Optimal. Leaf size=95 \[ -\frac{10-97 x}{726 \left (3 x^2+2\right )^{3/2}}-\frac{16 \sqrt{3 x^2+2}}{1331 (2 x+1)}+\frac{887 x+24}{7986 \sqrt{3 x^2+2}}-\frac{32 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{1331 \sqrt{11}} \]
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Rubi [A] time = 0.169063, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1647, 807, 725, 206} \[ -\frac{10-97 x}{726 \left (3 x^2+2\right )^{3/2}}-\frac{16 \sqrt{3 x^2+2}}{1331 (2 x+1)}+\frac{887 x+24}{7986 \sqrt{3 x^2+2}}-\frac{32 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{1331 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 1647
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx &=-\frac{10-97 x}{726 \left (2+3 x^2\right )^{3/2}}-\frac{1}{18} \int \frac{-\frac{798}{121}-\frac{1968 x}{121}-\frac{2328 x^2}{121}}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac{24+887 x}{7986 \sqrt{2+3 x^2}}+\frac{1}{108} \int \frac{\frac{10368}{1331}+\frac{1728 x}{1331}}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac{24+887 x}{7986 \sqrt{2+3 x^2}}-\frac{16 \sqrt{2+3 x^2}}{1331 (1+2 x)}+\frac{32 \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx}{1331}\\ &=-\frac{10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac{24+887 x}{7986 \sqrt{2+3 x^2}}-\frac{16 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{32 \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )}{1331}\\ &=-\frac{10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac{24+887 x}{7986 \sqrt{2+3 x^2}}-\frac{16 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{32 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{1331 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.0632395, size = 91, normalized size = 0.96 \[ \frac{11 \left (4458 x^4+2805 x^3+4602 x^2+2717 x-446\right )-192 \sqrt{33 x^2+22} \left (6 x^3+3 x^2+4 x+2\right ) \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{87846 (2 x+1) \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 143, normalized size = 1.5 \begin{align*}{\frac{x}{6} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{1}{22} \left ( x+{\frac{1}{2}} \right ) ^{-1} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{4}{363} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{10\,x}{121} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{98\,x}{1331}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{16}{1331}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{32\,\sqrt{11}}{14641}{\it Artanh} \left ({\frac{ \left ( 8-6\,x \right ) \sqrt{11}}{11}{\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-12\,x+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50504, size = 144, normalized size = 1.52 \begin{align*} \frac{32}{14641} \, \sqrt{11} \operatorname{arsinh}\left (\frac{\sqrt{6} x}{2 \,{\left | 2 \, x + 1 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{743 \, x}{7986 \, \sqrt{3 \, x^{2} + 2}} + \frac{16}{1331 \, \sqrt{3 \, x^{2} + 2}} + \frac{61 \, x}{726 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{1}{11 \,{\left (2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x +{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} + \frac{4}{363 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65366, size = 356, normalized size = 3.75 \begin{align*} \frac{96 \, \sqrt{11}{\left (18 \, x^{5} + 9 \, x^{4} + 24 \, x^{3} + 12 \, x^{2} + 8 \, x + 4\right )} \log \left (-\frac{\sqrt{11} \sqrt{3 \, x^{2} + 2}{\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \,{\left (4458 \, x^{4} + 2805 \, x^{3} + 4602 \, x^{2} + 2717 \, x - 446\right )} \sqrt{3 \, x^{2} + 2}}{87846 \,{\left (18 \, x^{5} + 9 \, x^{4} + 24 \, x^{3} + 12 \, x^{2} + 8 \, x + 4\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{4 \, x^{2} + 3 \, x + 1}{{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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