Optimal. Leaf size=115 \[ -\frac {(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {(12473 x+10832) (2 x+3)^{3/2}}{18 \left (3 x^2+5 x+2\right )}-\frac {3983}{9} \sqrt {2 x+3}+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {818, 824, 826, 1166, 207} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {(12473 x+10832) (2 x+3)^{3/2}}{18 \left (3 x^2+5 x+2\right )}-\frac {3983}{9} \sqrt {2 x+3}+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 818
Rule 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {1}{6} \int \frac {(3+2 x)^{5/2} (-416+131 x)}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac {1}{18} \int \frac {(5709-11949 x) \sqrt {3+2 x}}{2+5 x+3 x^2} \, dx\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac {1}{54} \int \frac {99177+46203 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {59745+46203 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}-5886 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {68375}{9} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {3983}{9} \sqrt {3+2 x}-\frac {(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+1962 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 86, normalized size = 0.75 \begin {gather*} \frac {1}{54} \left (-\frac {3 \sqrt {2 x+3} \left (192 x^4-45083 x^3-112467 x^2-90465 x-23327\right )}{\left (3 x^2+5 x+2\right )^2}-27350 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right )+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 122, normalized size = 1.06 \begin {gather*} \frac {-96 (2 x+3)^{9/2}+46235 (2 x+3)^{7/2}-185997 (2 x+3)^{5/2}+239865 (2 x+3)^{3/2}-99575 \sqrt {2 x+3}}{9 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+1962 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 175, normalized size = 1.52 \begin {gather*} \frac {13675 \, \sqrt {5} \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 52974 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 52974 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 3 \, {\left (192 \, x^{4} - 45083 \, x^{3} - 112467 \, x^{2} - 90465 \, x - 23327\right )} \sqrt {2 \, x + 3}}{54 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 129, normalized size = 1.12 \begin {gather*} \frac {13675}{54} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {32}{27} \, \sqrt {2 \, x + 3} + \frac {137169 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 554983 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 717035 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 297925 \, \sqrt {2 \, x + 3}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 981 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 981 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 133, normalized size = 1.16 \begin {gather*} -\frac {13675 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{27}-981 \ln \left (-1+\sqrt {2 x +3}\right )+981 \ln \left (\sqrt {2 x +3}+1\right )-\frac {32 \sqrt {2 x +3}}{27}+\frac {\frac {9625 \left (2 x +3\right )^{\frac {3}{2}}}{3}-\frac {165625 \sqrt {2 x +3}}{27}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {104}{\sqrt {2 x +3}+1}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {104}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 143, normalized size = 1.24 \begin {gather*} \frac {13675}{54} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {32}{27} \, \sqrt {2 \, x + 3} + \frac {137169 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 554983 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 717035 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 297925 \, \sqrt {2 \, x + 3}}{27 \, {\left (9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 981 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 981 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 116, normalized size = 1.01 \begin {gather*} \frac {\frac {297925\,\sqrt {2\,x+3}}{243}-\frac {717035\,{\left (2\,x+3\right )}^{3/2}}{243}+\frac {554983\,{\left (2\,x+3\right )}^{5/2}}{243}-\frac {15241\,{\left (2\,x+3\right )}^{7/2}}{27}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {32\,\sqrt {2\,x+3}}{27}-\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,1962{}\mathrm {i}+\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,13675{}\mathrm {i}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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