Optimal. Leaf size=68 \[ -\frac{27 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{550 (5 x+3)^2}-\frac{2313 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]
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Rubi [A] time = 0.0154342, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {89, 78, 63, 206} \[ -\frac{27 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{550 (5 x+3)^2}-\frac{2313 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x}}{550 (3+5 x)^2}+\frac{1}{550} \int \frac{729+990 x}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x}}{550 (3+5 x)^2}-\frac{27 \sqrt{1-2 x}}{1210 (3+5 x)}+\frac{2313 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac{\sqrt{1-2 x}}{550 (3+5 x)^2}-\frac{27 \sqrt{1-2 x}}{1210 (3+5 x)}-\frac{2313 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{6050}\\ &=-\frac{\sqrt{1-2 x}}{550 (3+5 x)^2}-\frac{27 \sqrt{1-2 x}}{1210 (3+5 x)}-\frac{2313 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0295907, size = 53, normalized size = 0.78 \[ \frac{-\frac{55 \sqrt{1-2 x} (675 x+416)}{(5 x+3)^2}-4626 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{332750} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 48, normalized size = 0.7 \begin{align*} 50\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{27\, \left ( 1-2\,x \right ) ^{3/2}}{6050}}-{\frac{137\,\sqrt{1-2\,x}}{13750}} \right ) }-{\frac{2313\,\sqrt{55}}{166375}{\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 = 1.48961, size = 100, normalized size = 1.47 \begin{align*} \frac{2313}{332750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{675 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1507 \, \sqrt{-2 \, x + 1}}{3025 \,{\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 = 1.67636, size = 204, normalized size = 3. \begin{align*} \frac{2313 \, \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 (675 \, x + 416\right )} \sqrt{-2 \, x + 1}}{332750 \,{\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 = 2.2198, size = 92, normalized size = 1.35 \begin{align*} \frac{2313}{332750} \, \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{675 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1507 \, \sqrt{-2 \, x + 1}}{12100 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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