Integrand size = 26, antiderivative size = 200 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {1850}{729} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {3304795 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{326592 \sqrt {7}} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {1850}{729} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {3304795 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{326592 \sqrt {7}}+\frac {1165 \sqrt {1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac {3485 \sqrt {1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac {249575 \sqrt {1-2 x} \sqrt {5 x+3}}{108864} \]
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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)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac {1}{108} \int \frac {\left (-\frac {3655}{4}-1225 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {1}{648} \int \frac {\left (\frac {20765}{8}-\frac {3975 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = -\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {\int \frac {\left (\frac {1308195}{16}-\frac {748725 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{13608} \\ & = \frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac {\int \frac {-\frac {13271205}{8}-3108000 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{81648} \\ & = \frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac {3304795 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{653184}+\frac {9250}{729} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac {3304795 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{326592}+\frac {1}{729} \left (3700 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = \frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {1850}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {3304795 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{326592 \sqrt {7}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (5093072+25998852 x+45563928 x^2+29475315 x^3+3628800 x^4\right )}{(2+3 x)^4}-5801600 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+3304795 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2286144} \]
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Time = 1.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (3628800 x^{4}+29475315 x^{3}+45563928 x^{2}+25998852 x +5093072\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{108864 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {925 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{729}+\frac {3304795 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{4572288}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (267688395 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-469929600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+713835720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-1253145600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-152409600 x^{4} \sqrt {-10 x^{2}-x +3}+713835720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-1253145600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-1237963230 x^{3} \sqrt {-10 x^{2}-x +3}+317260320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -556953600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1913684976 x^{2} \sqrt {-10 x^{2}-x +3}+52876720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-92825600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1091951784 x \sqrt {-10 x^{2}-x +3}-213909024 \sqrt {-10 x^{2}-x +3}\right )}{4572288 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(332\) |
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Time = 0.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {3304795 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5801600 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (3628800 \, x^{4} + 29475315 \, x^{3} + 45563928 \, x^{2} + 25998852 \, x + 5093072\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4572288 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {5755}{49392} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1151 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {182225}{98784} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1488395}{1778112} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {44881 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{197568 \, {\left (3 \, x + 2\right )}} - \frac {28675}{127008} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {925}{729} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {3304795}{4572288} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {1643795}{762048} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (152) = 304\).
Time = 0.65 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=-\frac {660959}{9144576} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {925}{729} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {20}{243} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {55 \, {\left (8191 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 7386792 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2164545600 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 2731201984000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{54432 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^5} \,d x \]
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