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 = 0.06, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 51
Rule 63
Rule 97
Rule 145
Rule 149
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*}
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Mathematica [C] time = 0.04, 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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fricas [A] time = 0.94, size = 145, normalized size = 0.87 \[ \frac {118585 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (86448465 \, x^{6} + 413031555 \, x^{5} + 863203932 \, x^{4} + 473987484 \, x^{3} - 306463011 \, x^{2} - 361589428 \, x - 88036937\right )} \sqrt {-2 \, x + 1}}{1000604745 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 148, normalized size = 0.89 \[ -\frac {23717}{200120949} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {86448465 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 1344753900 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 8879858253 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 27592763184 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 36746543883 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 8458290820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13951406665 \, \sqrt {-2 \, x + 1}}{3049462080 \, {\left (3 \, x + 2\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 93, normalized size = 0.56 \[ \frac {47434 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{200120949}+\frac {-\frac {426906 \left (-2 x +1\right )^{\frac {13}{2}}}{117649}+\frac {948680 \left (-2 x +1\right )^{\frac {11}{2}}}{16807}-\frac {13423822 \left (-2 x +1\right )^{\frac {9}{2}}}{36015}+\frac {41712416 \left (-2 x +1\right )^{\frac {7}{2}}}{36015}-\frac {10203122 \left (-2 x +1\right )^{\frac {5}{2}}}{6615}+\frac {201304 \left (-2 x +1\right )^{\frac {3}{2}}}{567}+\frac {47434 \sqrt {-2 x +1}}{81}}{\left (-6 x -4\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 164, normalized size = 0.98 \[ -\frac {23717}{200120949} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (86448465 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 1344753900 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 8879858253 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 27592763184 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 36746543883 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8458290820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13951406665 \, \sqrt {-2 \, x + 1}\right )}}{47647845 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 144, normalized size = 0.86 \[ \frac {47434\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{200120949}-\frac {\frac {47434\,\sqrt {1-2\,x}}{177147}+\frac {201304\,{\left (1-2\,x\right )}^{3/2}}{1240029}-\frac {10203122\,{\left (1-2\,x\right )}^{5/2}}{14467005}+\frac {41712416\,{\left (1-2\,x\right )}^{7/2}}{78764805}-\frac {13423822\,{\left (1-2\,x\right )}^{9/2}}{78764805}+\frac {948680\,{\left (1-2\,x\right )}^{11/2}}{36756909}-\frac {47434\,{\left (1-2\,x\right )}^{13/2}}{28588707}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}} \]
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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