Optimal. Leaf size=181 \[ \frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}+\frac {288770 \sqrt {1-2 x}}{189 (3 x+2) (5 x+3)}+\frac {22109 \sqrt {1-2 x}}{216 (3 x+2)^2 (5 x+3)}+\frac {287 \sqrt {1-2 x}}{27 (3 x+2)^3 (5 x+3)}-\frac {7738475 \sqrt {1-2 x}}{504 (5 x+3)}-\frac {53384095 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 181, 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} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}+\frac {288770 \sqrt {1-2 x}}{189 (3 x+2) (5 x+3)}+\frac {22109 \sqrt {1-2 x}}{216 (3 x+2)^2 (5 x+3)}+\frac {287 \sqrt {1-2 x}}{27 (3 x+2)^3 (5 x+3)}-\frac {7738475 \sqrt {1-2 x}}{504 (5 x+3)}-\frac {53384095 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \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 149
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {1}{12} \int \frac {(230-229 x) \sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}-\frac {1}{108} \int \frac {-25717+38806 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}-\frac {\int \frac {-2810990+3869075 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx}{1512}\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac {288770 \sqrt {1-2 x}}{189 (2+3 x) (3+5 x)}-\frac {\int \frac {-211977465+242566800 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{10584}\\ &=-\frac {7738475 \sqrt {1-2 x}}{504 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac {288770 \sqrt {1-2 x}}{189 (2+3 x) (3+5 x)}+\frac {\int \frac {-8756550495+5362763175 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{116424}\\ &=-\frac {7738475 \sqrt {1-2 x}}{504 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac {288770 \sqrt {1-2 x}}{189 (2+3 x) (3+5 x)}+\frac {53384095}{168} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-514250 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {7738475 \sqrt {1-2 x}}{504 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac {288770 \sqrt {1-2 x}}{189 (2+3 x) (3+5 x)}-\frac {53384095}{168} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+514250 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {7738475 \sqrt {1-2 x}}{504 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac {288770 \sqrt {1-2 x}}{189 (2+3 x) (3+5 x)}-\frac {53384095 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 100, normalized size = 0.55 \begin {gather*} -\frac {\sqrt {1-2 x} \left (208938825 x^4+550239720 x^3+543154477 x^2+238179048 x+39145938\right )}{168 (3 x+2)^4 (5 x+3)}-\frac {53384095 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 124, normalized size = 0.69 \begin {gather*} \frac {\sqrt {1-2 x} \left (208938825 (1-2 x)^4-1936234740 (1-2 x)^3+6727689178 (1-2 x)^2-10387861820 (1-2 x)+6013803565\right )}{84 (3 (1-2 x)-7)^4 (5 (1-2 x)-11)}-\frac {53384095 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \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 = 1.92, size = 170, normalized size = 0.94 \begin {gather*} \frac {32986800 \, \sqrt {55} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 53384095 \, \sqrt {21} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (208938825 \, x^{4} + 550239720 \, x^{3} + 543154477 \, x^{2} + 238179048 \, x + 39145938\right )} \sqrt {-2 \, x + 1}}{3528 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 155, normalized size = 0.86 \begin {gather*} -9350 \, \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 {53384095}{3528} \, \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 {3025 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {33554925 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 236586273 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 556108595 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 435783215 \, \sqrt {-2 \, x + 1}}{1344 \, {\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 = 0.02, size = 100, normalized size = 0.55 \begin {gather*} -\frac {53384095 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1764}+18700 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {1210 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {\frac {11184975 \left (-2 x +1\right )^{\frac {7}{2}}}{28}-\frac {11266013 \left (-2 x +1\right )^{\frac {5}{2}}}{4}+\frac {79444085 \left (-2 x +1\right )^{\frac {3}{2}}}{12}-\frac {62254745 \sqrt {-2 x +1}}{12}}{\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 = 164, normalized size = 0.91 \begin {gather*} -9350 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {53384095}{3528} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {208938825 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 1936234740 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 6727689178 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 10387861820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 6013803565 \, \sqrt {-2 \, x + 1}}{84 \, {\left (405 \, {\left (2 \, x - 1\right )}^{5} + 4671 \, {\left (2 \, x - 1\right )}^{4} + 21546 \, {\left (2 \, x - 1\right )}^{3} + 49686 \, {\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 126, normalized size = 0.70 \begin {gather*} 18700\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {53384095\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1764}-\frac {\frac {171822959\,\sqrt {1-2\,x}}{972}-\frac {74199013\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {480549227\,{\left (1-2\,x\right )}^{5/2}}{2430}-\frac {32270579\,{\left (1-2\,x\right )}^{7/2}}{567}+\frac {1547695\,{\left (1-2\,x\right )}^{9/2}}{252}}{\frac {114562\,x}{405}+\frac {16562\,{\left (2\,x-1\right )}^2}{135}+\frac {266\,{\left (2\,x-1\right )}^3}{5}+\frac {173\,{\left (2\,x-1\right )}^4}{15}+{\left (2\,x-1\right )}^5-\frac {686}{9}} \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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