Optimal. Leaf size=93 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{1375}-\frac{21 \sqrt{1-2 x} (375 x+1144)}{6875}-\frac{266 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]
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Rubi [A] time = 0.0272588, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{1375}-\frac{21 \sqrt{1-2 x} (375 x+1144)}{6875}-\frac{266 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]
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)^4}{\sqrt{1-2 x} (3+5 x)^2} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac{1}{55} \int \frac{(-77-84 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{1375}-\frac{\sqrt{1-2 x} (2+3 x)^3}{55 (3+5 x)}+\frac{\int \frac{(2+3 x) (4858+7875 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{1375}\\ &=-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{1375}-\frac{\sqrt{1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac{21 \sqrt{1-2 x} (1144+375 x)}{6875}+\frac{133 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{6875}\\ &=-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{1375}-\frac{\sqrt{1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac{21 \sqrt{1-2 x} (1144+375 x)}{6875}-\frac{133 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{6875}\\ &=-\frac{84 \sqrt{1-2 x} (2+3 x)^2}{1375}-\frac{\sqrt{1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac{21 \sqrt{1-2 x} (1144+375 x)}{6875}-\frac{266 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0552487, size = 63, normalized size = 0.68 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (22275 x^3+82665 x^2+171765 x+78112\right )}{5 x+3}-266 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 63, normalized size = 0.7 \begin{align*} -{\frac{81}{500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{333}{250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{12393}{2500}\sqrt{1-2\,x}}+{\frac{2}{34375}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{266\,\sqrt{55}}{378125}{\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 = 1.5226, size = 108, normalized size = 1.16 \begin{align*} -\frac{81}{500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{333}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{133}{378125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{12393}{2500} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{6875 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71003, size = 215, normalized size = 2.31 \begin{align*} \frac{133 \, \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 (22275 \, x^{3} + 82665 \, x^{2} + 171765 \, x + 78112\right )} \sqrt{-2 \, x + 1}}{378125 \,{\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 = 1.42927, size = 122, normalized size = 1.31 \begin{align*} -\frac{81}{500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{333}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{133}{378125} \, \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{12393}{2500} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{6875 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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