Optimal. Leaf size=108 \[ \frac {(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac {137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac {137 \sqrt {1-2 x}}{10584 (3 x+2)}+\frac {137 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac {(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac {137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac {137 \sqrt {1-2 x}}{10584 (3 x+2)}+\frac {137 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}+\frac {137}{84} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}-\frac {137}{252} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1512}\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{10584}\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{10584}\\ &=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.39 \[ \frac {(1-2 x)^{5/2} \left (\frac {12005}{(3 x+2)^4}-2192 \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{1008420} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 99, normalized size = 0.92 \[ \frac {137 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (3699 \, x^{3} - 13245 \, x^{2} - 7990 \, x + 970\right )} \sqrt {-2 \, x + 1}}{222264 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 100, normalized size = 0.93 \[ \frac {137}{222264} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3699 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 15393 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 46991 \, \sqrt {-2 \, x + 1}}{84672 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.61 \[ -\frac {137 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{111132}-\frac {1296 \left (-\frac {137 \left (-2 x +1\right )^{\frac {7}{2}}}{254016}-\frac {733 \left (-2 x +1\right )^{\frac {5}{2}}}{326592}+\frac {1507 \left (-2 x +1\right )^{\frac {3}{2}}}{139968}-\frac {959 \sqrt {-2 x +1}}{139968}\right )}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 110, normalized size = 1.02 \[ \frac {137}{222264} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3699 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 15393 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 46991 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 89, normalized size = 0.82 \[ \frac {\frac {959\,\sqrt {1-2\,x}}{8748}-\frac {1507\,{\left (1-2\,x\right )}^{3/2}}{8748}+\frac {733\,{\left (1-2\,x\right )}^{5/2}}{20412}+\frac {137\,{\left (1-2\,x\right )}^{7/2}}{15876}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {137\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{111132} \]
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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