Optimal. Leaf size=121 \[ \frac{995 \sqrt{1-2 x}}{22 (5 x+3)}-\frac{15 \sqrt{1-2 x}}{2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0412471, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{995 \sqrt{1-2 x}}{22 (5 x+3)}-\frac{15 \sqrt{1-2 x}}{2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^2 (3+5 x)^3} \, dx &=\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}-\int \frac{-18+25 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{1}{22} \int \frac{-1298+1485 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}-\frac{1}{242} \int \frac{-53614+32835 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}-936 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{33325}{22} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}+936 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{33325}{22} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0902104, size = 94, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} \left (14925 x^2+18410 x+5662\right )}{22 (3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 82, normalized size = 0.7 \begin{align*} -6\,{\frac{\sqrt{1-2\,x}}{-2\,x-4/3}}+{\frac{624\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+250\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{133\, \left ( 1-2\,x \right ) ^{3/2}}{110}}+{\frac{131\,\sqrt{1-2\,x}}{50}} \right ) }-{\frac{6665\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47884, size = 173, normalized size = 1.43 \begin{align*} \frac{6665}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{312}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{14925 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 66670 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 74393 \, \sqrt{-2 \, x + 1}}{11 \,{\left (75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6037, size = 428, normalized size = 3.54 \begin{align*} \frac{46655 \, \sqrt{11} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 75504 \, \sqrt{7} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (14925 \, x^{2} + 18410 \, x + 5662\right )} \sqrt{-2 \, x + 1}}{1694 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 85.0684, size = 474, normalized size = 3.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46266, size = 166, normalized size = 1.37 \begin{align*} \frac{6665}{242} \, \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{312}{7} \, \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{9 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{5 \,{\left (665 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1441 \, \sqrt{-2 \, x + 1}\right )}}{44 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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