Optimal. Leaf size=68 \[ \frac {121}{14 \sqrt {1-2 x} (2+3 x)}-\frac {1091 \sqrt {1-2 x}}{294 (2+3 x)}+\frac {134 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 79, 65, 212}
\begin {gather*} -\frac {1091 \sqrt {1-2 x}}{294 (3 x+2)}+\frac {121}{14 \sqrt {1-2 x} (3 x+2)}+\frac {134 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)}-\frac {1}{14} \int \frac {-247+175 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)}-\frac {1091 \sqrt {1-2 x}}{294 (2+3 x)}-\frac {67}{147} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)}-\frac {1091 \sqrt {1-2 x}}{294 (2+3 x)}+\frac {67}{147} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)}-\frac {1091 \sqrt {1-2 x}}{294 (2+3 x)}+\frac {134 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 61, normalized size = 0.90 \begin {gather*} \frac {-2541+1091 (1-2 x)}{147 (-7+3 (1-2 x)) \sqrt {1-2 x}}+\frac {134 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 45, normalized size = 0.66
method | result | size |
risch | \(\frac {1091 x +725}{147 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {134 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(41\) |
derivativedivides | \(\frac {121}{49 \sqrt {1-2 x}}+\frac {2 \sqrt {1-2 x}}{441 \left (-\frac {4}{3}-2 x \right )}+\frac {134 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(45\) |
default | \(\frac {121}{49 \sqrt {1-2 x}}+\frac {2 \sqrt {1-2 x}}{441 \left (-\frac {4}{3}-2 x \right )}+\frac {134 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(45\) |
trager | \(-\frac {\left (1091 x +725\right ) \sqrt {1-2 x}}{147 \left (6 x^{2}+x -2\right )}+\frac {67 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{3087}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 65, normalized size = 0.96 \begin {gather*} -\frac {67}{3087} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1091 \, x + 725\right )}}{147 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 66, normalized size = 0.97 \begin {gather*} \frac {67 \, \sqrt {21} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1091 \, x + 725\right )} \sqrt {-2 \, x + 1}}{3087 \, {\left (6 \, x^{2} + x - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.58, size = 68, normalized size = 1.00 \begin {gather*} -\frac {67}{3087} \, \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 {2 \, {\left (1091 \, x + 725\right )}}{147 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 46, normalized size = 0.68 \begin {gather*} \frac {\frac {2182\,x}{441}+\frac {1450}{441}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}+\frac {134\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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