Optimal. Leaf size=175 \[ -\frac {2 \sqrt {2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 \sqrt {2 x+3} (2607 x+2152)}{25 \sqrt {3 x^2+5 x+2}}+\frac {916 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3 x^2+5 x+2}}-\frac {3476 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.11, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {822, 843, 718, 424, 419} \[ -\frac {2 \sqrt {2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 \sqrt {2 x+3} (2607 x+2152)}{25 \sqrt {3 x^2+5 x+2}}+\frac {916 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3 x^2+5 x+2}}-\frac {3476 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 822
Rule 843
Rubi steps
\begin {align*} \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {696+423 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {-6579-7821 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {5214}{25} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {1374}{5} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {\left (3476 \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 )}{25 \sqrt {2+5 x+3 x^2}}+\frac {\left (916 \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 )}{5 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {3476 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}}+\frac {916 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 196, normalized size = 1.12 \[ \frac {-\frac {6952 \left (3 x^2+5 x+2\right )}{\sqrt {2 x+3}}+\frac {2 \sqrt {2 x+3} \left (15642 x^3+38982 x^2+31713 x+8423\right )}{3 x^2+5 x+2}+\frac {728 (x+1) \sqrt {\frac {3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {x+1}{10 x+15}}}-\frac {3476 (x+1) \sqrt {\frac {3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {x+1}{10 x+15}}}}{25 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{54 \, x^{7} + 351 \, x^{6} + 963 \, x^{5} + 1447 \, x^{4} + 1287 \, x^{3} + 678 \, x^{2} + 196 \, x + 24}, x\right ) \]
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 \[ \int -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {2 \, x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 308, normalized size = 1.76 \[ \frac {2 \left (156420 x^{4}+624450 x^{3}+2607 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+828 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+901860 x^{2}+4345 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1380 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+559925 x +1738 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+552 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+126345\right ) \sqrt {3 x^{2}+5 x +2}}{125 \left (x +1\right )^{2} \left (3 x +2\right )^{2} \sqrt {2 x +3}} \]
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 \[ -\int \frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {2 \, x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x-5}{\sqrt {2\,x+3}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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 \[ - \int \frac {x}{9 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{9 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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