Optimal. Leaf size=124 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}+\frac{7103 \sqrt{1-2 x}}{30 (5 x+3)}-\frac{1133 \sqrt{1-2 x}}{30 (5 x+3)^2}+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7209}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0442867, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}+\frac{7103 \sqrt{1-2 x}}{30 (5 x+3)}-\frac{1133 \sqrt{1-2 x}}{30 (5 x+3)^2}+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7209}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx &=\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac{1}{3} \int \frac{(166-101 x) \sqrt{1-2 x}}{(2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{1133 \sqrt{1-2 x}}{30 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac{1}{30} \int \frac{-9266+10601 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{1133 \sqrt{1-2 x}}{30 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac{7103 \sqrt{1-2 x}}{30 (3+5 x)}-\frac{1}{330} \int \frac{-382734+234399 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{1133 \sqrt{1-2 x}}{30 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac{7103 \sqrt{1-2 x}}{30 (3+5 x)}-4900 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{79299}{10} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{1133 \sqrt{1-2 x}}{30 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac{7103 \sqrt{1-2 x}}{30 (3+5 x)}+4900 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{79299}{10} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{1133 \sqrt{1-2 x}}{30 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac{7103 \sqrt{1-2 x}}{30 (3+5 x)}+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7209}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.114982, size = 93, normalized size = 0.75 \[ \frac{1}{50} \left (\frac{5 \sqrt{1-2 x} \left (35515 x^2+43806 x+13474\right )}{(3 x+2) (5 x+3)^2}-14418 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 82, normalized size = 0.7 \begin{align*} -{\frac{98}{3}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}+{\frac{1400\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+550\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{141\, \left ( 1-2\,x \right ) ^{3/2}}{50}}+{\frac{1529\,\sqrt{1-2\,x}}{250}} \right ) }-{\frac{7209\,\sqrt{55}}{25}{\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 = 2.61532, size = 173, normalized size = 1.4 \begin{align*} \frac{7209}{50} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{700}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{35515 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 158642 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 177023 \, \sqrt{-2 \, x + 1}}{5 \,{\left (75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43198, size = 428, normalized size = 3.45 \begin{align*} \frac{21627 \, \sqrt{11} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 35000 \, \sqrt{7} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 15 \,{\left (35515 \, x^{2} + 43806 \, x + 13474\right )} \sqrt{-2 \, x + 1}}{150 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.21184, size = 166, normalized size = 1.34 \begin{align*} \frac{7209}{50} \, \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{700}{3} \, \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{49 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{11 \,{\left (705 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1529 \, \sqrt{-2 \, x + 1}\right )}}{20 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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