Optimal. Leaf size=73 \[ \frac{41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac{2137 x+312}{7350 \sqrt{3 x^2+2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0425624, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {823, 12, 725, 206} \[ \frac{41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac{2137 x+312}{7350 \sqrt{3 x^2+2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 823
Rule 12
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}-\frac{1}{630} \int \frac{-1206-492 x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}+\frac{\int \frac{11232}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{132300}\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}+\frac{104 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1225}\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}-\frac{104 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1225}\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1225 \sqrt{35}}\\ \end{align*}
Mathematica [A] time = 0.0398818, size = 63, normalized size = 0.86 \[ \frac{\frac{35 \left (6411 x^3+936 x^2+5709 x+1534\right )}{\left (3 x^2+2\right )^{3/2}}-624 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257250} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 122, normalized size = 1.7 \begin{align*} -{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{12}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{13}{105} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{39\,x}{140} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{1833\,x}{4900}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{52}{1225}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{104\,\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.50479, size = 109, normalized size = 1.49 \begin{align*} \frac{104}{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{2137 \, x}{7350 \, \sqrt{3 \, x^{2} + 2}} + \frac{52}{1225 \, \sqrt{3 \, x^{2} + 2}} + \frac{41 \, x}{210 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{13}{105 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5448, size = 281, normalized size = 3.85 \begin{align*} \frac{312 \, \sqrt{35}{\left (9 \, x^{4} + 12 \, x^{2} + 4\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 (6411 \, x^{3} + 936 \, x^{2} + 5709 \, x + 1534\right )} \sqrt{3 \, x^{2} + 2}}{257250 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25445, size = 126, normalized size = 1.73 \begin{align*} \frac{104}{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{3 \,{\left ({\left (2137 \, x + 312\right )} x + 1903\right )} x + 1534}{7350 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]