Optimal. Leaf size=97 \[ \frac{219 \sqrt{1-2 x}}{98 (3 x+2)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0359745, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{219 \sqrt{1-2 x}}{98 (3 x+2)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)} \, dx &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2}+\frac{1}{14} \int \frac{43-45 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2}+\frac{219 \sqrt{1-2 x}}{98 (2+3 x)}+\frac{1}{98} \int \frac{1793-1095 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2}+\frac{219 \sqrt{1-2 x}}{98 (2+3 x)}-\frac{7569}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2}+\frac{219 \sqrt{1-2 x}}{98 (2+3 x)}+\frac{7569}{98} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2}+\frac{219 \sqrt{1-2 x}}{98 (2+3 x)}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0541704, size = 82, normalized size = 0.85 \[ \frac{9 \sqrt{1-2 x} (73 x+51)}{98 (3 x+2)^2}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 66, normalized size = 0.7 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{73\, \left ( 1-2\,x \right ) ^{3/2}}{882}}-{\frac{25\,\sqrt{1-2\,x}}{126}} \right ) }+{\frac{2523\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{50\,\sqrt{55}}{11}{\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 = 1.83043, size = 149, normalized size = 1.54 \begin{align*} \frac{25}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2523}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9 \,{\left (73 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 175 \, \sqrt{-2 \, x + 1}\right )}}{49 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70095, size = 358, normalized size = 3.69 \begin{align*} \frac{17150 \, \sqrt{11} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 27753 \, \sqrt{7} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 693 \,{\left (73 \, x + 51\right )} \sqrt{-2 \, x + 1}}{7546 \,{\left (9 \, x^{2} + 12 \, x + 4\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 = 1.61165, size = 144, normalized size = 1.48 \begin{align*} \frac{25}{11} \, \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{2523}{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{9 \,{\left (73 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 175 \, \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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