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 = 0.0747664, antiderivative size = 181, normalized size of antiderivative = 1., 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 97
Rule 12
Rule 149
Rule 145
Rule 51
Rule 63
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*}
Mathematica [C] time = 0.0348526, 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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Maple [A] time = 0.011, size = 93, normalized size = 0.5 \begin{align*} 69984\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{7}} \left ( -{\frac{33935\, \left ( 1-2\,x \right ) ^{13/2}}{112021056}}-{\frac{176975\, \left ( 1-2\,x \right ) ^{11/2}}{108020304}}+{\frac{4931597\, \left ( 1-2\,x \right ) ^{9/2}}{185177664}}-{\frac{96613\, \left ( 1-2\,x \right ) ^{7/2}}{964467}}+{\frac{1920721\, \left ( 1-2\,x \right ) ^{5/2}}{11337408}}-{\frac{1187725\, \left ( 1-2\,x \right ) ^{3/2}}{8503056}}+{\frac{1662815\,\sqrt{1-2\,x}}{34012224}} \right ) }+{\frac{33935\,\sqrt{21}}{24504606}{\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 = 5.28164, size = 221, normalized size = 1.22 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39861, size = 501, normalized size = 2.77 \begin{align*} \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 )}} \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 = 2.40031, size = 200, normalized size = 1.1 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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