Optimal. Leaf size=172 \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac {113875 \sqrt {1-2 x}}{6 (5 x+3)}+\frac {1256 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac {581 \sqrt {1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac {169975 \sqrt {1-2 x}}{54 (5 x+3)^2}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac {113875 \sqrt {1-2 x}}{6 (5 x+3)}+\frac {1256 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac {581 \sqrt {1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac {169975 \sqrt {1-2 x}}{54 (5 x+3)^2}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 149
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {1}{9} \int \frac {(232-233 x) \sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}-\frac {1}{54} \int \frac {-26260+39738 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}-\frac {1}{378} \int \frac {-2861390+3956400 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {\int \frac {-205876440+235585350 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{8316}\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}-\frac {\int \frac {-8504523720+5208414750 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{91476}\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}-392785 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {1271325}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}+392785 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {1271325}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 98, normalized size = 0.57 \[ \frac {\sqrt {1-2 x} \left (5124375 x^4+13153400 x^3+12649336 x^2+5401374 x+864074\right )}{2 (3 x+2)^3 (5 x+3)^2}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 170, normalized size = 0.99 \[ \frac {485415 \, \sqrt {55} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 785570 \, \sqrt {21} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (5124375 \, x^{4} + 13153400 \, x^{3} + 12649336 \, x^{2} + 5401374 \, x + 864074\right )} \sqrt {-2 \, x + 1}}{42 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 151, normalized size = 0.88 \[ \frac {23115}{2} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {392785}{21} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {55 \, {\left (1365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2981 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {61947 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 291200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 342265 \, \sqrt {-2 \, x + 1}}{4 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 103, normalized size = 0.60 \[ \frac {785570 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21}-23115 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {-75075 \left (-2 x +1\right )^{\frac {3}{2}}+163955 \sqrt {-2 x +1}}{\left (-10 x -6\right )^{2}}-\frac {108 \left (\frac {6883 \left (-2 x +1\right )^{\frac {5}{2}}}{6}-\frac {145600 \left (-2 x +1\right )^{\frac {3}{2}}}{27}+\frac {342265 \sqrt {-2 x +1}}{54}\right )}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 163, normalized size = 0.95 \[ \frac {23115}{2} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {392785}{21} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {5124375 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 46804300 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 160263994 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 243823580 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 139064695 \, \sqrt {-2 \, x + 1}}{675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 125, normalized size = 0.73 \[ \frac {785570\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21}-23115\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {27812939\,\sqrt {1-2\,x}}{135}-\frac {48764716\,{\left (1-2\,x\right )}^{3/2}}{135}+\frac {160263994\,{\left (1-2\,x\right )}^{5/2}}{675}-\frac {1872172\,{\left (1-2\,x\right )}^{7/2}}{27}+\frac {22775\,{\left (1-2\,x\right )}^{9/2}}{3}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \]
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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