Optimal. Leaf size=100 \[ -\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {7 (619 x+546) \sqrt {2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1582 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1225 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {818, 826, 1166, 207} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {7 (619 x+546) \sqrt {2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1582 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1225 \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 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {1}{6} \int \frac {(-658-147 x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {7 \sqrt {3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}+\frac {1}{18} \int \frac {26649+12411 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {7 \sqrt {3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {16065+12411 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {7 \sqrt {3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}-4746 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+6125 \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {7 \sqrt {3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}+1582 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-1225 \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 = 80, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2 x+3} \left (12443 x^3+30979 x^2+25073 x+6555\right )}{6 \left (3 x^2+5 x+2\right )^2}+1582 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1225 \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 = 0.28, size = 100, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2 x+3} \left (12443 (2 x+3)^3-50029 (2 x+3)^2+64505 (2 x+3)-26775\right )}{3 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+1582 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1225 \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 = 169, normalized size = 1.69 \begin {gather*} \frac {1225 \, \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 ) + 4746 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 4746 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + {\left (12443 \, x^{3} + 30979 \, x^{2} + 25073 \, x + 6555\right )} \sqrt {2 \, x + 3}}{6 \, {\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 = 1.20 \begin {gather*} \frac {1225}{6} \, \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 {12443 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 50029 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 64505 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 26775 \, \sqrt {2 \, x + 3}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 791 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 791 \, \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 = 1.24 \begin {gather*} -\frac {1225 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}-791 \ln \left (-1+\sqrt {2 x +3}\right )+791 \ln \left (\sqrt {2 x +3}+1\right )+\frac {\frac {7475 \left (2 x +3\right )^{\frac {3}{2}}}{3}-4625 \sqrt {2 x +3}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {92}{\sqrt {2 x +3}+1}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {92}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 134, normalized size = 1.34 \begin {gather*} \frac {1225}{6} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {12443 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 50029 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 64505 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 26775 \, \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 )}} + 791 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 791 \, \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 = 2.42, size = 101, normalized size = 1.01 \begin {gather*} 1582\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {\frac {2975\,\sqrt {2\,x+3}}{3}-\frac {64505\,{\left (2\,x+3\right )}^{3/2}}{27}+\frac {50029\,{\left (2\,x+3\right )}^{5/2}}{27}-\frac {12443\,{\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 {1225\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{3} \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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