Optimal. Leaf size=73 \[ -\frac {6}{55} \sqrt {1-2 x} (11+3 x)-\frac {\sqrt {1-2 x} (2+3 x)^2}{55 (3+5 x)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 152, 65,
212} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^2}{55 (5 x+3)}-\frac {6}{55} \sqrt {1-2 x} (3 x+11)-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{55 (3+5 x)}-\frac {1}{55} \int \frac {(-74-90 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {6}{55} \sqrt {1-2 x} (11+3 x)-\frac {\sqrt {1-2 x} (2+3 x)^2}{55 (3+5 x)}+\frac {4}{55} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {6}{55} \sqrt {1-2 x} (11+3 x)-\frac {\sqrt {1-2 x} (2+3 x)^2}{55 (3+5 x)}-\frac {4}{55} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {6}{55} \sqrt {1-2 x} (11+3 x)-\frac {\sqrt {1-2 x} (2+3 x)^2}{55 (3+5 x)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 58, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {1-2 x} \left (202+396 x+99 x^2\right )}{55 (3+5 x)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 54, normalized size = 0.74
method | result | size |
risch | \(\frac {198 x^{3}+693 x^{2}+8 x -202}{55 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {8 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\) | \(51\) |
derivativedivides | \(\frac {9 \left (1-2 x \right )^{\frac {3}{2}}}{50}-\frac {351 \sqrt {1-2 x}}{250}+\frac {2 \sqrt {1-2 x}}{6875 \left (-\frac {6}{5}-2 x \right )}-\frac {8 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\) | \(54\) |
default | \(\frac {9 \left (1-2 x \right )^{\frac {3}{2}}}{50}-\frac {351 \sqrt {1-2 x}}{250}+\frac {2 \sqrt {1-2 x}}{6875 \left (-\frac {6}{5}-2 x \right )}-\frac {8 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\) | \(54\) |
trager | \(-\frac {\left (99 x^{2}+396 x +202\right ) \sqrt {1-2 x}}{55 \left (3+5 x \right )}-\frac {4 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{3025}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 71, normalized size = 0.97 \begin {gather*} \frac {9}{50} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4}{3025} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {351}{250} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{1375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 64, normalized size = 0.88 \begin {gather*} \frac {4 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (99 \, x^{2} + 396 \, x + 202\right )} \sqrt {-2 \, x + 1}}{3025 \, {\left (5 \, x + 3\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 = 1.35, size = 74, normalized size = 1.01 \begin {gather*} \frac {9}{50} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4}{3025} \, \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 {351}{250} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{1375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 55, normalized size = 0.75 \begin {gather*} \frac {9\,{\left (1-2\,x\right )}^{3/2}}{50}-\frac {351\,\sqrt {1-2\,x}}{250}-\frac {2\,\sqrt {1-2\,x}}{6875\,\left (2\,x+\frac {6}{5}\right )}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,8{}\mathrm {i}}{3025} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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