Optimal. Leaf size=128 \[ \frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}-\frac {43 (1-2 x)^{5/2}}{315 (3 x+2)^4}+\frac {43 (1-2 x)^{3/2}}{567 (3 x+2)^3}+\frac {43 \sqrt {1-2 x}}{7938 (3 x+2)}-\frac {43 \sqrt {1-2 x}}{1134 (3 x+2)^2}+\frac {43 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}-\frac {43 (1-2 x)^{5/2}}{315 (3 x+2)^4}+\frac {43 (1-2 x)^{3/2}}{567 (3 x+2)^3}+\frac {43 \sqrt {1-2 x}}{7938 (3 x+2)}-\frac {43 \sqrt {1-2 x}}{1134 (3 x+2)^2}+\frac {43 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \]
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)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}+\frac {172}{105} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}-\frac {43}{63} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac {43 (1-2 x)^{3/2}}{567 (2+3 x)^3}+\frac {43}{189} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac {43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac {43 \sqrt {1-2 x}}{1134 (2+3 x)^2}-\frac {43 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1134}\\ &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac {43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac {43 \sqrt {1-2 x}}{1134 (2+3 x)^2}+\frac {43 \sqrt {1-2 x}}{7938 (2+3 x)}-\frac {43 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{7938}\\ &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac {43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac {43 \sqrt {1-2 x}}{1134 (2+3 x)^2}+\frac {43 \sqrt {1-2 x}}{7938 (2+3 x)}+\frac {43 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{7938}\\ &=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac {43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac {43 \sqrt {1-2 x}}{1134 (2+3 x)^2}+\frac {43 \sqrt {1-2 x}}{7938 (2+3 x)}+\frac {43 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.33 \[ \frac {(1-2 x)^{7/2} \left (\frac {117649}{(3 x+2)^5}-5504 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{12353145} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 115, normalized size = 0.90 \[ \frac {215 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (17415 \, x^{4} - 116415 \, x^{3} - 53772 \, x^{2} + 3322 \, x - 7018\right )} \sqrt {-2 \, x + 1}}{833490 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 116, normalized size = 0.91 \[ -\frac {43}{166698} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {17415 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 163170 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 809088 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 1032430 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 516215 \, \sqrt {-2 \, x + 1}}{635040 \, {\left (3 \, x + 2\right )}^{5}} \]
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 \[ \frac {43 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{83349}+\frac {-\frac {43 \left (-2 x +1\right )^{\frac {9}{2}}}{49}-\frac {74 \left (-2 x +1\right )^{\frac {7}{2}}}{9}+\frac {5504 \left (-2 x +1\right )^{\frac {5}{2}}}{135}-\frac {4214 \left (-2 x +1\right )^{\frac {3}{2}}}{81}+\frac {2107 \sqrt {-2 x +1}}{81}}{\left (-6 x -4\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 128, normalized size = 1.00 \[ -\frac {43}{166698} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17415 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 163170 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 809088 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 1032430 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 516215 \, \sqrt {-2 \, x + 1}}{19845 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 107, normalized size = 0.84 \[ \frac {43\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{83349}+\frac {\frac {4214\,{\left (1-2\,x\right )}^{3/2}}{19683}-\frac {2107\,\sqrt {1-2\,x}}{19683}-\frac {5504\,{\left (1-2\,x\right )}^{5/2}}{32805}+\frac {74\,{\left (1-2\,x\right )}^{7/2}}{2187}+\frac {43\,{\left (1-2\,x\right )}^{9/2}}{11907}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
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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