Optimal. Leaf size=68 \[ -\frac {1091 \sqrt {1-2 x}}{294 (3 x+2)}+\frac {121}{14 \sqrt {1-2 x} (3 x+2)}+\frac {134 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
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Rubi [A] time = 0.02, 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 = {89, 78, 63, 206} \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 63
Rule 78
Rule 89
Rule 206
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} \operatorname {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 = 0.05, size = 62, normalized size = 0.91 \begin {gather*} \frac {21 (1091 x+725)+134 \sqrt {21-42 x} (3 x+2) \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3087 \sqrt {1-2 x} (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 61, normalized size = 0.90 \begin {gather*} \frac {1091 (1-2 x)-2541}{147 (3 (1-2 x)-7) \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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fricas [A] time = 1.05, 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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giac [A] time = 1.24, 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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maple [A] time = 0.01, size = 45, normalized size = 0.66 \begin {gather*} \frac {134 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{3087}+\frac {121}{49 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{441 \left (-2 x -\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, 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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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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sympy [F(-1)] 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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