Optimal. Leaf size=117 \[ \frac{3840 \sqrt{1-2 x}}{343 (3 x+2)}+\frac{55 \sqrt{1-2 x}}{49 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0477497, antiderivative size = 117, 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, 151, 156, 63, 206} \[ \frac{3840 \sqrt{1-2 x}}{343 (3 x+2)}+\frac{55 \sqrt{1-2 x}}{49 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^4 (3+5 x)} \, dx &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3}+\frac{1}{21} \int \frac{60-75 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3}+\frac{55 \sqrt{1-2 x}}{49 (2+3 x)^2}+\frac{1}{294} \int \frac{4380-4950 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3}+\frac{55 \sqrt{1-2 x}}{49 (2+3 x)^2}+\frac{3840 \sqrt{1-2 x}}{343 (2+3 x)}+\frac{\int \frac{188130-115200 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{2058}\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3}+\frac{55 \sqrt{1-2 x}}{49 (2+3 x)^2}+\frac{3840 \sqrt{1-2 x}}{343 (2+3 x)}-\frac{132465}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+625 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3}+\frac{55 \sqrt{1-2 x}}{49 (2+3 x)^2}+\frac{3840 \sqrt{1-2 x}}{343 (2+3 x)}+\frac{132465}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-625 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3}+\frac{55 \sqrt{1-2 x}}{49 (2+3 x)^2}+\frac{3840 \sqrt{1-2 x}}{343 (2+3 x)}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0721906, size = 87, normalized size = 0.74 \[ \frac{3 \sqrt{1-2 x} \left (11520 x^2+15745 x+5393\right )}{343 (3 x+2)^3}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 75, normalized size = 0.6 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1280\, \left ( 1-2\,x \right ) ^{5/2}}{1029}}-{\frac{7790\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{1318\,\sqrt{1-2\,x}}{189}} \right ) }+{\frac{88310\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{250\,\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 = 1.50917, size = 173, normalized size = 1.48 \begin{align*} \frac{125}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{44155}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12 \,{\left (5760 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 27265 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 32291 \, \sqrt{-2 \, x + 1}\right )}}{343 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71767, size = 425, normalized size = 3.63 \begin{align*} \frac{300125 \, \sqrt{11} \sqrt{5}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 485705 \, \sqrt{7} \sqrt{3}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \,{\left (11520 \, x^{2} + 15745 \, x + 5393\right )} \sqrt{-2 \, x + 1}}{26411 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 = 1.44843, size = 166, normalized size = 1.42 \begin{align*} \frac{125}{11} \, \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{44155}{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{3 \,{\left (5760 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 27265 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 32291 \, \sqrt{-2 \, x + 1}\right )}}{686 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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