Optimal. Leaf size=82 \[ \frac {25}{42} (1-2 x)^{7/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {2}{81} (1-2 x)^{3/2}+\frac {14}{81} \sqrt {1-2 x}-\frac {14}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ \frac {25}{42} (1-2 x)^{7/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {2}{81} (1-2 x)^{3/2}+\frac {14}{81} \sqrt {1-2 x}-\frac {14}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx &=\int \left (\frac {155}{18} (1-2 x)^{3/2}-\frac {25}{6} (1-2 x)^{5/2}+\frac {(1-2 x)^{3/2}}{9 (2+3 x)}\right ) \, dx\\ &=-\frac {31}{18} (1-2 x)^{5/2}+\frac {25}{42} (1-2 x)^{7/2}+\frac {1}{9} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {2}{81} (1-2 x)^{3/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {25}{42} (1-2 x)^{7/2}+\frac {7}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {14}{81} \sqrt {1-2 x}+\frac {2}{81} (1-2 x)^{3/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {25}{42} (1-2 x)^{7/2}+\frac {49}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {14}{81} \sqrt {1-2 x}+\frac {2}{81} (1-2 x)^{3/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {25}{42} (1-2 x)^{7/2}-\frac {49}{81} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {14}{81} \sqrt {1-2 x}+\frac {2}{81} (1-2 x)^{3/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {25}{42} (1-2 x)^{7/2}-\frac {14}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.68 \[ \frac {3 \sqrt {1-2 x} \left (-2700 x^3+144 x^2+1853 x-527\right )-98 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1701} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 61, normalized size = 0.74 \[ \frac {7}{243} \, \sqrt {7} \sqrt {3} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - \frac {1}{567} \, {\left (2700 \, x^{3} - 144 \, x^{2} - 1853 \, x + 527\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 90, normalized size = 1.10 \[ -\frac {25}{42} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {31}{18} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {7}{243} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {14}{81} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.68 \[ -\frac {14 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{243}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{81}-\frac {31 \left (-2 x +1\right )^{\frac {5}{2}}}{18}+\frac {25 \left (-2 x +1\right )^{\frac {7}{2}}}{42}+\frac {14 \sqrt {-2 x +1}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 73, normalized size = 0.89 \[ \frac {25}{42} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {31}{18} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {7}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {14}{81} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 57, normalized size = 0.70 \[ \frac {14\,\sqrt {1-2\,x}}{81}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {31\,{\left (1-2\,x\right )}^{5/2}}{18}+\frac {25\,{\left (1-2\,x\right )}^{7/2}}{42}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,14{}\mathrm {i}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 35.78, size = 114, normalized size = 1.39 \[ \frac {25 \left (1 - 2 x\right )^{\frac {7}{2}}}{42} - \frac {31 \left (1 - 2 x\right )^{\frac {5}{2}}}{18} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} + \frac {14 \sqrt {1 - 2 x}}{81} + \frac {98 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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