Optimal. Leaf size=113 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{55 (5 x+3)}-\frac{78 \sqrt{1-2 x} (3 x+2)^3}{1925}-\frac{1668 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{6 \sqrt{1-2 x} (19875 x+59708)}{34375}-\frac{332 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]
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Rubi [A] time = 0.0368678, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{55 (5 x+3)}-\frac{78 \sqrt{1-2 x} (3 x+2)^3}{1925}-\frac{1668 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{6 \sqrt{1-2 x} (19875 x+59708)}{34375}-\frac{332 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)^2} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{55 (3+5 x)}-\frac{1}{55} \int \frac{(-80-78 x) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{78 \sqrt{1-2 x} (2+3 x)^3}{1925}-\frac{\sqrt{1-2 x} (2+3 x)^4}{55 (3+5 x)}+\frac{\int \frac{(2+3 x)^2 (7238+11676 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{1925}\\ &=-\frac{1668 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{78 \sqrt{1-2 x} (2+3 x)^3}{1925}-\frac{\sqrt{1-2 x} (2+3 x)^4}{55 (3+5 x)}-\frac{\int \frac{(-502012-834750 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{48125}\\ &=-\frac{1668 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{78 \sqrt{1-2 x} (2+3 x)^3}{1925}-\frac{\sqrt{1-2 x} (2+3 x)^4}{55 (3+5 x)}-\frac{6 \sqrt{1-2 x} (59708+19875 x)}{34375}+\frac{166 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{34375}\\ &=-\frac{1668 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{78 \sqrt{1-2 x} (2+3 x)^3}{1925}-\frac{\sqrt{1-2 x} (2+3 x)^4}{55 (3+5 x)}-\frac{6 \sqrt{1-2 x} (59708+19875 x)}{34375}-\frac{166 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{34375}\\ &=-\frac{1668 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{78 \sqrt{1-2 x} (2+3 x)^3}{1925}-\frac{\sqrt{1-2 x} (2+3 x)^4}{55 (3+5 x)}-\frac{6 \sqrt{1-2 x} (59708+19875 x)}{34375}-\frac{332 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0611738, size = 68, normalized size = 0.6 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (1670625 x^4+6994350 x^3+13532310 x^2+20175210 x+8527768\right )}{5 x+3}-2324 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{13234375} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 72, normalized size = 0.6 \begin{align*}{\frac{243}{1400} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{8829}{5000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{35703}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{434043}{25000}\sqrt{1-2\,x}}+{\frac{2}{171875}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{332\,\sqrt{55}}{1890625}{\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 = 1.8859, size = 120, normalized size = 1.06 \begin{align*} \frac{243}{1400} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{8829}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{35703}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{166}{1890625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{434043}{25000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{34375 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74695, size = 250, normalized size = 2.21 \begin{align*} \frac{1162 \, \sqrt{55}{\left (5 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (1670625 \, x^{4} + 6994350 \, x^{3} + 13532310 \, x^{2} + 20175210 \, x + 8527768\right )} \sqrt{-2 \, x + 1}}{13234375 \,{\left (5 \, x + 3\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 = 2.74584, size = 143, normalized size = 1.27 \begin{align*} -\frac{243}{1400} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{8829}{5000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{35703}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{166}{1890625} \, \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{434043}{25000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{34375 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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