Optimal. Leaf size=72 \[ -\frac {4}{15} \sqrt {1-2 x}+\frac {14}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {22}{5} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 72, 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 = {84, 156, 63, 206} \[ -\frac {4}{15} \sqrt {1-2 x}+\frac {14}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {22}{5} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 84
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)} \, dx &=-\frac {4}{15} \sqrt {1-2 x}+\frac {1}{15} \int \frac {-9-136 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {4}{15} \sqrt {1-2 x}-\frac {49}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {121}{5} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {4}{15} \sqrt {1-2 x}+\frac {49}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {121}{5} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {4}{15} \sqrt {1-2 x}+\frac {14}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {22}{5} \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 = 0.03, size = 66, normalized size = 0.92 \[ -\frac {2}{225} \left (30 \sqrt {1-2 x}-175 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+99 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 83, normalized size = 1.15 \[ \frac {11}{25} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {7}{9} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - \frac {4}{15} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 88, normalized size = 1.22 \[ \frac {11}{25} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7}{9} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4}{15} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.65 \[ \frac {14 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{9}-\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{25}-\frac {4 \sqrt {-2 x +1}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 82, normalized size = 1.14 \[ \frac {11}{25} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7}{9} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4}{15} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 46, normalized size = 0.64 \[ \frac {14\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9}-\frac {22\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{25}-\frac {4\,\sqrt {1-2\,x}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.26, size = 146, normalized size = 2.03 \[ - \frac {4 \sqrt {1 - 2 x}}{15} - \frac {98 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{3} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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