Optimal. Leaf size=153 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{584179 \sqrt{1-2 x}}{196 (3 x+2)}+\frac{25159 \sqrt{1-2 x}}{84 (3 x+2)^2}+\frac{1201 \sqrt{1-2 x}}{30 (3 x+2)^3}+\frac{63 \sqrt{1-2 x}}{10 (3 x+2)^4}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0687126, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{584179 \sqrt{1-2 x}}{196 (3 x+2)}+\frac{25159 \sqrt{1-2 x}}{84 (3 x+2)^2}+\frac{1201 \sqrt{1-2 x}}{30 (3 x+2)^3}+\frac{63 \sqrt{1-2 x}}{10 (3 x+2)^4}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{(228-225 x) \sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}-\frac{1}{180} \int \frac{-25182+37890 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}-\frac{\int \frac{-2761290+3783150 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{3780}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}-\frac{\int \frac{-209531070+237752550 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{52920}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}-\frac{\int \frac{-9014096070+5520491550 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{370440}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}-\frac{20149879}{196} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+166375 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}+\frac{20149879}{196} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-166375 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.125893, size = 93, normalized size = 0.61 \[ \frac{\sqrt{1-2 x} \left (709777485 x^4+1916515215 x^3+1941349752 x^2+874383298 x+147756688\right )}{2940 (3 x+2)^5}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 93, normalized size = 0.6 \begin{align*} -486\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{584179\, \left ( 1-2\,x \right ) ^{9/2}}{588}}-{\frac{504319\, \left ( 1-2\,x \right ) ^{7/2}}{54}}+{\frac{13335122\, \left ( 1-2\,x \right ) ^{5/2}}{405}}-{\frac{75232787\, \left ( 1-2\,x \right ) ^{3/2}}{1458}}+{\frac{29479429\,\sqrt{1-2\,x}}{972}} \right ) }+{\frac{20149879\,\sqrt{21}}{2058}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-6050\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.82177, size = 221, normalized size = 1.44 \begin{align*} 3025 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20149879}{4116} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{709777485 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 6672140370 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 23523155208 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 36864065630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 21667380315 \, \sqrt{-2 \, x + 1}}{1470 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3695, size = 552, normalized size = 3.61 \begin{align*} \frac{62254500 \, \sqrt{55}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 100749395 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \,{\left (709777485 \, x^{4} + 1916515215 \, x^{3} + 1941349752 \, x^{2} + 874383298 \, x + 147756688\right )} \sqrt{-2 \, x + 1}}{20580 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 1.70704, size = 209, normalized size = 1.37 \begin{align*} 3025 \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{20149879}{4116} \, \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{709777485 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 6672140370 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 23523155208 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 36864065630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 21667380315 \, \sqrt{-2 \, x + 1}}{47040 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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