Optimal. Leaf size=121 \[ -\frac{991 \sqrt{3 x^2+2}}{171500 (2 x+3)}-\frac{87 \sqrt{3 x^2+2}}{4900 (2 x+3)^2}-\frac{97 \sqrt{3 x^2+2}}{2100 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+2}}{140 (2 x+3)^4}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]
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Rubi [A] time = 0.075237, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {835, 807, 725, 206} \[ -\frac{991 \sqrt{3 x^2+2}}{171500 (2 x+3)}-\frac{87 \sqrt{3 x^2+2}}{4900 (2 x+3)^2}-\frac{97 \sqrt{3 x^2+2}}{2100 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+2}}{140 (2 x+3)^4}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]
Antiderivative was successfully verified.
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Rule 835
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^5 \sqrt{2+3 x^2}} \, dx &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{1}{140} \int \frac{-164+117 x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}+\frac{\int \frac{3024-4074 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx}{14700}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{\int \frac{-21840+54810 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{1029000}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{991 \sqrt{2+3 x^2}}{171500 (3+2 x)}-\frac{27 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{42875}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{991 \sqrt{2+3 x^2}}{171500 (3+2 x)}+\frac{27 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{42875}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{991 \sqrt{2+3 x^2}}{171500 (3+2 x)}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{42875 \sqrt{35}}\\ \end{align*}
Mathematica [A] time = 0.0813856, size = 70, normalized size = 0.58 \[ \frac{81 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{35 \sqrt{3 x^2+2} \left (5946 x^3+35892 x^2+79423 x+70389\right )}{(2 x+3)^4}}{4501875} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 116, normalized size = 1. \begin{align*} -{\frac{97}{16800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{87}{19600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{991}{343000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{27\,\sqrt{35}}{1500625}{\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}{2240}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61529, size = 185, normalized size = 1.53 \begin{align*} -\frac{27}{1500625} \, \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{13 \, \sqrt{3 \, x^{2} + 2}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{97 \, \sqrt{3 \, x^{2} + 2}}{2100 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{87 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{991 \, \sqrt{3 \, x^{2} + 2}}{171500 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77651, size = 339, normalized size = 2.8 \begin{align*} \frac{81 \, \sqrt{35}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) - 70 \,{\left (5946 \, x^{3} + 35892 \, x^{2} + 79423 \, x + 70389\right )} \sqrt{3 \, x^{2} + 2}}{9003750 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 2}{\left (2 \, x + 3\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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