Optimal. Leaf size=154 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac{6649 \sqrt{1-2 x}}{27 (3 x+2) (5 x+3)}+\frac{917 \sqrt{1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac{44545 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0622437, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac{6649 \sqrt{1-2 x}}{27 (3 x+2) (5 x+3)}+\frac{917 \sqrt{1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac{44545 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^4 (3+5 x)^2} \, dx &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{1}{9} \int \frac{(197-163 x) \sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{917 \sqrt{1-2 x}}{54 (2+3 x)^2 (3+5 x)}-\frac{1}{54} \int \frac{-16180+22273 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{917 \sqrt{1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac{6649 \sqrt{1-2 x}}{27 (2+3 x) (3+5 x)}-\frac{1}{378} \int \frac{-1220205+1396290 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{44545 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{917 \sqrt{1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac{6649 \sqrt{1-2 x}}{27 (2+3 x) (3+5 x)}+\frac{\int \frac{-50405355+30869685 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{4158}\\ &=-\frac{44545 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{917 \sqrt{1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac{6649 \sqrt{1-2 x}}{27 (2+3 x) (3+5 x)}+\frac{307295}{6} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-82885 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{44545 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{917 \sqrt{1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac{6649 \sqrt{1-2 x}}{27 (2+3 x) (3+5 x)}-\frac{307295}{6} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+82885 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{44545 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac{917 \sqrt{1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac{6649 \sqrt{1-2 x}}{27 (2+3 x) (3+5 x)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.124485, size = 95, normalized size = 0.62 \[ -\frac{\sqrt{1-2 x} \left (400905 x^3+788512 x^2+516513 x+112668\right )}{6 (3 x+2)^3 (5 x+3)}-\frac{307295 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}+3014 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 91, normalized size = 0.6 \begin{align*} 108\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{6731\, \left ( 1-2\,x \right ) ^{5/2}}{36}}-{\frac{71365\, \left ( 1-2\,x \right ) ^{3/2}}{81}}+{\frac{336385\,\sqrt{1-2\,x}}{324}} \right ) }-{\frac{307295\,\sqrt{21}}{63}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+242\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+3014\,{\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 = 2.20014, size = 197, normalized size = 1.28 \begin{align*} -1507 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{307295}{126} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{400905 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 2779739 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 6422815 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 4945325 \, \sqrt{-2 \, x + 1}}{3 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41438, size = 459, normalized size = 2.98 \begin{align*} \frac{189882 \, \sqrt{55}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 307295 \, \sqrt{21}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (400905 \, x^{3} + 788512 \, x^{2} + 516513 \, x + 112668\right )} \sqrt{-2 \, x + 1}}{126 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 = 1.71335, size = 188, normalized size = 1.22 \begin{align*} -1507 \, \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{307295}{126} \, \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{605 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{60579 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 285460 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 336385 \, \sqrt{-2 \, x + 1}}{24 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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