Optimal. Leaf size=104 \[ \frac{41 x+26}{70 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{331 \sqrt{3 x^2+2}}{8575 (2 x+3)}+\frac{9 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]
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Rubi [A] time = 0.0566041, antiderivative size = 104, 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 = {823, 835, 807, 725, 206} \[ \frac{41 x+26}{70 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{331 \sqrt{3 x^2+2}}{8575 (2 x+3)}+\frac{9 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]
Antiderivative was successfully verified.
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Rule 823
Rule 835
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}-\frac{1}{210} \int \frac{-468-492 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}+\frac{\int \frac{12360+1620 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{14700}\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{331 \sqrt{2+3 x^2}}{8575 (3+2 x)}+\frac{1962 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{8575}\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{331 \sqrt{2+3 x^2}}{8575 (3+2 x)}-\frac{1962 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{8575}\\ &=\frac{26+41 x}{70 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{9 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{331 \sqrt{2+3 x^2}}{8575 (3+2 x)}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{8575 \sqrt{35}}\\ \end{align*}
Mathematica [A] time = 0.0817641, size = 70, normalized size = 0.67 \[ \frac{-\frac{35 \left (3972 x^3+4068 x^2-7397 x-3658\right )}{(2 x+3)^2 \sqrt{3 x^2+2}}-3924 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{600250} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 107, normalized size = 1. \begin{align*} -{\frac{103}{980} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{981}{8575}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{993\,x}{17150}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{1962\,\sqrt{35}}{300125}{\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{13}{280} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\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 = 1.50509, size = 173, normalized size = 1.66 \begin{align*} \frac{1962}{300125} \, \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{993 \, x}{17150 \, \sqrt{3 \, x^{2} + 2}} + \frac{981}{8575 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{70 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{103}{490 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57375, size = 332, normalized size = 3.19 \begin{align*} \frac{1962 \, \sqrt{35}{\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\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 (3972 \, x^{3} + 4068 \, x^{2} - 7397 \, x - 3658\right )} \sqrt{3 \, x^{2} + 2}}{600250 \,{\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\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 = 1.27443, size = 269, normalized size = 2.59 \begin{align*} \frac{1962}{300125} \, \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{3 \,{\left (157 \, x - 1478\right )}}{85750 \, \sqrt{3 \, x^{2} + 2}} - \frac{768 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 2461 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 6168 \, \sqrt{3} x + 856 \, \sqrt{3} + 6168 \, \sqrt{3 \, x^{2} + 2}}{6125 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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