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {1}{48} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {40787}{560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {101}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {165 \, \sqrt {10} {\left (89 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {21224 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {84896 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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