Optimal. Leaf size=80 \[ \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}+\frac {2 \sqrt {1-2 x} (2975 x+1978)}{147 (3 x+2)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
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Rubi [A] time = 0.02, antiderivative size = 80, 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 = {98, 146, 63, 206} \begin {gather*} \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}+\frac {2 \sqrt {1-2 x} (2975 x+1978)}{147 (3 x+2)}-\frac {68 \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 98
Rule 146
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)}-\frac {1}{7} \int \frac {(3+5 x) (124+170 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)}+\frac {2 \sqrt {1-2 x} (1978+2975 x)}{147 (2+3 x)}+\frac {34}{147} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)}+\frac {2 \sqrt {1-2 x} (1978+2975 x)}{147 (2+3 x)}-\frac {34}{147} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)}+\frac {2 \sqrt {1-2 x} (1978+2975 x)}{147 (2+3 x)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.99 \begin {gather*} \frac {68 \sqrt {21} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )-21 \sqrt {2 x-1} \left (6125 x^2-4968 x-6035\right )}{3087 \sqrt {-(1-2 x)^2} (3 x+2)} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.13, size = 70, normalized size = 0.88 \begin {gather*} \frac {6125 (1-2 x)^2-2314 (1-2 x)-27951}{294 (3 (1-2 x)-7) \sqrt {1-2 x}}-\frac {68 \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 = 1.11, size = 70, normalized size = 0.88 \begin {gather*} \frac {34 \, \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 (6125 \, x^{2} - 4968 \, x - 6035\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 = 1.34, size = 77, normalized size = 0.96 \begin {gather*} \frac {34}{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 {125}{18} \, \sqrt {-2 \, x + 1} - \frac {35933 \, x + 23960}{441 \, {\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 = 54, normalized size = 0.68 \begin {gather*} -\frac {68 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{3087}+\frac {125 \sqrt {-2 x +1}}{18}+\frac {1331}{98 \sqrt {-2 x +1}}-\frac {2 \sqrt {-2 x +1}}{1323 \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 = 1.34, size = 74, normalized size = 0.92 \begin {gather*} \frac {34}{3087} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {125}{18} \, \sqrt {-2 \, x + 1} - \frac {35933 \, x + 23960}{441 \, {\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 = 0.06, size = 55, normalized size = 0.69 \begin {gather*} \frac {\frac {35933\,x}{1323}+\frac {23960}{1323}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}-\frac {68\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087}+\frac {125\,\sqrt {1-2\,x}}{18} \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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