Optimal. Leaf size=120 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac{201 \sqrt{1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac{1512 \sqrt{1-2 x} (3 x+2)^2}{75625}-\frac{189 \sqrt{1-2 x} (2875 x+8976)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]
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Rubi [A] time = 0.0402599, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac{201 \sqrt{1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac{1512 \sqrt{1-2 x} (3 x+2)^2}{75625}-\frac{189 \sqrt{1-2 x} (2875 x+8976)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
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)^3} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{1}{110} \int \frac{(-150-183 x) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{\int \frac{(-6237-3024 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}+\frac{\int \frac{(2+3 x) (348138+543375 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{151250}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{189 \sqrt{1-2 x} (8976+2875 x)}{756250}+\frac{22113 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{756250}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{189 \sqrt{1-2 x} (8976+2875 x)}{756250}-\frac{22113 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{756250}\\ &=-\frac{1512 \sqrt{1-2 x} (2+3 x)^2}{75625}-\frac{\sqrt{1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac{201 \sqrt{1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac{189 \sqrt{1-2 x} (8976+2875 x)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0669388, size = 68, normalized size = 0.57 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (7350750 x^4+32506650 x^3+76970520 x^2+63610155 x+16525496\right )}{(5 x+3)^2}-44226 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{41593750} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -{\frac{243}{2500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{513}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{39393}{12500}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{333}{2420} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{67}{220}\sqrt{1-2\,x}} \right ) }-{\frac{22113\,\sqrt{55}}{20796875}{\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 = 1.51166, size = 136, normalized size = 1.13 \begin{align*} -\frac{243}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{513}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{22113}{41593750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{39393}{12500} \, \sqrt{-2 \, x + 1} + \frac{333 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{75625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62659, size = 281, normalized size = 2.34 \begin{align*} \frac{22113 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (7350750 \, x^{4} + 32506650 \, x^{3} + 76970520 \, x^{2} + 63610155 \, x + 16525496\right )} \sqrt{-2 \, x + 1}}{41593750 \,{\left (25 \, x^{2} + 30 \, x + 9\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 = 2.72346, size = 138, normalized size = 1.15 \begin{align*} -\frac{243}{2500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{513}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{22113}{41593750} \, \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{39393}{12500} \, \sqrt{-2 \, x + 1} + \frac{333 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{302500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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