Optimal. Leaf size=54 \[ \frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
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Rubi [A]
time = 0.02, 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 = {89, 65, 212}
\begin {gather*} -\frac {217}{242 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \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)^2}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac {49}{22 (1-2 x)^{5/2}}-\frac {217}{242 (1-2 x)^{3/2}}+\frac {1}{121 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}+\frac {1}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}-\frac {1}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 46, normalized size = 0.85 \begin {gather*} \frac {7 (-8+93 x)}{363 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 38, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {49}{66 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6655}-\frac {217}{242 \sqrt {1-2 x}}\) | \(38\) |
default | \(\frac {49}{66 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6655}-\frac {217}{242 \sqrt {1-2 x}}\) | \(38\) |
trager | \(\frac {7 \left (93 x -8\right ) \sqrt {1-2 x}}{363 \left (-1+2 x \right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{6655}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 51, normalized size = 0.94 \begin {gather*} \frac {1}{6655} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {7 \, {\left (93 \, x - 8\right )}}{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 [A]
time = 0.89, size = 69, normalized size = 1.28 \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 ) + 385 \, {\left (93 \, x - 8\right )} \sqrt {-2 \, x + 1}}{19965 \, {\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 = 20.46, size = 83, normalized size = 1.54 \begin {gather*} \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 )}{121} - \frac {217}{242 \sqrt {1 - 2 x}} + \frac {49}{66 \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 = 1.13, size = 61, normalized size = 1.13 \begin {gather*} \frac {1}{6655} \, \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 {7 \, {\left (93 \, x - 8\right )}}{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 = 1.19, size = 32, normalized size = 0.59 \begin {gather*} \frac {\frac {217\,x}{121}-\frac {56}{363}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{6655} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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