Optimal. Leaf size=95 \[ \frac {-10+97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}-\frac {16 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {32 \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{1331 \sqrt {11}} \]
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Rubi [A]
time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 = {1661, 821, 739,
212} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 821
Rule 1661
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 \text {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*}
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Mathematica [A]
time = 0.72, size = 91, normalized size = 0.96 \begin {gather*} \frac {11 \left (-446+2717 x+4602 x^2+2805 x^3+4458 x^4\right )-192 \sqrt {22+33 x^2} \left (2+4 x+3 x^2+6 x^3\right ) \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{87846 (1+2 x) \left (2+3 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 143, normalized size = 1.51
method | result | size |
risch | \(\frac {4458 x^{4}+2805 x^{3}+4602 x^{2}+2717 x -446}{7986 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (2 x +1\right )}-\frac {32 \sqrt {11}\, \arctanh \left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{14641}\) | \(70\) |
trager | \(\frac {4458 x^{4}+2805 x^{3}+4602 x^{2}+2717 x -446}{7986 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (2 x +1\right )}+\frac {32 \RootOf \left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \RootOf \left (\textit {\_Z}^{2}-11\right )}{2 x +1}\right )}{14641}\) | \(86\) |
default | \(\frac {x}{6 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {x}{6 \sqrt {3 x^{2}+2}}+\frac {4}{363 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {10 x}{121 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {98 x}{1331 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}+\frac {16}{1331 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}-\frac {32 \sqrt {11}\, \arctanh \left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{14641}-\frac {1}{22 \left (x +\frac {1}{2}\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 107, normalized size = 1.13 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 134, normalized size = 1.41 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs.
\(2 (76) = 152\).
time = 5.20, size = 233, normalized size = 2.45 \begin {gather*} -\frac {1}{263538} \, \sqrt {11} {\left (743 \, \sqrt {11} \sqrt {3} - 576 \, \log \left (\sqrt {11} \sqrt {3} - 3\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {32 \, \sqrt {11} \log \left (\sqrt {11} {\left (\sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {11}}{2 \, x + 1}\right )} - 3\right )}{14641 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {\frac {\frac {\frac {11 \, {\left (\frac {731}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {528}{{\left (2 \, x + 1\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}\right )}}{2 \, x + 1} - \frac {14163}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} + \frac {6111}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} - \frac {2229}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{7986 \, {\left (\frac {6}{2 \, x + 1} - \frac {11}{{\left (2 \, x + 1\right )}^{2}} - 3\right )} \sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.31, size = 270, normalized size = 2.84 \begin {gather*} \frac {\sqrt {11}\,\left (8\,\ln \left (x+\frac {1}{2}\right )-8\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{14641}+\frac {\sqrt {11}\,\left (\frac {48\,\ln \left (x+\frac {1}{2}\right )}{1331}-\frac {48\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{1331}\right )}{22}-\frac {8\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x+\frac {1}{2}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {291}{1936}+\frac {\sqrt {6}\,15{}\mathrm {i}}{1936}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {97}{968}+\frac {\sqrt {6}\,5{}\mathrm {i}}{968}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {291}{1936}+\frac {\sqrt {6}\,15{}\mathrm {i}}{1936}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {97}{968}+\frac {\sqrt {6}\,5{}\mathrm {i}}{968}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-288+\sqrt {6}\,2481{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1149984\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (288+\sqrt {6}\,2481{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1149984\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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