Optimal. Leaf size=173 \[ -\frac {2 \sqrt {2 x+3} (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 \sqrt {2 x+3} (2139 x+1759)}{15 \sqrt {3 x^2+5 x+2}}+\frac {748 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2852 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.10, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {820, 822, 843, 718, 424, 419} \[ -\frac {2 \sqrt {2 x+3} (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 \sqrt {2 x+3} (2139 x+1759)}{15 \sqrt {3 x^2+5 x+2}}+\frac {748 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2852 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 820
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} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{3} \int \frac {181+105 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (1759+2139 x)}{15 \sqrt {2+5 x+3 x^2}}+\frac {4}{15} \int \frac {-1806-2139 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (1759+2139 x)}{15 \sqrt {2+5 x+3 x^2}}-\frac {1426}{5} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+374 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (1759+2139 x)}{15 \sqrt {2+5 x+3 x^2}}-\frac {\left (2852 \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 )}{5 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (748 \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 )}{\sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 \sqrt {3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (1759+2139 x)}{15 \sqrt {2+5 x+3 x^2}}-\frac {2852 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {748 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 196, normalized size = 1.13 \[ -\frac {\frac {5704 \left (3 x^2+5 x+2\right )}{\sqrt {2 x+3}}-\frac {6 \sqrt {2 x+3} \left (4278 x^3+10648 x^2+8657 x+2297\right )}{3 x^2+5 x+2}-\frac {608 (x+1) \sqrt {\frac {3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {x+1}{10 x+15}}}+\frac {2852 (x+1) \sqrt {\frac {3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {x+1}{10 x+15}}}}{15 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]
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 \[ \int -\frac {\sqrt {2 \, x + 3} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 308, normalized size = 1.78 \[ \frac {2 \left (128340 x^{4}+511950 x^{3}+2139 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+666 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+738870 x^{2}+3565 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1110 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+458475 x +1426 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+444 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+103365\right ) \sqrt {3 x^{2}+5 x +2}}{75 \left (x +1\right )^{2} \left (3 x +2\right )^{2} \sqrt {2 x +3}} \]
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 \[ -\int \frac {\sqrt {2 \, x + 3} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {2\,x+3}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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 \[ - \int \left (- \frac {5 \sqrt {2 x + 3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {x \sqrt {2 x + 3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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