Optimal. Leaf size=133 \[ \frac {318643 \sqrt {1-2 x}}{1176 (3 x+2)}+\frac {13723 \sqrt {1-2 x}}{504 (3 x+2)^2}+\frac {131 \sqrt {1-2 x}}{36 (3 x+2)^3}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}+\frac {10990843 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}-550 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \begin {gather*} \frac {318643 \sqrt {1-2 x}}{1176 (3 x+2)}+\frac {13723 \sqrt {1-2 x}}{504 (3 x+2)^2}+\frac {131 \sqrt {1-2 x}}{36 (3 x+2)^3}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4}+\frac {10990843 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}-550 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
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)^5 (3+5 x)} \, dx &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {153-229 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {1}{252} \int \frac {16737-22925 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {13723 \sqrt {1-2 x}}{504 (2+3 x)^2}+\frac {\int \frac {1269891-1440915 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{3528}\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {13723 \sqrt {1-2 x}}{504 (2+3 x)^2}+\frac {318643 \sqrt {1-2 x}}{1176 (2+3 x)}+\frac {\int \frac {54630891-33457515 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{24696}\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {13723 \sqrt {1-2 x}}{504 (2+3 x)^2}+\frac {318643 \sqrt {1-2 x}}{1176 (2+3 x)}-\frac {10990843 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1176}+15125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {13723 \sqrt {1-2 x}}{504 (2+3 x)^2}+\frac {318643 \sqrt {1-2 x}}{1176 (2+3 x)}+\frac {10990843 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1176}-15125 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4}+\frac {131 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {13723 \sqrt {1-2 x}}{504 (2+3 x)^2}+\frac {318643 \sqrt {1-2 x}}{1176 (2+3 x)}+\frac {10990843 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}-550 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 90, normalized size = 0.68 \begin {gather*} \frac {21 \left (\frac {\sqrt {1-2 x} \left (8603361 x^3+17494905 x^2+11868230 x+2686470\right )}{(3 x+2)^4}-646800 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+21981686 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 104, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-2 x} \left (8603361 (1-2 x)^3-60799893 (1-2 x)^2+143262623 (1-2 x)-112557851\right )}{588 (3 (1-2 x)-7)^4}+\frac {10990843 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}-550 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.70, size = 150, normalized size = 1.13 \begin {gather*} \frac {6791400 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 10990843 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (8603361 \, x^{3} + 17494905 \, x^{2} + 11868230 \, x + 2686470\right )} \sqrt {-2 \, x + 1}}{24696 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 139, normalized size = 1.05 \begin {gather*} 275 \, \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 {10990843}{24696} \, \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 {8603361 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 60799893 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 143262623 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 112557851 \, \sqrt {-2 \, x + 1}}{9408 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.63 \begin {gather*} \frac {10990843 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{12348}-550 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )-\frac {162 \left (\frac {318643 \left (-2 x +1\right )^{\frac {7}{2}}}{3528}-\frac {2895233 \left (-2 x +1\right )^{\frac {5}{2}}}{4536}+\frac {2923727 \left (-2 x +1\right )^{\frac {3}{2}}}{1944}-\frac {2297099 \sqrt {-2 x +1}}{1944}\right )}{\left (-6 x -4\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 146, normalized size = 1.10 \begin {gather*} 275 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {10990843}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8603361 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 60799893 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 143262623 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 112557851 \, \sqrt {-2 \, x + 1}}{588 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 107, normalized size = 0.80 \begin {gather*} \frac {10990843\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-550\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {2297099\,\sqrt {1-2\,x}}{972}-\frac {2923727\,{\left (1-2\,x\right )}^{3/2}}{972}+\frac {2895233\,{\left (1-2\,x\right )}^{5/2}}{2268}-\frac {318643\,{\left (1-2\,x\right )}^{7/2}}{1764}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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