Optimal. Leaf size=120 \[ \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}+\frac {\sqrt {1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac {39185 \sqrt {1-2 x}}{57624 (3 x+2)}-\frac {39185 \sqrt {1-2 x}}{24696 (3 x+2)^2}-\frac {39185 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 145, 51, 63, 206} \[ \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}+\frac {\sqrt {1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac {39185 \sqrt {1-2 x}}{57624 (3 x+2)}-\frac {39185 \sqrt {1-2 x}}{24696 (3 x+2)^2}-\frac {39185 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 98
Rule 145
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{7} \int \frac {(-173-325 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac {39185 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1764}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac {39185 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{8232}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac {39185 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{57624}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac {39185 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{57624}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac {39185 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.49 \[ \frac {62696 (3 x+2)^4 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+49 \left (44100 x^2+58389 x+19333\right )}{259308 \sqrt {1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 114, normalized size = 0.95 \[ \frac {39185 \, \sqrt {21} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2115990 \, x^{4} + 4819755 \, x^{3} + 4093057 \, x^{2} + 1534434 \, x + 213998\right )} \sqrt {-2 \, x + 1}}{1210104 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 109, normalized size = 0.91 \[ \frac {39185}{1210104} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {5324}{16807 \, \sqrt {-2 \, x + 1}} - \frac {2231037 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 15062817 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 33905795 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 25445455 \, \sqrt {-2 \, x + 1}}{3226944 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.62 \[ -\frac {39185 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{605052}+\frac {5324}{16807 \sqrt {-2 x +1}}+\frac {\frac {743679 \left (-2 x +1\right )^{\frac {7}{2}}}{67228}-\frac {717277 \left (-2 x +1\right )^{\frac {5}{2}}}{9604}+\frac {691955 \left (-2 x +1\right )^{\frac {3}{2}}}{4116}-\frac {74185 \sqrt {-2 x +1}}{588}}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 119, normalized size = 0.99 \[ \frac {39185}{1210104} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1057995 \, {\left (2 \, x - 1\right )}^{4} + 9051735 \, {\left (2 \, x - 1\right )}^{3} + 28993349 \, {\left (2 \, x - 1\right )}^{2} + 82402418 \, x - 19287625}{28812 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2401 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 98, normalized size = 0.82 \[ \frac {\frac {840841\,x}{23814}+\frac {591701\,{\left (2\,x-1\right )}^2}{47628}+\frac {431035\,{\left (2\,x-1\right )}^3}{111132}+\frac {39185\,{\left (2\,x-1\right )}^4}{86436}-\frac {393625}{47628}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {39185\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{605052} \]
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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