Integrand size = 26, antiderivative size = 151 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}-\frac {240911 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \]
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Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {240911 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}}+\frac {3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac {1991 (5 x+3)^{3/2} \sqrt {1-2 x}}{224 (3 x+2)^2}-\frac {21901 \sqrt {5 x+3} \sqrt {1-2 x}}{3136 (3 x+2)} \]
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181}{56} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx \\ & = \frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991}{112} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx \\ & = \frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {21901}{448} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = -\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {240911 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272} \\ & = -\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {240911 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136} \\ & = -\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}-\frac {240911 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (541680+2381420 x+3485960 x^2+1705089 x^3\right )}{(2+3 x)^4}-722733 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{65856} \]
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Time = 3.72 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1705089 x^{3}+3485960 x^{2}+2381420 x +541680\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{9408 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {240911 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43904 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (58541373 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+156110328 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+156110328 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+23871246 x^{3} \sqrt {-10 x^{2}-x +3}+69382368 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +48803440 x^{2} \sqrt {-10 x^{2}-x +3}+11563728 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33339880 x \sqrt {-10 x^{2}-x +3}+7583520 \sqrt {-10 x^{2}-x +3}\right )}{131712 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
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Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {722733 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1705089 \, x^{3} + 3485960 \, x^{2} + 2381420 \, x + 541680\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{131712 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{5}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {240911}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {9955}{2352} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{168 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {5973 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {73667 \, \sqrt {-10 \, x^{2} - x + 3}}{9408 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).
Time = 0.51 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {240911}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (543 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 696920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 156094400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {11919936000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {47679744000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4704 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^5} \,d x \]
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