Integrand size = 24, antiderivative size = 128 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {8051 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
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Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 43, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {8051 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}}+\frac {347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac {8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (3 x+2)} \]
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Rule 43
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {(1-2 x)^{5/2} (1403+2625 x)}{(2+3 x)^5} \, dx \\ & = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}+\frac {8051 \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4} \, dx}{2940} \\ & = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}-\frac {8051 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3} \, dx}{5292} \\ & = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {8051 \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx}{10584} \\ & = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {8051 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{31752} \\ & = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {8051 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31752} \\ & = -\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {8051 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (503276+2919346 x+8277204 x^2+12406455 x^3+7323345 x^4\right )}{2 (2+3 x)^5}+40255 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1666980} \]
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Time = 1.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
method | result | size |
risch | \(\frac {14646690 x^{5}+17489565 x^{4}+4147953 x^{3}-2438512 x^{2}-1912794 x -503276}{158760 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {8051 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(61\) |
pseudoelliptic | \(\frac {80510 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-21 \sqrt {1-2 x}\, \left (7323345 x^{4}+12406455 x^{3}+8277204 x^{2}+2919346 x +503276\right )}{3333960 \left (2+3 x \right )^{5}}\) | \(65\) |
derivativedivides | \(-\frac {3888 \left (-\frac {54247 \left (1-2 x \right )^{\frac {9}{2}}}{2286144}+\frac {12269 \left (1-2 x \right )^{\frac {7}{2}}}{69984}-\frac {16102 \left (1-2 x \right )^{\frac {5}{2}}}{32805}+\frac {394499 \left (1-2 x \right )^{\frac {3}{2}}}{629856}-\frac {394499 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}+\frac {8051 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(75\) |
default | \(-\frac {3888 \left (-\frac {54247 \left (1-2 x \right )^{\frac {9}{2}}}{2286144}+\frac {12269 \left (1-2 x \right )^{\frac {7}{2}}}{69984}-\frac {16102 \left (1-2 x \right )^{\frac {5}{2}}}{32805}+\frac {394499 \left (1-2 x \right )^{\frac {3}{2}}}{629856}-\frac {394499 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}+\frac {8051 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(75\) |
trager | \(-\frac {\left (7323345 x^{4}+12406455 x^{3}+8277204 x^{2}+2919346 x +503276\right ) \sqrt {1-2 x}}{158760 \left (2+3 x \right )^{5}}-\frac {8051 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{666792}\) | \(82\) |
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Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {40255 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (7323345 \, x^{4} + 12406455 \, x^{3} + 8277204 \, x^{2} + 2919346 \, x + 503276\right )} \sqrt {-2 \, x + 1}}{3333960 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {8051}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7323345 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 54106290 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 151487616 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{79380 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {8051}{666792} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7323345 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 54106290 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 151487616 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{2540160 \, {\left (3 \, x + 2\right )}^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {8051\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396}-\frac {\frac {394499\,\sqrt {1-2\,x}}{78732}-\frac {394499\,{\left (1-2\,x\right )}^{3/2}}{39366}+\frac {257632\,{\left (1-2\,x\right )}^{5/2}}{32805}-\frac {12269\,{\left (1-2\,x\right )}^{7/2}}{4374}+\frac {54247\,{\left (1-2\,x\right )}^{9/2}}{142884}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
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