Optimal. Leaf size=106 \[ -\frac{340 \sqrt{1-2 x}}{77 (5 x+3)}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{650}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0364174, antiderivative size = 106, 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} \[ -\frac{340 \sqrt{1-2 x}}{77 (5 x+3)}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{650}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^2 (3+5 x)^2} \, dx &=\frac{3 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}+\frac{1}{7} \int \frac{41-45 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{340 \sqrt{1-2 x}}{77 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}-\frac{1}{77} \int \frac{1663-1020 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{340 \sqrt{1-2 x}}{77 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}+\frac{639}{7} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{1625}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{340 \sqrt{1-2 x}}{77 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}-\frac{639}{7} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{1625}{11} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{340 \sqrt{1-2 x}}{77 (3+5 x)}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{650}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.070332, size = 100, normalized size = 0.94 \[ \frac{4550 \sqrt{55} \left (15 x^2+19 x+6\right ) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-11 \sqrt{1-2 x} (1020 x+647)}{847 (3 x+2) (5 x+3)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 70, normalized size = 0.7 \begin{align*}{\frac{6}{7}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{426\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{10}{11}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{650\,\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 = 1.76163, size = 149, normalized size = 1.41 \begin{align*} -\frac{325}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{213}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1157 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68955, size = 365, normalized size = 3.44 \begin{align*} \frac{15925 \, \sqrt{11} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 25773 \, \sqrt{7} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (1020 \, x + 647\right )} \sqrt{-2 \, x + 1}}{5929 \,{\left (15 \, x^{2} + 19 \, x + 6\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 = 2.06648, size = 157, normalized size = 1.48 \begin{align*} -\frac{325}{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{213}{49} \, \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{4 \,{\left (510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1157 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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