Optimal. Leaf size=94 \[ -\frac {2}{21} (2 x+3)^{7/2}+\frac {62}{45} (2 x+3)^{5/2}+\frac {526}{81} (2 x+3)^{3/2}+\frac {3278}{81} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {4250}{81} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {824, 826, 1166, 207} \[ -\frac {2}{21} (2 x+3)^{7/2}+\frac {62}{45} (2 x+3)^{5/2}+\frac {526}{81} (2 x+3)^{3/2}+\frac {3278}{81} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {4250}{81} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{7/2}}{2+5 x+3 x^2} \, dx &=-\frac {2}{21} (3+2 x)^{7/2}+\frac {1}{3} \int \frac {(3+2 x)^{5/2} (49+31 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {62}{45} (3+2 x)^{5/2}-\frac {2}{21} (3+2 x)^{7/2}+\frac {1}{9} \int \frac {(3+2 x)^{3/2} (317+263 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {526}{81} (3+2 x)^{3/2}+\frac {62}{45} (3+2 x)^{5/2}-\frac {2}{21} (3+2 x)^{7/2}+\frac {1}{27} \int \frac {\sqrt {3+2 x} (1801+1639 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {3278}{81} \sqrt {3+2 x}+\frac {526}{81} (3+2 x)^{3/2}+\frac {62}{45} (3+2 x)^{5/2}-\frac {2}{21} (3+2 x)^{7/2}+\frac {1}{81} \int \frac {9653+9167 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {3278}{81} \sqrt {3+2 x}+\frac {526}{81} (3+2 x)^{3/2}+\frac {62}{45} (3+2 x)^{5/2}-\frac {2}{21} (3+2 x)^{7/2}+\frac {2}{81} \operatorname {Subst}\left (\int \frac {-8195+9167 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {3278}{81} \sqrt {3+2 x}+\frac {526}{81} (3+2 x)^{3/2}+\frac {62}{45} (3+2 x)^{5/2}-\frac {2}{21} (3+2 x)^{7/2}-36 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {21250}{81} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {3278}{81} \sqrt {3+2 x}+\frac {526}{81} (3+2 x)^{3/2}+\frac {62}{45} (3+2 x)^{5/2}-\frac {2}{21} (3+2 x)^{7/2}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {4250}{81} \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 = 0.07, size = 70, normalized size = 0.74 \[ -\frac {8 \sqrt {2 x+3} \left (270 x^3-738 x^2-8639 x-24728\right )}{2835}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {4250}{81} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 86, normalized size = 0.91 \[ \frac {2125}{243} \, \sqrt {5} \sqrt {3} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) - \frac {8}{2835} \, {\left (270 \, x^{3} - 738 \, x^{2} - 8639 \, x - 24728\right )} \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 101, normalized size = 1.07 \[ -\frac {2}{21} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {62}{45} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {526}{81} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {2125}{243} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {3278}{81} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 80, normalized size = 0.85 \[ -\frac {4250 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{243}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right )-\frac {2 \left (2 x +3\right )^{\frac {7}{2}}}{21}+\frac {62 \left (2 x +3\right )^{\frac {5}{2}}}{45}+\frac {526 \left (2 x +3\right )^{\frac {3}{2}}}{81}+\frac {3278 \sqrt {2 x +3}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 97, normalized size = 1.03 \[ -\frac {2}{21} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {62}{45} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {526}{81} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {2125}{243} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {3278}{81} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 71, normalized size = 0.76 \[ \frac {3278\,\sqrt {2\,x+3}}{81}+\frac {526\,{\left (2\,x+3\right )}^{3/2}}{81}+\frac {62\,{\left (2\,x+3\right )}^{5/2}}{45}-\frac {2\,{\left (2\,x+3\right )}^{7/2}}{21}-\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,12{}\mathrm {i}+\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,4250{}\mathrm {i}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 113.48, size = 138, normalized size = 1.47 \[ - \frac {2 \left (2 x + 3\right )^{\frac {7}{2}}}{21} + \frac {62 \left (2 x + 3\right )^{\frac {5}{2}}}{45} + \frac {526 \left (2 x + 3\right )^{\frac {3}{2}}}{81} + \frac {3278 \sqrt {2 x + 3}}{81} + \frac {21250 \left (\begin {cases} - \frac {\sqrt {15} \operatorname {acoth}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 > \frac {5}{3} \\- \frac {\sqrt {15} \operatorname {atanh}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 < \frac {5}{3} \end {cases}\right )}{81} - 6 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 6 \log {\left (\sqrt {2 x + 3} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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