Optimal. Leaf size=102 \[ -\frac {(139 x+121) (2 x+3)^{3/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {25 (131 x+112) \sqrt {2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1250 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {2905}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {818, 820, 826, 1166, 207} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{3/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {25 (131 x+112) \sqrt {2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1250 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {2905}{3} \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 820
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {1}{6} \int \frac {(-900-425 x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-\frac {1}{6} \int \frac {-7025-3275 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-4225-3275 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-3750 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {14525}{3} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}+1250 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {2905}{3} \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 = 0.08, size = 82, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2 x+3} \left (9825 x^3+24497 x^2+19891 x+5237\right )}{6 \left (3 x^2+5 x+2\right )^2}+1250 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {2905}{3} \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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IntegrateAlgebraic [A] time = 0.29, size = 102, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2 x+3} \left (9825 (2 x+3)^3-39431 (2 x+3)^2+50875 (2 x+3)-21125\right )}{3 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+1250 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {2905}{3} \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 [B] time = 0.40, size = 170, normalized size = 1.67 \begin {gather*} \frac {2905 \, \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 ) + 11250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 11250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 3 \, {\left (9825 \, x^{3} + 24497 \, x^{2} + 19891 \, x + 5237\right )} \sqrt {2 \, x + 3}}{18 \, {\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 = 0.19, size = 120, normalized size = 1.18 \begin {gather*} \frac {2905}{18} \, \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 {9825 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 39431 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 50875 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 21125 \, \sqrt {2 \, x + 3}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 625 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 625 \, \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 = 124, normalized size = 1.22 \begin {gather*} -\frac {2905 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{9}-625 \ln \left (-1+\sqrt {2 x +3}\right )+625 \ln \left (\sqrt {2 x +3}+1\right )+\frac {1835 \left (2 x +3\right )^{\frac {3}{2}}-\frac {10025 \sqrt {2 x +3}}{3}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {80}{\sqrt {2 x +3}+1}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {80}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 134, normalized size = 1.31 \begin {gather*} \frac {2905}{18} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {9825 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 39431 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 50875 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 21125 \, \sqrt {2 \, x + 3}}{3 \, {\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 )}} + 625 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 625 \, \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 = 0.07, size = 101, normalized size = 0.99 \begin {gather*} 1250\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {\frac {21125\,\sqrt {2\,x+3}}{27}-\frac {50875\,{\left (2\,x+3\right )}^{3/2}}{27}+\frac {39431\,{\left (2\,x+3\right )}^{5/2}}{27}-\frac {3275\,{\left (2\,x+3\right )}^{7/2}}{9}}{\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 {2905\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{9} \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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