Optimal. Leaf size=202 \[ -\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}+\frac {23464 \sqrt {3 x^2+5 x+2}}{125 \sqrt {2 x+3}}+\frac {4 (2409 x+2054)}{25 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}+\frac {3212 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3 x^2+5 x+2}}-\frac {11732 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{125 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {822, 834, 843, 718, 424, 419} \[ -\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}+\frac {23464 \sqrt {3 x^2+5 x+2}}{125 \sqrt {2 x+3}}+\frac {4 (2409 x+2054)}{25 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}+\frac {3212 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3 x^2+5 x+2}}-\frac {11732 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{125 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 822
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {918+705 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {6441+7227 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {8}{375} \int \frac {10764+\frac {26397 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {17598}{125} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {4818}{25} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {\left (11732 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{125 \sqrt {2+5 x+3 x^2}}+\frac {\left (3212 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{25 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {11732 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{125 \sqrt {2+5 x+3 x^2}}+\frac {3212 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 215, normalized size = 1.06 \[ \frac {2 \left (5 \left (14454 x^3+36414 x^2+29941 x+8031\right )+1048 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {2 x+3} \sqrt {\frac {3 x+2}{2 x+3}} \left (6 x^3+19 x^2+19 x+6\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )-5866 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {2 x+3} \sqrt {\frac {3 x+2}{2 x+3}} \left (6 x^3+19 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )}{125 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{108 \, x^{8} + 864 \, x^{7} + 2979 \, x^{6} + 5783 \, x^{5} + 6915 \, x^{4} + 5217 \, x^{3} + 2426 \, x^{2} + 636 \, x + 72}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 308, normalized size = 1.52 \[ \frac {2 \sqrt {3 x^{2}+5 x +2}\, \left (527940 x^{4}+2121150 x^{3}+8799 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3246 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3080770 x^{2}+14665 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+5410 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1921725 x +5866 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+2164 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+435415\right )}{625 \left (3 x +2\right )^{2} \left (x +1\right )^{2} \sqrt {2 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^{3/2}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{18 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{18 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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