Optimal. Leaf size=80 \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac {\sqrt {1-2 x} (12425 x+8329)}{882 (3 x+2)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]
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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 {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac {\sqrt {1-2 x} (12425 x+8329)}{882 (3 x+2)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \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}{\sqrt {1-2 x} (2+3 x)^3} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {1}{42} \int \frac {(-191-355 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}-\frac {2381}{882} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}+\frac {2381}{882} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.72 \begin {gather*} \frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}}-\frac {\sqrt {1-2 x} \left (36750 x^2+49207 x+16469\right )}{882 (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 70, normalized size = 0.88 \begin {gather*} \frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}}-\frac {\left (18375 (1-2 x)^2-85957 (1-2 x)+100520\right ) \sqrt {1-2 x}}{441 (3 (1-2 x)-7)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 75, normalized size = 0.94 \begin {gather*} \frac {2381 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (36750 \, x^{2} + 49207 \, x + 16469\right )} \sqrt {-2 \, x + 1}}{18522 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 77, normalized size = 0.96 \begin {gather*} -\frac {2381}{18522} \, \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}{27} \, \sqrt {-2 \, x + 1} + \frac {621 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1435 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
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 \begin {gather*} \frac {2381 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{9261}-\frac {125 \sqrt {-2 x +1}}{27}-\frac {2 \left (-\frac {69 \left (-2 x +1\right )^{\frac {3}{2}}}{98}+\frac {205 \sqrt {-2 x +1}}{126}\right )}{3 \left (-6 x -4\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 83, normalized size = 1.04 \begin {gather*} -\frac {2381}{18522} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {125}{27} \, \sqrt {-2 \, x + 1} + \frac {621 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1435 \, \sqrt {-2 \, x + 1}}{1323 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 63, normalized size = 0.79 \begin {gather*} \frac {2381\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9261}-\frac {125\,\sqrt {1-2\,x}}{27}-\frac {\frac {205\,\sqrt {1-2\,x}}{1701}-\frac {23\,{\left (1-2\,x\right )}^{3/2}}{441}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \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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