Optimal. Leaf size=133 \[ \frac{337955 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{14555 \sqrt{1-2 x}}{1176 (3 x+2)^2}+\frac{139 \sqrt{1-2 x}}{84 (3 x+2)^3}+\frac{\sqrt{1-2 x}}{4 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.058789, 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 = {99, 151, 156, 63, 206} \[ \frac{337955 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{14555 \sqrt{1-2 x}}{1176 (3 x+2)^2}+\frac{139 \sqrt{1-2 x}}{84 (3 x+2)^3}+\frac{\sqrt{1-2 x}}{4 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 99
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}-\frac{1}{4} \int \frac{-23+35 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}-\frac{1}{84} \int \frac{-2535+3475 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}-\frac{\int \frac{-192405+218325 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{1176}\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}-\frac{\int \frac{-8277405+5069325 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{8232}\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}-\frac{11656955 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2744}+6875 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}+\frac{11656955 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2744}-6875 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.185854, size = 88, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (9124785 x^3+18555225 x^2+12587542 x+2849254\right )}{2744 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 84, normalized size = 0.6 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{337955\, \left ( 1-2\,x \right ) ^{7/2}}{8232}}-{\frac{3070705\, \left ( 1-2\,x \right ) ^{5/2}}{10584}}+{\frac{3100927\, \left ( 1-2\,x \right ) ^{3/2}}{4536}}-{\frac{116015\,\sqrt{1-2\,x}}{216}} \right ) }+{\frac{11656955\,\sqrt{21}}{28812}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-250\,{\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 = 1.83038, size = 197, normalized size = 1.48 \begin{align*} 125 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11656955}{57624} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9124785 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 64484805 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 151945423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 119379435 \, \sqrt{-2 \, x + 1}}{1372 \,{\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 = 1.66639, size = 466, normalized size = 3.5 \begin{align*} \frac{7203000 \, \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 ) + 11656955 \, \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 (9124785 \, x^{3} + 18555225 \, x^{2} + 12587542 \, x + 2849254\right )} \sqrt{-2 \, x + 1}}{57624 \,{\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 = 2.15542, size = 188, normalized size = 1.41 \begin{align*} 125 \, \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{11656955}{57624} \, \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{9124785 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 64484805 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 151945423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 119379435 \, \sqrt{-2 \, x + 1}}{21952 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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