Optimal. Leaf size=154 \[ -\frac {46555 \sqrt {1-2 x}}{42 (5 x+3)}+\frac {6949 \sqrt {1-2 x}}{63 (3 x+2) (5 x+3)}+\frac {133 \sqrt {1-2 x}}{18 (3 x+2)^2 (5 x+3)}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac {321161 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+1350 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 154, 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 {46555 \sqrt {1-2 x}}{42 (5 x+3)}+\frac {6949 \sqrt {1-2 x}}{63 (3 x+2) (5 x+3)}+\frac {133 \sqrt {1-2 x}}{18 (3 x+2)^2 (5 x+3)}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac {321161 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+1350 \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)^4 (3+5 x)^2} \, dx &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {1}{9} \int \frac {155-233 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {133 \sqrt {1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac {1}{126} \int \frac {16912-23275 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {133 \sqrt {1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac {6949 \sqrt {1-2 x}}{63 (2+3 x) (3+5 x)}+\frac {1}{882} \int \frac {1275267-1459290 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {46555 \sqrt {1-2 x}}{42 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {133 \sqrt {1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac {6949 \sqrt {1-2 x}}{63 (2+3 x) (3+5 x)}-\frac {\int \frac {52679781-32262615 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{9702}\\ &=-\frac {46555 \sqrt {1-2 x}}{42 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {133 \sqrt {1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac {6949 \sqrt {1-2 x}}{63 (2+3 x) (3+5 x)}+\frac {321161}{14} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-37125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {46555 \sqrt {1-2 x}}{42 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {133 \sqrt {1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac {6949 \sqrt {1-2 x}}{63 (2+3 x) (3+5 x)}-\frac {321161}{14} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+37125 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {46555 \sqrt {1-2 x}}{42 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac {133 \sqrt {1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac {6949 \sqrt {1-2 x}}{63 (2+3 x) (3+5 x)}-\frac {321161 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+1350 \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 = 95, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {1-2 x} \left (418995 x^3+824092 x^2+539819 x+117752\right )}{14 (3 x+2)^3 (5 x+3)}-\frac {321161 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+1350 \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 = 0.40, size = 124, normalized size = 0.81 \begin {gather*} \frac {418995 (1-2 x)^{7/2}-2905169 (1-2 x)^{5/2}+6712629 (1-2 x)^{3/2}-5168471 \sqrt {1-2 x}}{7 (3 (1-2 x)-7)^3 (5 (1-2 x)-11)}-\frac {321161 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+1350 \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 = 1.59, size = 150, normalized size = 0.97 \begin {gather*} \frac {198450 \, \sqrt {55} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 321161 \, \sqrt {21} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (418995 \, x^{3} + 824092 \, x^{2} + 539819 \, x + 117752\right )} \sqrt {-2 \, x + 1}}{294 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 139, normalized size = 0.90 \begin {gather*} -675 \, \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 {321161}{294} \, \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 {275 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {63009 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 296884 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 349811 \, \sqrt {-2 \, x + 1}}{56 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 91, normalized size = 0.59 \begin {gather*} -\frac {321161 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{147}+1350 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {110 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {\frac {63009 \left (-2 x +1\right )^{\frac {5}{2}}}{7}-42412 \left (-2 x +1\right )^{\frac {3}{2}}+49973 \sqrt {-2 x +1}}{\left (-6 x -4\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 146, normalized size = 0.95 \begin {gather*} -675 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {321161}{294} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {418995 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 2905169 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 6712629 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 5168471 \, \sqrt {-2 \, x + 1}}{7 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 108, normalized size = 0.70 \begin {gather*} 1350\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {321161\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{147}-\frac {\frac {738353\,\sqrt {1-2\,x}}{135}-\frac {319649\,{\left (1-2\,x\right )}^{3/2}}{45}+\frac {2905169\,{\left (1-2\,x\right )}^{5/2}}{945}-\frac {9311\,{\left (1-2\,x\right )}^{7/2}}{21}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \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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