Optimal. Leaf size=161 \[ \frac {11 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac {11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {209 \sqrt {1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}} \]
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Rubi [A] time = 0.06, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 12, 149, 146, 63, 206} \[ \frac {11 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac {11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {209 \sqrt {1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 97
Rule 146
Rule 149
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}-\frac {11}{3} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11}{36} \int \frac {\sqrt {1-2 x} (3+5 x)^2 (24+18 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11}{324} \int \frac {(-162-72 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \int \frac {(-9522-3330 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{13608}\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}+\frac {146971 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{15876}\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac {146971 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15876}\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 89, normalized size = 0.55 \[ -\frac {21 \left (52920000 x^6+226697490 x^5+288394965 x^4+106869513 x^3-43687652 x^2-40879074 x-7933096\right )-1469710 (3 x+2)^5 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )}{1666980 \sqrt {1-2 x} (3 x+2)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 119, normalized size = 0.74 \[ \frac {734855 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (26460000 \, x^{5} + 126578745 \, x^{4} + 207486855 \, x^{3} + 157178184 \, x^{2} + 56745266 \, x + 7933096\right )} \sqrt {-2 \, x + 1}}{1666980 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 125, normalized size = 0.78 \[ \frac {146971}{333396} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1000}{729} \, \sqrt {-2 \, x + 1} + \frac {345408705 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 2999598210 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 9762357024 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 14111613390 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7644051695 \, \sqrt {-2 \, x + 1}}{11430720 \, {\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.52 \[ -\frac {146971 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{166698}+\frac {1000 \sqrt {-2 x +1}}{729}+\frac {-\frac {284287 \left (-2 x +1\right )^{\frac {9}{2}}}{294}+\frac {226727 \left (-2 x +1\right )^{\frac {7}{2}}}{27}-\frac {11068432 \left (-2 x +1\right )^{\frac {5}{2}}}{405}+\frac {9599737 \left (-2 x +1\right )^{\frac {3}{2}}}{243}-\frac {31200211 \sqrt {-2 x +1}}{1458}}{\left (-6 x -4\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 137, normalized size = 0.85 \[ \frac {146971}{333396} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1000}{729} \, \sqrt {-2 \, x + 1} + \frac {345408705 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 2999598210 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 9762357024 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 14111613390 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7644051695 \, \sqrt {-2 \, x + 1}}{357210 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 116, normalized size = 0.72 \[ \frac {1000\,\sqrt {1-2\,x}}{729}-\frac {146971\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{166698}+\frac {\frac {31200211\,\sqrt {1-2\,x}}{354294}-\frac {9599737\,{\left (1-2\,x\right )}^{3/2}}{59049}+\frac {11068432\,{\left (1-2\,x\right )}^{5/2}}{98415}-\frac {226727\,{\left (1-2\,x\right )}^{7/2}}{6561}+\frac {284287\,{\left (1-2\,x\right )}^{9/2}}{71442}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
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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