Optimal. Leaf size=80 \[ -\frac {\sqrt {1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac {9 \sqrt {1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac {1347 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
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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} \[ -\frac {\sqrt {1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac {9 \sqrt {1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac {1347 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 146
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {1}{110} \int \frac {(-144-195 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}+\frac {1347 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac {1347 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6050}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac {1347 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.72 \[ \frac {-\frac {55 \sqrt {1-2 x} \left (32670 x^2+39405 x+11884\right )}{(5 x+3)^2}-2694 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{332750} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 74, normalized size = 0.92 \[ \frac {1347 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (32670 \, x^{2} + 39405 \, x + 11884\right )} \sqrt {-2 \, x + 1}}{332750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 77, normalized size = 0.96 \[ \frac {1347}{332750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {27}{125} \, \sqrt {-2 \, x + 1} + \frac {1005 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2233 \, \sqrt {-2 \, x + 1}}{60500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.71 \[ -\frac {1347 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{166375}-\frac {27 \sqrt {-2 x +1}}{125}+\frac {\frac {201 \left (-2 x +1\right )^{\frac {3}{2}}}{3025}-\frac {203 \sqrt {-2 x +1}}{1375}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 83, normalized size = 1.04 \[ \frac {1347}{332750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{125} \, \sqrt {-2 \, x + 1} + \frac {1005 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2233 \, \sqrt {-2 \, x + 1}}{15125 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 63, normalized size = 0.79 \[ -\frac {1347\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{166375}-\frac {27\,\sqrt {1-2\,x}}{125}-\frac {\frac {203\,\sqrt {1-2\,x}}{34375}-\frac {201\,{\left (1-2\,x\right )}^{3/2}}{75625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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