Optimal. Leaf size=67 \[ \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, 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 = {89, 65, 212}
\begin {gather*} -\frac {27}{20} \sqrt {1-2 x}-\frac {784}{121 \sqrt {1-2 x}}+\frac {343}{132 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 89
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac {343}{44 (1-2 x)^{5/2}}-\frac {784}{121 (1-2 x)^{3/2}}+\frac {27}{20 \sqrt {1-2 x}}+\frac {1}{605 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}+\frac {1}{605} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {1}{605} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 51, normalized size = 0.76 \begin {gather*} \frac {-\frac {55 \left (9494-33321 x+9801 x^2\right )}{(1-2 x)^{3/2}}-6 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{99825} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 47, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {343}{132 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{33275}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27 \sqrt {1-2 x}}{20}\) | \(47\) |
default | \(\frac {343}{132 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{33275}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27 \sqrt {1-2 x}}{20}\) | \(47\) |
trager | \(-\frac {\left (9801 x^{2}-33321 x +9494\right ) \sqrt {1-2 x}}{1815 \left (-1+2 x \right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{33275}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 60, normalized size = 0.90 \begin {gather*} \frac {1}{33275} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{20} \, \sqrt {-2 \, x + 1} + \frac {49 \, {\left (384 \, x - 115\right )}}{1452 \, {\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 [A]
time = 1.18, size = 74, normalized size = 1.10 \begin {gather*} \frac {3 \, \sqrt {55} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (9801 \, x^{2} - 33321 \, x + 9494\right )} \sqrt {-2 \, x + 1}}{99825 \, {\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 = 28.40, size = 95, normalized size = 1.42 \begin {gather*} - \frac {27 \sqrt {1 - 2 x}}{20} + \frac {2 \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 )}{605} - \frac {784}{121 \sqrt {1 - 2 x}} + \frac {343}{132 \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 = 2.95, size = 70, normalized size = 1.04 \begin {gather*} \frac {1}{33275} \, \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 {27}{20} \, \sqrt {-2 \, x + 1} - \frac {49 \, {\left (384 \, x - 115\right )}}{1452 \, {\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 = 1.20, size = 41, normalized size = 0.61 \begin {gather*} \frac {\frac {1568\,x}{121}-\frac {5635}{1452}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{33275}-\frac {27\,\sqrt {1-2\,x}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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