Optimal. Leaf size=54 \[ -\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {90, 65, 212}
\begin {gather*} \frac {3}{10} (1-2 x)^{3/2}-\frac {111}{50} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx &=\int \left (\frac {111}{50 \sqrt {1-2 x}}-\frac {9}{10} \sqrt {1-2 x}+\frac {1}{25 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}+\frac {1}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {1}{25} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 0.85 \begin {gather*} -\frac {3}{25} \sqrt {1-2 x} (16+5 x)-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 38, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {3 \left (1-2 x \right )^{\frac {3}{2}}}{10}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}-\frac {111 \sqrt {1-2 x}}{50}\) | \(38\) |
default | \(\frac {3 \left (1-2 x \right )^{\frac {3}{2}}}{10}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}-\frac {111 \sqrt {1-2 x}}{50}\) | \(38\) |
risch | \(\frac {3 \left (5 x +16\right ) \left (-1+2 x \right )}{25 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(39\) |
trager | \(\left (-\frac {3 x}{5}-\frac {48}{25}\right ) \sqrt {1-2 x}+\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1375}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 55, normalized size = 1.02 \begin {gather*} \frac {3}{10} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {111}{50} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 45, normalized size = 0.83 \begin {gather*} -\frac {3}{25} \, {\left (5 \, x + 16\right )} \sqrt {-2 \, x + 1} + \frac {1}{1375} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 10.92, size = 90, normalized size = 1.67 \begin {gather*} \frac {3 \left (1 - 2 x\right )^{\frac {3}{2}}}{10} - \frac {111 \sqrt {1 - 2 x}}{50} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} > \frac {5}{11} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} < \frac {5}{11} \end {cases}\right )}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 58, normalized size = 1.07 \begin {gather*} \frac {3}{10} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1375} \, \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 {111}{50} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 37, normalized size = 0.69 \begin {gather*} \frac {3\,{\left (1-2\,x\right )}^{3/2}}{10}-\frac {111\,\sqrt {1-2\,x}}{50}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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