Optimal. Leaf size=97 \[ \frac{315 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{5 \sqrt{1-2 x}}{22 (5 x+3)^2}+18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2115}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0351119, 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} \[ \frac{315 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{5 \sqrt{1-2 x}}{22 (5 x+3)^2}+18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2115}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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+5 x)^3} \, dx &=-\frac{5 \sqrt{1-2 x}}{22 (3+5 x)^2}-\frac{1}{22} \int \frac{36-45 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{5 \sqrt{1-2 x}}{22 (3+5 x)^2}+\frac{315 \sqrt{1-2 x}}{242 (3+5 x)}+\frac{1}{242} \int \frac{1548-945 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{5 \sqrt{1-2 x}}{22 (3+5 x)^2}+\frac{315 \sqrt{1-2 x}}{242 (3+5 x)}-27 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{10575}{242} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{5 \sqrt{1-2 x}}{22 (3+5 x)^2}+\frac{315 \sqrt{1-2 x}}{242 (3+5 x)}+27 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{10575}{242} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{5 \sqrt{1-2 x}}{22 (3+5 x)^2}+\frac{315 \sqrt{1-2 x}}{242 (3+5 x)}+18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2115}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0752661, size = 81, normalized size = 0.84 \[ 18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{5 \left (\frac{11 \sqrt{1-2 x} (315 x+178)}{(5 x+3)^2}-846 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{2662} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.7 \begin{align*}{\frac{18\,\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{63\, \left ( 1-2\,x \right ) ^{3/2}}{1210}}+{\frac{61\,\sqrt{1-2\,x}}{550}} \right ) }-{\frac{2115\,\sqrt{55}}{1331}{\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 = 1.61259, size = 149, normalized size = 1.54 \begin{align*} \frac{2115}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5 \,{\left (315 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 671 \, \sqrt{-2 \, x + 1}\right )}}{121 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71076, size = 366, normalized size = 3.77 \begin{align*} \frac{14805 \, \sqrt{11} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 23958 \, \sqrt{7} \sqrt{3}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 385 \,{\left (315 \, x + 178\right )} \sqrt{-2 \, x + 1}}{18634 \,{\left (25 \, x^{2} + 30 \, x + 9\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 = 1.76396, size = 144, normalized size = 1.48 \begin{align*} \frac{2115}{2662} \, \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{9}{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{5 \,{\left (315 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 671 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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