Optimal. Leaf size=81 \[ \frac {213}{6655 \sqrt {1-2 x}}-\frac {71}{1210 \sqrt {1-2 x} (5 x+3)}-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}-\frac {213 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \]
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Rubi [A] time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \begin {gather*} -\frac {213 \sqrt {1-2 x}}{2662 (5 x+3)}+\frac {71}{605 \sqrt {1-2 x} (5 x+3)}-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}-\frac {213 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}+\frac {71}{110} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}+\frac {71}{605 \sqrt {1-2 x} (3+5 x)}+\frac {213}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}+\frac {71}{605 \sqrt {1-2 x} (3+5 x)}-\frac {213 \sqrt {1-2 x}}{2662 (3+5 x)}+\frac {213 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}+\frac {71}{605 \sqrt {1-2 x} (3+5 x)}-\frac {213 \sqrt {1-2 x}}{2662 (3+5 x)}-\frac {213 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2662}\\ &=-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}+\frac {71}{605 \sqrt {1-2 x} (3+5 x)}-\frac {213 \sqrt {1-2 x}}{2662 (3+5 x)}-\frac {213 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.59 \begin {gather*} \frac {284 (5 x+3)^2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )-121}{13310 \sqrt {1-2 x} (5 x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 70, normalized size = 0.86 \begin {gather*} \frac {1065 (1-2 x)^2-3905 (1-2 x)+3388}{1331 (5 (1-2 x)-11)^2 \sqrt {1-2 x}}-\frac {213 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 84, normalized size = 1.04 \begin {gather*} \frac {213 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (2130 \, x^{2} + 1775 \, x + 274\right )} \sqrt {-2 \, x + 1}}{146410 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 77, normalized size = 0.95 \begin {gather*} \frac {213}{146410} \, \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 {28}{1331 \, \sqrt {-2 \, x + 1}} + \frac {5 \, {\left (73 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 165 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.70 \begin {gather*} -\frac {213 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{73205}+\frac {28}{1331 \sqrt {-2 x +1}}+\frac {\frac {365 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}-\frac {75 \sqrt {-2 x +1}}{121}}{\left (-10 x -6\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 83, normalized size = 1.02 \begin {gather*} \frac {213}{146410} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1065 \, {\left (2 \, x - 1\right )}^{2} + 7810 \, x - 517}{1331 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 62, normalized size = 0.77 \begin {gather*} \frac {\frac {142\,x}{605}+\frac {213\,{\left (2\,x-1\right )}^2}{6655}-\frac {47}{3025}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}-\frac {213\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{73205} \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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