Optimal. Leaf size=48 \[ -\frac {\sqrt {1-2 x}}{55 (3+5 x)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {79, 65, 212}
\begin {gather*} -\frac {\sqrt {1-2 x}}{55 (5 x+3)}-\frac {68 \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 79
Rule 212
Rubi steps
\begin {align*} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x}}{55 (3+5 x)}+\frac {34}{55} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {\sqrt {1-2 x}}{55 (3+5 x)}-\frac {34}{55} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {\sqrt {1-2 x}}{55 (3+5 x)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 48, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-2 x}}{55 (3+5 x)}-\frac {68 \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 = 0.11, size = 36, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{275 \left (-\frac {6}{5}-2 x \right )}-\frac {68 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\) | \(36\) |
default | \(\frac {2 \sqrt {1-2 x}}{275 \left (-\frac {6}{5}-2 x \right )}-\frac {68 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\) | \(36\) |
risch | \(\frac {-1+2 x}{55 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {68 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\) | \(41\) |
trager | \(-\frac {\sqrt {1-2 x}}{55 \left (3+5 x \right )}-\frac {34 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{3025}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 53, normalized size = 1.10 \begin {gather*} \frac {34}{3025} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {\sqrt {-2 \, x + 1}}{55 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 54, normalized size = 1.12 \begin {gather*} \frac {34 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs.
\(2 (39) = 78\).
time = 223.96, size = 190, normalized size = 3.96 \begin {gather*} \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (-1 + \frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{4} + \frac {\log {\left (1 + \frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{4} - \frac {1}{4 \cdot \left (1 + \frac {\sqrt {55}}{5 \sqrt {1 - 2 x}}\right )} - \frac {1}{4 \left (-1 + \frac {\sqrt {55}}{5 \sqrt {1 - 2 x}}\right )}\right )}{275} & \text {for}\: \frac {1}{\sqrt {1 - 2 x}} > - \frac {\sqrt {55}}{11} \wedge \frac {1}{\sqrt {1 - 2 x}} < \frac {\sqrt {55}}{11} \end {cases}\right )}{11} + \frac {14 \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 )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 56, normalized size = 1.17 \begin {gather*} \frac {34}{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 {\sqrt {-2 \, x + 1}}{55 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 35, normalized size = 0.73 \begin {gather*} -\frac {68\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{3025}-\frac {2\,\sqrt {1-2\,x}}{275\,\left (2\,x+\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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