Optimal. Leaf size=167 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac{2 \sqrt{1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac{23717 \sqrt{1-2 x}}{9529569 (3 x+2)}+\frac{23717 \sqrt{1-2 x}}{4084101 (3 x+2)^2}+\frac{47434 \sqrt{1-2 x}}{2917215 (3 x+2)^3}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}} \]
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Rubi [A] time = 0.0552626, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 145, 51, 63, 206} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac{2 \sqrt{1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac{23717 \sqrt{1-2 x}}{9529569 (3 x+2)}+\frac{23717 \sqrt{1-2 x}}{4084101 (3 x+2)^2}+\frac{47434 \sqrt{1-2 x}}{2917215 (3 x+2)^3}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 145
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}+\frac{1}{21} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^7} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}+\frac{\int \frac{(148-3640 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^6} \, dx}{2646}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{47434 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx}{138915}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{47434 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{583443}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{23717 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{1361367}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}+\frac{23717 \sqrt{1-2 x}}{9529569 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}-\frac{23717 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{9529569}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}+\frac{23717 \sqrt{1-2 x}}{9529569 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}+\frac{23717 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{9529569}\\ &=\frac{47434 \sqrt{1-2 x}}{2917215 (2+3 x)^3}+\frac{23717 \sqrt{1-2 x}}{4084101 (2+3 x)^2}+\frac{23717 \sqrt{1-2 x}}{9529569 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac{2 \sqrt{1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0296301, size = 52, normalized size = 0.31 \[ \frac{(1-2 x)^{3/2} \left (\frac{235298 \left (165375 x^2+219414 x+72797\right )}{(3 x+2)^7}-24286208 \, _2F_1\left (\frac{3}{2},6;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{6537284334} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 93, normalized size = 0.6 \begin{align*} 69984\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{7}} \left ( -{\frac{23717\, \left ( 1-2\,x \right ) ^{13/2}}{457419312}}+{\frac{118585\, \left ( 1-2\,x \right ) ^{11/2}}{147027636}}-{\frac{6711911\, \left ( 1-2\,x \right ) ^{9/2}}{1260236880}}+{\frac{1303513\, \left ( 1-2\,x \right ) ^{7/2}}{78764805}}-{\frac{5101561\, \left ( 1-2\,x \right ) ^{5/2}}{231472080}}+{\frac{25163\, \left ( 1-2\,x \right ) ^{3/2}}{4960116}}+{\frac{23717\,\sqrt{1-2\,x}}{2834352}} \right ) }+{\frac{47434\,\sqrt{21}}{200120949}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.9789, size = 221, normalized size = 1.32 \begin{align*} -\frac{23717}{200120949} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (86448465 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 1344753900 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 8879858253 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 27592763184 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 36746543883 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 8458290820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13951406665 \, \sqrt{-2 \, x + 1}\right )}}{47647845 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.594, size = 509, normalized size = 3.05 \begin{align*} \frac{118585 \, \sqrt{21}{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (86448465 \, x^{6} + 413031555 \, x^{5} + 863203932 \, x^{4} + 473987484 \, x^{3} - 306463011 \, x^{2} - 361589428 \, x - 88036937\right )} \sqrt{-2 \, x + 1}}{1000604745 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.66353, size = 200, normalized size = 1.2 \begin{align*} -\frac{23717}{200120949} \, \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{86448465 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + 1344753900 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 8879858253 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 27592763184 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 36746543883 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 8458290820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13951406665 \, \sqrt{-2 \, x + 1}}{3049462080 \,{\left (3 \, x + 2\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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