Optimal. Leaf size=127 \[ -\frac {335 \sqrt {1-2 x}}{2 (5 x+3)}+\frac {50 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 151, 156, 63, 206} \[ -\frac {335 \sqrt {1-2 x}}{2 (5 x+3)}+\frac {50 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {1}{6} \int \frac {122-167 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}+\frac {1}{42} \int \frac {9177-10500 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-\frac {1}{462} \int \frac {379071-232155 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}+\frac {6933}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-5610 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-\frac {6933}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+5610 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 90, normalized size = 0.71 \[ -\frac {\sqrt {1-2 x} \left (3015 x^2+3920 x+1271\right )}{2 (3 x+2)^2 (5 x+3)}-2311 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 136, normalized size = 1.07 \[ \frac {2311 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 1428 \, \sqrt {55} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 7 \, {\left (3015 \, x^{2} + 3920 \, x + 1271\right )} \sqrt {-2 \, x + 1}}{14 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 123, normalized size = 0.97 \[ -102 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2311}{14} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {55 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} + \frac {405 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 959 \, \sqrt {-2 \, x + 1}}{4 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 82, normalized size = 0.65 \[ -\frac {2311 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{7}+204 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {22 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {405 \left (-2 x +1\right )^{\frac {3}{2}}-959 \sqrt {-2 x +1}}{\left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 128, normalized size = 1.01 \[ -102 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2311}{14} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3015 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 13870 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15939 \, \sqrt {-2 \, x + 1}}{45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 90, normalized size = 0.71 \[ 204\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {2311\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {\frac {1771\,\sqrt {1-2\,x}}{5}-\frac {2774\,{\left (1-2\,x\right )}^{3/2}}{9}+67\,{\left (1-2\,x\right )}^{5/2}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \]
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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