Optimal. Leaf size=181 \[ \frac {11 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {3223 \sqrt {1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac {11 \sqrt {1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac {33935 \sqrt {1-2 x}}{2333772 (3 x+2)}+\frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}} \]
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Rubi [A] time = 0.07, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {97, 12, 149, 145, 51, 63, 206} \[ \frac {11 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {3223 \sqrt {1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac {11 \sqrt {1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac {33935 \sqrt {1-2 x}}{2333772 (3 x+2)}+\frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 97
Rule 145
Rule 149
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {1}{21} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}-\frac {55}{21} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {55}{378} \int \frac {(42-18 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \int \frac {(3+5 x)^2 (-2016+2250 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{1134}\\ &=-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \int \frac {(3+5 x) (-137988+156780 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{95256}\\ &=-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}-\frac {33935 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{333396}\\ &=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}-\frac {33935 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2333772}\\ &=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}+\frac {33935 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2333772}\\ &=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}+\frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 52, normalized size = 0.29 \[ \frac {(1-2 x)^{7/2} \left (\frac {823543 \left (18375 x^2+24448 x+8133\right )}{(3 x+2)^7}-4343680 \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{1089547389} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 145, normalized size = 0.80 \[ \frac {33935 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (24738615 \, x^{6} - 141112395 \, x^{5} - 283697388 \, x^{4} - 164222766 \, x^{3} - 39606312 \, x^{2} - 12384752 \, x - 4005436\right )} \sqrt {-2 \, x + 1}}{49009212 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 148, normalized size = 0.82 \[ -\frac {33935}{49009212} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {24738615 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - 133793100 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 2174834277 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 8180415936 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 13834953363 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 11406910900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3992418815 \, \sqrt {-2 \, x + 1}}{149361408 \, {\left (3 \, x + 2\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 93, normalized size = 0.51 \[ \frac {33935 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{24504606}+\frac {-\frac {101805 \left (-2 x +1\right )^{\frac {13}{2}}}{4802}-\frac {353950 \left (-2 x +1\right )^{\frac {11}{2}}}{3087}+\frac {4931597 \left (-2 x +1\right )^{\frac {9}{2}}}{2646}-\frac {3091616 \left (-2 x +1\right )^{\frac {7}{2}}}{441}+\frac {1920721 \left (-2 x +1\right )^{\frac {5}{2}}}{162}-\frac {2375450 \left (-2 x +1\right )^{\frac {3}{2}}}{243}+\frac {1662815 \sqrt {-2 x +1}}{486}}{\left (-6 x -4\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 164, normalized size = 0.91 \[ -\frac {33935}{49009212} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {24738615 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + 133793100 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2174834277 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 8180415936 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 13834953363 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 11406910900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3992418815 \, \sqrt {-2 \, x + 1}}{1166886 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 143, normalized size = 0.79 \[ \frac {\frac {2375450\,{\left (1-2\,x\right )}^{3/2}}{531441}-\frac {1662815\,\sqrt {1-2\,x}}{1062882}-\frac {1920721\,{\left (1-2\,x\right )}^{5/2}}{354294}+\frac {3091616\,{\left (1-2\,x\right )}^{7/2}}{964467}-\frac {4931597\,{\left (1-2\,x\right )}^{9/2}}{5786802}+\frac {353950\,{\left (1-2\,x\right )}^{11/2}}{6751269}+\frac {33935\,{\left (1-2\,x\right )}^{13/2}}{3500658}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}+\frac {33935\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{24504606} \]
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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