Optimal. Leaf size=133 \[ \frac{318643 \sqrt{1-2 x}}{1176 (3 x+2)}+\frac{13723 \sqrt{1-2 x}}{504 (3 x+2)^2}+\frac{131 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4}+\frac{10990843 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}-550 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0554824, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 151, 156, 63, 206} \[ \frac{318643 \sqrt{1-2 x}}{1176 (3 x+2)}+\frac{13723 \sqrt{1-2 x}}{504 (3 x+2)^2}+\frac{131 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4}+\frac{10990843 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}-550 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{153-229 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{131 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{1}{252} \int \frac{16737-22925 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{131 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{13723 \sqrt{1-2 x}}{504 (2+3 x)^2}+\frac{\int \frac{1269891-1440915 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{3528}\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{131 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{13723 \sqrt{1-2 x}}{504 (2+3 x)^2}+\frac{318643 \sqrt{1-2 x}}{1176 (2+3 x)}+\frac{\int \frac{54630891-33457515 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{24696}\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{131 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{13723 \sqrt{1-2 x}}{504 (2+3 x)^2}+\frac{318643 \sqrt{1-2 x}}{1176 (2+3 x)}-\frac{10990843 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1176}+15125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{131 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{13723 \sqrt{1-2 x}}{504 (2+3 x)^2}+\frac{318643 \sqrt{1-2 x}}{1176 (2+3 x)}+\frac{10990843 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1176}-15125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4}+\frac{131 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{13723 \sqrt{1-2 x}}{504 (2+3 x)^2}+\frac{318643 \sqrt{1-2 x}}{1176 (2+3 x)}+\frac{10990843 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}-550 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.136453, size = 90, normalized size = 0.68 \[ \frac{21 \left (\frac{\sqrt{1-2 x} \left (8603361 x^3+17494905 x^2+11868230 x+2686470\right )}{(3 x+2)^4}-646800 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )+21981686 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 84, normalized size = 0.6 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{318643\, \left ( 1-2\,x \right ) ^{7/2}}{3528}}-{\frac{2895233\, \left ( 1-2\,x \right ) ^{5/2}}{4536}}+{\frac{2923727\, \left ( 1-2\,x \right ) ^{3/2}}{1944}}-{\frac{2297099\,\sqrt{1-2\,x}}{1944}} \right ) }+{\frac{10990843\,\sqrt{21}}{12348}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-550\,{\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 = 1.61228, size = 197, normalized size = 1.48 \begin{align*} 275 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{10990843}{24696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8603361 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 60799893 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 143262623 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 112557851 \, \sqrt{-2 \, x + 1}}{588 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55114, size = 466, normalized size = 3.5 \begin{align*} \frac{6791400 \, \sqrt{55}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 10990843 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (8603361 \, x^{3} + 17494905 \, x^{2} + 11868230 \, x + 2686470\right )} \sqrt{-2 \, x + 1}}{24696 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 2.65172, size = 188, normalized size = 1.41 \begin{align*} 275 \, \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{10990843}{24696} \, \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{8603361 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 60799893 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 143262623 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 112557851 \, \sqrt{-2 \, x + 1}}{9408 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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