Optimal. Leaf size=80 \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10 (1450 x+969)}{1029 \sqrt {1-2 x} (3 x+2)}-\frac {200 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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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, 144, 63, 206} \begin {gather*} \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10 (1450 x+969)}{1029 \sqrt {1-2 x} (3 x+2)}-\frac {200 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 144
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {1}{21} \int \frac {(3+5 x) (130+180 x)}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {10 (969+1450 x)}{1029 \sqrt {1-2 x} (2+3 x)}+\frac {100 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1029}\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {10 (969+1450 x)}{1029 \sqrt {1-2 x} (2+3 x)}-\frac {100 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1029}\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {10 (969+1450 x)}{1029 \sqrt {1-2 x} (2+3 x)}-\frac {200 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 65, normalized size = 0.81 \begin {gather*} \frac {200 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )-\frac {21 \left (42475 x^2+21050 x-4839\right )}{6 x^2+x-2}}{21609 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 70, normalized size = 0.88 \begin {gather*} \frac {-42475 (1-2 x)^2+127050 (1-2 x)-65219}{2058 (3 (1-2 x)-7) (1-2 x)^{3/2}}-\frac {200 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 84, normalized size = 1.05 \begin {gather*} \frac {100 \, \sqrt {21} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (42475 \, x^{2} + 21050 \, x - 4839\right )} \sqrt {-2 \, x + 1}}{21609 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.36, size = 77, normalized size = 0.96 \begin {gather*} \frac {100}{21609} \, \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 {121 \, {\left (117 \, x - 20\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {\sqrt {-2 \, x + 1}}{1029 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 54, normalized size = 0.68 \begin {gather*} -\frac {200 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21609}+\frac {1331}{294 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {4719}{686 \sqrt {-2 x +1}}-\frac {2 \sqrt {-2 x +1}}{3087 \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 = 1.10, size = 74, normalized size = 0.92 \begin {gather*} \frac {100}{21609} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {42475 \, {\left (2 \, x - 1\right )}^{2} + 254100 \, x - 61831}{2058 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\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 {6050\,x}{147}+\frac {42475\,{\left (2\,x-1\right )}^2}{6174}-\frac {8833}{882}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}-\frac {200\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609} \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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