Optimal. Leaf size=54 \[ \frac {25}{18} (1-2 x)^{3/2}-\frac {155}{18} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
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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 = {88, 63, 206} \[ \frac {25}{18} (1-2 x)^{3/2}-\frac {155}{18} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 206
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} \operatorname {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 = 0.05, size = 46, normalized size = 0.85 \[ -\frac {5}{9} \sqrt {1-2 x} (5 x+13)-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 45, normalized size = 0.83 \[ -\frac {5}{9} \, {\left (5 \, x + 13\right )} \sqrt {-2 \, x + 1} + \frac {1}{189} \, \sqrt {21} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 58, normalized size = 1.07 \[ \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{189} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {155}{18} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.70 \[ -\frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{189}+\frac {25 \left (-2 x +1\right )^{\frac {3}{2}}}{18}-\frac {155 \sqrt {-2 x +1}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.58, size = 55, normalized size = 1.02 \[ \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{189} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {155}{18} \, \sqrt {-2 \, x + 1} \]
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 \[ \frac {25\,{\left (1-2\,x\right )}^{3/2}}{18}-\frac {155\,\sqrt {1-2\,x}}{18}-\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{189} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.53, size = 90, normalized size = 1.67 \[ \frac {25 \left (1 - 2 x\right )^{\frac {3}{2}}}{18} - \frac {155 \sqrt {1 - 2 x}}{18} + \frac {2 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} > \frac {3}{7} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} < \frac {3}{7} \end {cases}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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