Optimal. Leaf size=181 \[ -\frac {8110915 \sqrt {1-2 x}}{1176 (5 x+3)}+\frac {302668 \sqrt {1-2 x}}{441 (3 x+2) (5 x+3)}+\frac {23173 \sqrt {1-2 x}}{504 (3 x+2)^2 (5 x+3)}+\frac {83 \sqrt {1-2 x}}{18 (3 x+2)^3 (5 x+3)}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac {55953383 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {8110915 \sqrt {1-2 x}}{1176 (5 x+3)}+\frac {302668 \sqrt {1-2 x}}{441 (3 x+2) (5 x+3)}+\frac {23173 \sqrt {1-2 x}}{504 (3 x+2)^2 (5 x+3)}+\frac {83 \sqrt {1-2 x}}{18 (3 x+2)^3 (5 x+3)}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac {55953383 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \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)^2} \, dx &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {1}{12} \int \frac {188-299 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {1}{252} \int \frac {26957-40670 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {\int \frac {2946286-4055275 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx}{3528}\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {302668 \sqrt {1-2 x}}{441 (2+3 x) (3+5 x)}+\frac {\int \frac {222179601-254241120 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{24696}\\ &=-\frac {8110915 \sqrt {1-2 x}}{1176 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {302668 \sqrt {1-2 x}}{441 (2+3 x) (3+5 x)}-\frac {\int \frac {9177988743-5620864095 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{271656}\\ &=-\frac {8110915 \sqrt {1-2 x}}{1176 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {302668 \sqrt {1-2 x}}{441 (2+3 x) (3+5 x)}+\frac {55953383}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-231000 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {8110915 \sqrt {1-2 x}}{1176 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {302668 \sqrt {1-2 x}}{441 (2+3 x) (3+5 x)}-\frac {55953383}{392} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+231000 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {8110915 \sqrt {1-2 x}}{1176 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {302668 \sqrt {1-2 x}}{441 (2+3 x) (3+5 x)}-\frac {55953383 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \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 (218994705 x^4+576721848 x^3+569295605 x^2+249642200 x+41029970\right )}{392 (3 x+2)^4 (5 x+3)}-\frac {55953383 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \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 = 0.44, size = 124, normalized size = 0.69 \begin {gather*} \frac {\sqrt {1-2 x} \left (218994705 (1-2 x)^4-2029422516 (1-2 x)^3+7051481738 (1-2 x)^2-10887812348 (1-2 x)+6303237941\right )}{196 (3 (1-2 x)-7)^4 (5 (1-2 x)-11)}-\frac {55953383 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \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 = 1.65, size = 170, normalized size = 0.94 \begin {gather*} \frac {34574400 \, \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 ) + 55953383 \, \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 (218994705 \, x^{4} + 576721848 \, x^{3} + 569295605 \, x^{2} + 249642200 \, x + 41029970\right )} \sqrt {-2 \, x + 1}}{8232 \, {\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 = 1.04, size = 155, normalized size = 0.86 \begin {gather*} -4200 \, \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 {55953383}{8232} \, \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 {1375 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {35067141 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 247239993 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 581129563 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 455372631 \, \sqrt {-2 \, x + 1}}{3136 \, {\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 {55953383 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{4116}+8400 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {550 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {\frac {35067141 \left (-2 x +1\right )^{\frac {7}{2}}}{196}-\frac {35319999 \left (-2 x +1\right )^{\frac {5}{2}}}{28}+\frac {11859787 \left (-2 x +1\right )^{\frac {3}{2}}}{4}-\frac {9293319 \sqrt {-2 x +1}}{4}}{\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 = 1.07, size = 164, normalized size = 0.91 \begin {gather*} -4200 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {55953383}{8232} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {218994705 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 2029422516 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 7051481738 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 10887812348 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 6303237941 \, \sqrt {-2 \, x + 1}}{196 \, {\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 = 1.20, size = 126, normalized size = 0.70 \begin {gather*} 8400\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {55953383\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4116}-\frac {\frac {128637509\,\sqrt {1-2\,x}}{1620}-\frac {55550063\,{\left (1-2\,x\right )}^{3/2}}{405}+\frac {503677267\,{\left (1-2\,x\right )}^{5/2}}{5670}-\frac {169118543\,{\left (1-2\,x\right )}^{7/2}}{6615}+\frac {1622183\,{\left (1-2\,x\right )}^{9/2}}{588}}{\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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