Optimal. Leaf size=56 \[ \frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}-\frac {2}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 53, 65, 212}
\begin {gather*} \frac {2}{121 \sqrt {1-2 x}}+\frac {7}{33 (1-2 x)^{3/2}}-\frac {2}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {1}{11} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}+\frac {5}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}-\frac {5}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}-\frac {2}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 45, normalized size = 0.80 \begin {gather*} \frac {\frac {913-132 x}{(1-2 x)^{3/2}}-6 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3993} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 38, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {7}{33 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {2}{121 \sqrt {1-2 x}}\) | \(38\) |
default | \(\frac {7}{33 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {2}{121 \sqrt {1-2 x}}\) | \(38\) |
trager | \(-\frac {\left (12 x -83\right ) \sqrt {1-2 x}}{363 \left (-1+2 x \right )^{2}}+\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 51, normalized size = 0.91 \begin {gather*} \frac {1}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {12 \, x - 83}{363 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (37) = 74\).
time = 1.04, size = 75, normalized size = 1.34 \begin {gather*} \frac {3 \, \sqrt {11} \sqrt {5} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (12 \, x - 83\right )} \sqrt {-2 \, x + 1}}{3993 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.92, size = 83, normalized size = 1.48 \begin {gather*} \frac {10 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{121} + \frac {2}{121 \sqrt {1 - 2 x}} + \frac {7}{33 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.79, size = 61, normalized size = 1.09 \begin {gather*} \frac {1}{1331} \, \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 {12 \, x - 83}{363 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 33, normalized size = 0.59 \begin {gather*} -\frac {\frac {4\,x}{121}-\frac {83}{363}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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