Integrand size = 24, antiderivative size = 174 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}-\frac {4369 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}} \]
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Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 154, 150, 44, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {4369 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}}+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^6}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{735 (3 x+2)^5}-\frac {\sqrt {1-2 x} (237807 x+146585)}{185220 (3 x+2)^4}-\frac {4369 \sqrt {1-2 x}}{1210104 (3 x+2)}-\frac {4369 \sqrt {1-2 x}}{518616 (3 x+2)^2} \]
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Rule 44
Rule 65
Rule 99
Rule 150
Rule 154
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx \\ & = -\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {1}{378} \int \frac {(3+5 x)^2 (-1674+2160 x)}{\sqrt {1-2 x} (2+3 x)^6} \, dx \\ & = -\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\int \frac {(3+5 x) (-118854+144450 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{39690} \\ & = -\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}+\frac {4369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{37044} \\ & = -\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}+\frac {4369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{172872} \\ & = -\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}+\frac {4369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1210104} \\ & = -\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}-\frac {4369 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1210104} \\ & = -\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}-\frac {4369 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.46 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (7033976+606784 x-98441652 x^2-182748162 x^3-42669876 x^4+76086135 x^5+15925005 x^6\right )}{2 (2+3 x)^7}-21845 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63530460} \]
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Time = 1.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\frac {31850010 x^{7}+136247265 x^{6}-161425887 x^{5}-322826448 x^{4}-14135142 x^{3}+99655220 x^{2}+13461168 x -7033976}{6050520 \left (2+3 x \right )^{7} \sqrt {1-2 x}}-\frac {4369 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12706092}\) | \(71\) |
pseudoelliptic | \(\frac {-43690 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{7} \sqrt {21}-21 \sqrt {1-2 x}\, \left (15925005 x^{6}+76086135 x^{5}-42669876 x^{4}-182748162 x^{3}-98441652 x^{2}+606784 x +7033976\right )}{127060920 \left (2+3 x \right )^{7}}\) | \(75\) |
trager | \(-\frac {\left (15925005 x^{6}+76086135 x^{5}-42669876 x^{4}-182748162 x^{3}-98441652 x^{2}+606784 x +7033976\right ) \sqrt {1-2 x}}{6050520 \left (2+3 x \right )^{7}}-\frac {4369 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{25412184}\) | \(92\) |
derivativedivides | \(\frac {\frac {353889 \left (1-2 x \right )^{\frac {13}{2}}}{67228}-\frac {196605 \left (1-2 x \right )^{\frac {11}{2}}}{2401}+\frac {5639843 \left (1-2 x \right )^{\frac {9}{2}}}{20580}+\frac {172608 \left (1-2 x \right )^{\frac {7}{2}}}{1715}-\frac {725323 \left (1-2 x \right )^{\frac {5}{2}}}{420}+\frac {21845 \left (1-2 x \right )^{\frac {3}{2}}}{9}-\frac {30583 \sqrt {1-2 x}}{36}}{\left (-4-6 x \right )^{7}}-\frac {4369 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12706092}\) | \(93\) |
default | \(\frac {\frac {353889 \left (1-2 x \right )^{\frac {13}{2}}}{67228}-\frac {196605 \left (1-2 x \right )^{\frac {11}{2}}}{2401}+\frac {5639843 \left (1-2 x \right )^{\frac {9}{2}}}{20580}+\frac {172608 \left (1-2 x \right )^{\frac {7}{2}}}{1715}-\frac {725323 \left (1-2 x \right )^{\frac {5}{2}}}{420}+\frac {21845 \left (1-2 x \right )^{\frac {3}{2}}}{9}-\frac {30583 \sqrt {1-2 x}}{36}}{\left (-4-6 x \right )^{7}}-\frac {4369 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12706092}\) | \(93\) |
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Time = 0.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {21845 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (15925005 \, x^{6} + 76086135 \, x^{5} - 42669876 \, x^{4} - 182748162 \, x^{3} - 98441652 \, x^{2} + 606784 \, x + 7033976\right )} \sqrt {-2 \, x + 1}}{127060920 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {4369}{25412184} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {15925005 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 247722300 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 829056921 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 304480512 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 5224501569 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 7342978300 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2570042405 \, \sqrt {-2 \, x + 1}}{3025260 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {4369}{25412184} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {15925005 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 247722300 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 829056921 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 304480512 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 5224501569 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 7342978300 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2570042405 \, \sqrt {-2 \, x + 1}}{387233280 \, {\left (3 \, x + 2\right )}^{7}} \]
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Time = 1.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {\frac {21845\,{\left (1-2\,x\right )}^{3/2}}{19683}-\frac {30583\,\sqrt {1-2\,x}}{78732}-\frac {725323\,{\left (1-2\,x\right )}^{5/2}}{918540}+\frac {57536\,{\left (1-2\,x\right )}^{7/2}}{1250235}+\frac {5639843\,{\left (1-2\,x\right )}^{9/2}}{45008460}-\frac {21845\,{\left (1-2\,x\right )}^{11/2}}{583443}+\frac {4369\,{\left (1-2\,x\right )}^{13/2}}{1815156}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}-\frac {4369\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12706092} \]
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