Optimal. Leaf size=82 \[ -\frac{13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}-\frac{41 (4-9 x) \sqrt{3 x^2+2}}{2450 (2 x+3)^2}-\frac{123 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]
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Rubi [A] time = 0.0339467, antiderivative size = 82, 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 = {807, 721, 725, 206} \[ -\frac{13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}-\frac{41 (4-9 x) \sqrt{3 x^2+2}}{2450 (2 x+3)^2}-\frac{123 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]
Antiderivative was successfully verified.
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Rule 807
Rule 721
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^4} \, dx &=-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}+\frac{41}{35} \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac{41 (4-9 x) \sqrt{2+3 x^2}}{2450 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}+\frac{123 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1225}\\ &=-\frac{41 (4-9 x) \sqrt{2+3 x^2}}{2450 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac{123 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1225}\\ &=-\frac{41 (4-9 x) \sqrt{2+3 x^2}}{2450 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac{123 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1225 \sqrt{35}}\\ \end{align*}
Mathematica [A] time = 0.0555901, size = 65, normalized size = 0.79 \[ \frac{-\frac{35 \sqrt{3 x^2+2} \left (516 x^2-2337 x+3296\right )}{(2 x+3)^3}-738 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257250} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 128, normalized size = 1.6 \begin{align*} -{\frac{13}{840} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{41}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{369}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{123}{42875}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{123\,\sqrt{35}}{42875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{1107\,x}{85750}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4866, size = 155, normalized size = 1.89 \begin{align*} \frac{123}{42875} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{123}{4900} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{41 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{1225 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{369 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44595, size = 288, normalized size = 3.51 \begin{align*} \frac{369 \, \sqrt{35}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \,{\left (516 \, x^{2} - 2337 \, x + 3296\right )} \sqrt{3 \, x^{2} + 2}}{257250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25969, size = 308, normalized size = 3.76 \begin{align*} \frac{123}{42875} \, \sqrt{35} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{4659 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 30 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 11610 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 25740 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 60 \, \sqrt{3} x - 1376 \, \sqrt{3} + 60 \, \sqrt{3 \, x^{2} + 2}}{9800 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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