Optimal. Leaf size=105 \[ -\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0480583, antiderivative size = 105, 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, 152, 156, 63, 206} \[ -\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 103
Rule 152
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx &=\frac{3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{1}{7} \int \frac{-10-75 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x)}-\frac{2 \int \frac{-555+\frac{4275 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1617}\\ &=-\frac{190}{1617 (1-2 x)^{3/2}}-\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{4 \int \frac{\frac{55065}{2}-\frac{30825 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{124509}\\ &=-\frac{190}{1617 (1-2 x)^{3/2}}-\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x)}-\frac{1080}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{625}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{190}{1617 (1-2 x)^{3/2}}-\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{1080}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{625}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{190}{1617 (1-2 x)^{3/2}}-\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0236097, size = 73, normalized size = 0.7 \[ -\frac{2640 (3 x+2) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-7 \left (350 (3 x+2) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+99\right )}{1617 (1-2 x)^{3/2} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 72, normalized size = 0.7 \begin{align*} -{\frac{18}{343}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}+{\frac{720\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{8}{1617} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{808}{41503}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{250\,\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 = 1.57606, size = 149, normalized size = 1.42 \begin{align*} \frac{125}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{360}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (6165 \,{\left (2 \, x - 1\right )}^{2} - 15120 \, x + 9716\right )}}{124509 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36061, size = 421, normalized size = 4.01 \begin{align*} \frac{900375 \, \sqrt{11} \sqrt{5}{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1437480 \, \sqrt{7} \sqrt{3}{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (24660 \, x^{2} - 39780 \, x + 15881\right )} \sqrt{-2 \, x + 1}}{9587193 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\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 = 2.20729, size = 157, normalized size = 1.5 \begin{align*} \frac{125}{1331} \, \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{360}{2401} \, \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{16 \,{\left (303 \, x - 190\right )}}{124509 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{27 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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