Optimal. Leaf size=128 \[ \frac {347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac {8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (3 x+2)}+\frac {8051 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
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Rubi [A] time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 47, 63, 206} \[ \frac {347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac {8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (3 x+2)}+\frac {8051 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \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 {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 \operatorname {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*}
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Mathematica [A] time = 0.13, size = 84, normalized size = 0.66 \[ -\frac {80510 (3 x+2)^5 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )-21 \left (14646690 x^5+17489565 x^4+4147953 x^3-2438512 x^2-1912794 x-503276\right )}{3333960 \sqrt {1-2 x} (3 x+2)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 115, normalized size = 0.90 \[ \frac {40255 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (7323345 \, x^{4} + 12406455 \, x^{3} + 8277204 \, x^{2} + 2919346 \, x + 503276\right )} \sqrt {-2 \, x + 1}}{3333960 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 116, normalized size = 0.91 \[ -\frac {8051}{666792} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7323345 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 54106290 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 151487616 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{2540160 \, {\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.59 \[ \frac {8051 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{333396}-\frac {3888 \left (-\frac {54247 \left (-2 x +1\right )^{\frac {9}{2}}}{2286144}+\frac {12269 \left (-2 x +1\right )^{\frac {7}{2}}}{69984}-\frac {16102 \left (-2 x +1\right )^{\frac {5}{2}}}{32805}+\frac {394499 \left (-2 x +1\right )^{\frac {3}{2}}}{629856}-\frac {394499 \sqrt {-2 x +1}}{1259712}\right )}{\left (-6 x -4\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 128, normalized size = 1.00 \[ -\frac {8051}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7323345 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 54106290 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 151487616 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{79380 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 108, normalized size = 0.84 \[ \frac {8051\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396}-\frac {\frac {394499\,\sqrt {1-2\,x}}{78732}-\frac {394499\,{\left (1-2\,x\right )}^{3/2}}{39366}+\frac {257632\,{\left (1-2\,x\right )}^{5/2}}{32805}-\frac {12269\,{\left (1-2\,x\right )}^{7/2}}{4374}+\frac {54247\,{\left (1-2\,x\right )}^{9/2}}{142884}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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