Optimal. Leaf size=147 \[ \frac{1020 \sqrt{1-2 x}}{5 x+3}-\frac{1015 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{45 \sqrt{1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-13665 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0582476, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 151, 156, 63, 206} \[ \frac{1020 \sqrt{1-2 x}}{5 x+3}-\frac{1015 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{45 \sqrt{1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-13665 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{1}{6} \int \frac{157-237 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{45 \sqrt{1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac{1}{42} \int \frac{17087-23625 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{1015 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{45 \sqrt{1-2 x}}{2 (2+3 x) (3+5 x)^2}-\frac{1}{924} \int \frac{1229382-1406790 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{1015 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{45 \sqrt{1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac{1020 \sqrt{1-2 x}}{3+5 x}+\frac{\int \frac{50784426-31101840 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{10164}\\ &=-\frac{1015 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{45 \sqrt{1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac{1020 \sqrt{1-2 x}}{3+5 x}-\frac{42219}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{68325}{2} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{1015 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{45 \sqrt{1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac{1020 \sqrt{1-2 x}}{3+5 x}+\frac{42219}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{68325}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{1015 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{45 \sqrt{1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac{1020 \sqrt{1-2 x}}{3+5 x}+14073 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-13665 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.087134, size = 119, normalized size = 0.81 \[ \frac{77 \sqrt{1-2 x} \left (91800 x^3+174435 x^2+110315 x+23219\right )+309606 \sqrt{21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-191310 \sqrt{55} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{154 (3 x+2)^2 (5 x+3)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 94, normalized size = 0.6 \begin{align*} -324\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{205\, \left ( 1-2\,x \right ) ^{3/2}}{36}}-{\frac{161\,\sqrt{1-2\,x}}{12}} \right ) }+{\frac{14073\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+500\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{203\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{2211\,\sqrt{1-2\,x}}{100}} \right ) }-{\frac{13665\,\sqrt{55}}{11}{\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 = 1.57136, size = 197, normalized size = 1.34 \begin{align*} \frac{13665}{22} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{14073}{14} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (45900 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 312135 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 707200 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 533841 \, \sqrt{-2 \, x + 1}\right )}}{225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59267, size = 497, normalized size = 3.38 \begin{align*} \frac{95655 \, \sqrt{11} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 154803 \, \sqrt{7} \sqrt{3}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (91800 \, x^{3} + 174435 \, x^{2} + 110315 \, x + 23219\right )} \sqrt{-2 \, x + 1}}{154 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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 = 2.2515, size = 200, normalized size = 1.36 \begin{align*} \frac{13665}{22} \, \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{14073}{14} \, \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{2 \,{\left (45900 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 312135 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 707200 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 533841 \, \sqrt{-2 \, x + 1}\right )}}{{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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