Optimal. Leaf size=133 \[ -\frac{35845 \sqrt{1-2 x}}{1078 (5 x+3)}+\frac{162 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0485905, antiderivative size = 133, 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 = {103, 151, 156, 63, 206} \[ -\frac{35845 \sqrt{1-2 x}}{1078 (5 x+3)}+\frac{162 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac{1}{14} \int \frac{58-75 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac{162 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)}+\frac{1}{98} \int \frac{4253-4860 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{35845 \sqrt{1-2 x}}{1078 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac{162 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)}-\frac{\int \frac{175579-107535 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{1078}\\ &=-\frac{35845 \sqrt{1-2 x}}{1078 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac{162 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)}+\frac{67437}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{12250}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{35845 \sqrt{1-2 x}}{1078 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac{162 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)}-\frac{67437}{98} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{12250}{11} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{35845 \sqrt{1-2 x}}{1078 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac{162 \sqrt{1-2 x}}{49 (2+3 x) (3+5 x)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0813622, size = 96, normalized size = 0.72 \[ -\frac{\sqrt{1-2 x} \left (322605 x^2+419448 x+136021\right )}{1078 (3 x+2)^2 (5 x+3)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 82, normalized size = 0.6 \begin{align*} 486\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{143\, \left ( 1-2\,x \right ) ^{3/2}}{882}}-{\frac{145\,\sqrt{1-2\,x}}{378}} \right ) }-{\frac{22479\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{50}{11}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{4900\,\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 = 1.48006, size = 173, normalized size = 1.3 \begin{align*} -\frac{2450}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22479}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{322605 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 1484106 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1705585 \, \sqrt{-2 \, x + 1}}{539 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7016, size = 436, normalized size = 3.28 \begin{align*} \frac{1680700 \, \sqrt{11} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 2719959 \, \sqrt{7} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (322605 \, x^{2} + 419448 \, x + 136021\right )} \sqrt{-2 \, x + 1}}{83006 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.03039, size = 166, normalized size = 1.25 \begin{align*} -\frac{2450}{121} \, \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{22479}{686} \, \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{125 \, \sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} + \frac{9 \,{\left (429 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1015 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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