Optimal. Leaf size=147 \[ \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {71 \sqrt {1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac {2721 \sqrt {1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac {377748 \sqrt {1-2 x} (3 x+2)^2}{831875}+\frac {63 \sqrt {1-2 x} (831375 x+2492512)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}} \]
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Rubi [A] time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 153, 147, 63, 206} \[ \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {71 \sqrt {1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac {2721 \sqrt {1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac {377748 \sqrt {1-2 x} (3 x+2)^2}{831875}+\frac {63 \sqrt {1-2 x} (831375 x+2492512)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{11} \int \frac {(2+3 x)^4 (173+312 x)}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\int \frac {(2+3 x)^3 (12450+21657 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (446523+755496 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{66550}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {\int \frac {(-31392102-52376625 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1663750}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}+\frac {33873 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{8318750}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8318750}\\ &=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 101, normalized size = 0.69 \[ \frac {\frac {13230 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )}{\sqrt {1-2 x}}-\frac {55 \left (22052250 x^5+129373200 x^4+516610710 x^3+69300960 x^2-377289427 x-154786070\right )}{\sqrt {1-2 x} (5 x+3)^2}-4956 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{41593750} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 99, normalized size = 0.67 \[ \frac {33873 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (242574750 \, x^{5} + 1423105200 \, x^{4} + 5682717810 \, x^{3} + 762244410 \, x^{2} - 4150263077 \, x - 1702670584\right )} \sqrt {-2 \, x + 1}}{457531250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 111, normalized size = 0.76 \[ \frac {729}{5000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8991}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33873}{457531250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {333639}{25000} \, \sqrt {-2 \, x + 1} + \frac {117649}{10648 \, \sqrt {-2 \, x + 1}} + \frac {403 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 891 \, \sqrt {-2 \, x + 1}}{3327500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.57 \[ -\frac {33873 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{228765625}+\frac {729 \left (-2 x +1\right )^{\frac {5}{2}}}{5000}-\frac {8991 \left (-2 x +1\right )^{\frac {3}{2}}}{5000}+\frac {333639 \sqrt {-2 x +1}}{25000}+\frac {117649}{10648 \sqrt {-2 x +1}}+\frac {\frac {403 \left (-2 x +1\right )^{\frac {3}{2}}}{831875}-\frac {81 \sqrt {-2 x +1}}{75625}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 110, normalized size = 0.75 \[ \frac {729}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8991}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33873}{457531250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {333639}{25000} \, \sqrt {-2 \, x + 1} + \frac {1838268849 \, {\left (2 \, x - 1\right )}^{2} + 16176751756 \, x + 808829747}{6655000 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 91, normalized size = 0.62 \[ \frac {\frac {367653449\,x}{3781250}+\frac {1838268849\,{\left (2\,x-1\right )}^2}{166375000}+\frac {73529977}{15125000}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}+\frac {333639\,\sqrt {1-2\,x}}{25000}-\frac {8991\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {729\,{\left (1-2\,x\right )}^{5/2}}{5000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,33873{}\mathrm {i}}{228765625} \]
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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