Optimal. Leaf size=63 \[ \frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 206} \[ -\frac {15 \sqrt {1-2 x}}{121 (5 x+3)}+\frac {2}{11 \sqrt {1-2 x} (5 x+3)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}+\frac {15}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {1-2 x}}{121 (3+5 x)}+\frac {15}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {1-2 x}}{121 (3+5 x)}-\frac {15}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15 \sqrt {1-2 x}}{121 (3+5 x)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.48 \[ \frac {4 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{121 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 71, normalized size = 1.13 \[ \frac {3 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (30 \, x + 7\right )} \sqrt {-2 \, x + 1}}{1331 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 68, normalized size = 1.08 \[ \frac {3}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.71 \[ -\frac {6 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {4}{121 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{121 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 65, normalized size = 1.03 \[ \frac {3}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 46, normalized size = 0.73 \[ \frac {\frac {12\,x}{121}+\frac {14}{605}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {6\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.52, size = 175, normalized size = 2.78 \[ \begin {cases} - \frac {6 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} + \frac {3 \sqrt {2}}{121 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{110 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {6 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} - \frac {3 \sqrt {2} i}{121 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{110 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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