Optimal. Leaf size=81 \[ \frac {213}{6655 \sqrt {1-2 x}}-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}-\frac {71}{1210 \sqrt {1-2 x} (3+5 x)}-\frac {213 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65,
212} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 212
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 \text {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 [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} \frac {\frac {55 \left (274+1775 x+2130 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-426 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{146410} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 57, normalized size = 0.70
method | result | size |
risch | \(\frac {2130 x^{2}+1775 x +274}{2662 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {213 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}\) | \(46\) |
derivativedivides | \(\frac {\frac {365 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {75 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {213 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {28}{1331 \sqrt {1-2 x}}\) | \(57\) |
default | \(\frac {\frac {365 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {75 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {213 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {28}{1331 \sqrt {1-2 x}}\) | \(57\) |
trager | \(-\frac {\left (2130 x^{2}+1775 x +274\right ) \sqrt {1-2 x}}{2662 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {213 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{146410}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, 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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Fricas [A]
time = 0.99, 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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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.18, 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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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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