Optimal. Leaf size=85 \[ -\frac{4}{45} (1-2 x)^{3/2}-\frac{272}{225} \sqrt{1-2 x}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.033054, antiderivative size = 85, 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 = {84, 154, 156, 63, 206} \[ -\frac{4}{45} (1-2 x)^{3/2}-\frac{272}{225} \sqrt{1-2 x}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 84
Rule 154
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)} \, dx &=-\frac{4}{45} (1-2 x)^{3/2}+\frac{1}{15} \int \frac{(-9-136 x) \sqrt{1-2 x}}{(2+3 x) (3+5 x)} \, dx\\ &=-\frac{272}{225} \sqrt{1-2 x}-\frac{4}{45} (1-2 x)^{3/2}+\frac{2}{225} \int \frac{-\frac{1767}{2}-3469 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{272}{225} \sqrt{1-2 x}-\frac{4}{45} (1-2 x)^{3/2}-\frac{343}{9} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{1331}{25} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{272}{225} \sqrt{1-2 x}-\frac{4}{45} (1-2 x)^{3/2}+\frac{343}{9} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{1331}{25} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{272}{225} \sqrt{1-2 x}-\frac{4}{45} (1-2 x)^{3/2}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0428353, size = 71, normalized size = 0.84 \[ \frac{2 \left (30 \sqrt{1-2 x} (10 x-73)+6125 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-3267 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{3375} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 56, normalized size = 0.7 \begin{align*} -{\frac{4}{45} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{242\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{98\,\sqrt{21}}{27}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{272}{225}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.3317, size = 123, normalized size = 1.45 \begin{align*} -\frac{4}{45} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{272}{225} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42813, size = 274, normalized size = 3.22 \begin{align*} \frac{121}{125} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac{49}{27} \, \sqrt{7} \sqrt{3} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac{4}{225} \,{\left (10 \, x - 73\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.8323, size = 158, normalized size = 1.86 \begin{align*} - \frac{4 \left (1 - 2 x\right )^{\frac{3}{2}}}{45} - \frac{272 \sqrt{1 - 2 x}}{225} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{9} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{25} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.0685, size = 131, normalized size = 1.54 \begin{align*} -\frac{4}{45} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{125} \, \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{49}{27} \, \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{272}{225} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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