Optimal. Leaf size=54 \[ -\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
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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 {25}{18} (1-2 x)^{3/2}-\frac {155}{18} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)} \, dx &=\int \left (\frac {155}{18 \sqrt {1-2 x}}-\frac {25}{6} \sqrt {1-2 x}+\frac {1}{9 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=-\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}+\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 0.85 \begin {gather*} -\frac {5}{9} \sqrt {1-2 x} (13+5 x)-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 38, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}-\frac {155 \sqrt {1-2 x}}{18}\) | \(38\) |
default | \(\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}-\frac {155 \sqrt {1-2 x}}{18}\) | \(38\) |
risch | \(\frac {5 \left (5 x +13\right ) \left (-1+2 x \right )}{9 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}\) | \(39\) |
trager | \(\left (-\frac {25 x}{9}-\frac {65}{9}\right ) \sqrt {1-2 x}+\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{189}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 55, normalized size = 1.02 \begin {gather*} \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{189} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {155}{18} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 45, normalized size = 0.83 \begin {gather*} -\frac {5}{9} \, {\left (5 \, x + 13\right )} \sqrt {-2 \, x + 1} + \frac {1}{189} \, \sqrt {21} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 10.58, size = 90, normalized size = 1.67 \begin {gather*} \frac {25 \left (1 - 2 x\right )^{\frac {3}{2}}}{18} - \frac {155 \sqrt {1 - 2 x}}{18} + \frac {2 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} > \frac {3}{7} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} < \frac {3}{7} \end {cases}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 58, normalized size = 1.07 \begin {gather*} \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{189} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {155}{18} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 37, normalized size = 0.69 \begin {gather*} \frac {25\,{\left (1-2\,x\right )}^{3/2}}{18}-\frac {155\,\sqrt {1-2\,x}}{18}-\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{189} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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