Optimal. Leaf size=101 \[ \frac {209 (1-2 x)^{5/2}}{2646 (3 x+2)^2}-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}-\frac {7559 (1-2 x)^{3/2}}{7938 (3 x+2)}-\frac {7559 \sqrt {1-2 x}}{3969}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]
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Rubi [A] time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {89, 78, 47, 50, 63, 206} \[ \frac {209 (1-2 x)^{5/2}}{2646 (3 x+2)^2}-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}-\frac {7559 (1-2 x)^{3/2}}{7938 (3 x+2)}-\frac {7559 \sqrt {1-2 x}}{3969}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2}}{189 (2+3 x)^3}+\frac {1}{189} \int \frac {(1-2 x)^{3/2} (841+1575 x)}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{189 (2+3 x)^3}+\frac {209 (1-2 x)^{5/2}}{2646 (2+3 x)^2}+\frac {7559 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2} \, dx}{2646}\\ &=-\frac {(1-2 x)^{5/2}}{189 (2+3 x)^3}+\frac {209 (1-2 x)^{5/2}}{2646 (2+3 x)^2}-\frac {7559 (1-2 x)^{3/2}}{7938 (2+3 x)}-\frac {7559 \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx}{2646}\\ &=-\frac {7559 \sqrt {1-2 x}}{3969}-\frac {(1-2 x)^{5/2}}{189 (2+3 x)^3}+\frac {209 (1-2 x)^{5/2}}{2646 (2+3 x)^2}-\frac {7559 (1-2 x)^{3/2}}{7938 (2+3 x)}-\frac {7559 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134}\\ &=-\frac {7559 \sqrt {1-2 x}}{3969}-\frac {(1-2 x)^{5/2}}{189 (2+3 x)^3}+\frac {209 (1-2 x)^{5/2}}{2646 (2+3 x)^2}-\frac {7559 (1-2 x)^{3/2}}{7938 (2+3 x)}+\frac {7559 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1134}\\ &=-\frac {7559 \sqrt {1-2 x}}{3969}-\frac {(1-2 x)^{5/2}}{189 (2+3 x)^3}+\frac {209 (1-2 x)^{5/2}}{2646 (2+3 x)^2}-\frac {7559 (1-2 x)^{3/2}}{7938 (2+3 x)}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 54, normalized size = 0.53 \[ \frac {(1-2 x)^{5/2} \left (245 (627 x+404)-30236 (3 x+2)^3 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{648270 (3 x+2)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 90, normalized size = 0.89 \[ \frac {7559 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (37800 \, x^{3} + 100809 \, x^{2} + 82493 \, x + 21424\right )} \sqrt {-2 \, x + 1}}{23814 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 93, normalized size = 0.92 \[ -\frac {7559}{23814} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {100}{81} \, \sqrt {-2 \, x + 1} - \frac {25209 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 114604 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 130291 \, \sqrt {-2 \, x + 1}}{4536 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.65 \[ \frac {7559 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{11907}-\frac {100 \sqrt {-2 x +1}}{81}-\frac {4 \left (-\frac {2801 \left (-2 x +1\right )^{\frac {5}{2}}}{84}+\frac {4093 \left (-2 x +1\right )^{\frac {3}{2}}}{27}-\frac {18613 \sqrt {-2 x +1}}{108}\right )}{3 \left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 101, normalized size = 1.00 \[ -\frac {7559}{23814} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {100}{81} \, \sqrt {-2 \, x + 1} - \frac {25209 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 114604 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 130291 \, \sqrt {-2 \, x + 1}}{567 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 81, normalized size = 0.80 \[ \frac {7559\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{11907}-\frac {100\,\sqrt {1-2\,x}}{81}-\frac {\frac {18613\,\sqrt {1-2\,x}}{2187}-\frac {16372\,{\left (1-2\,x\right )}^{3/2}}{2187}+\frac {2801\,{\left (1-2\,x\right )}^{5/2}}{1701}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]
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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