Optimal. Leaf size=89 \[ -\frac {(1-2 x)^{5/2}}{275 (5 x+3)}-\frac {9}{125} (1-2 x)^{5/2}+\frac {42 (1-2 x)^{3/2}}{1375}+\frac {126}{625} \sqrt {1-2 x}-\frac {126}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 89, 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 = {89, 80, 50, 63, 206} \[ -\frac {(1-2 x)^{5/2}}{275 (5 x+3)}-\frac {9}{125} (1-2 x)^{5/2}+\frac {42 (1-2 x)^{3/2}}{1375}+\frac {126}{625} \sqrt {1-2 x}-\frac {126}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {1}{275} \int \frac {(1-2 x)^{3/2} (360+495 x)}{3+5 x} \, dx\\ &=-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {63}{275} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {63}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}+\frac {693}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}-\frac {693}{625} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {126}{625} \sqrt {1-2 x}+\frac {42 (1-2 x)^{3/2}}{1375}-\frac {9}{125} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{275 (3+5 x)}-\frac {126}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.71 \[ \frac {\frac {5 \sqrt {1-2 x} \left (-900 x^3+160 x^2+935 x+298\right )}{5 x+3}-126 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 75, normalized size = 0.84 \[ \frac {63 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (900 \, x^{3} - 160 \, x^{2} - 935 \, x - 298\right )} \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 90, normalized size = 1.01 \[ -\frac {9}{125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {4}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {63}{3125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {128}{625} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.71 \[ -\frac {126 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{3125}-\frac {9 \left (-2 x +1\right )^{\frac {5}{2}}}{125}+\frac {4 \left (-2 x +1\right )^{\frac {3}{2}}}{125}+\frac {128 \sqrt {-2 x +1}}{625}+\frac {22 \sqrt {-2 x +1}}{3125 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 80, normalized size = 0.90 \[ -\frac {9}{125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {4}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {63}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {128}{625} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 64, normalized size = 0.72 \[ \frac {128\,\sqrt {1-2\,x}}{625}-\frac {22\,\sqrt {1-2\,x}}{3125\,\left (2\,x+\frac {6}{5}\right )}+\frac {4\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {9\,{\left (1-2\,x\right )}^{5/2}}{125}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,126{}\mathrm {i}}{3125} \]
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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