Optimal. Leaf size=202 \[ -\frac {2}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac {1}{99} \sqrt {2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac {\sqrt {2 x+3} (3987 x+1246) \sqrt {3 x^2+5 x+2}}{8910}+\frac {4153 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{3564 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {15283 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{17820 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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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 = {832, 814, 843, 718, 424, 419} \[ -\frac {2}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac {1}{99} \sqrt {2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac {\sqrt {2 x+3} (3987 x+1246) \sqrt {3 x^2+5 x+2}}{8910}+\frac {4153 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{3564 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {15283 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{17820 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 814
Rule 832
Rule 843
Rubi steps
\begin {align*} \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {2}{33} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{33} \int \frac {\left (287+\frac {381 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx\\ &=\frac {1}{99} \sqrt {3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{33} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\int \frac {\left (\frac {9807}{2}+\frac {9303 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{2079}\\ &=-\frac {\sqrt {3+2 x} (1246+3987 x) \sqrt {2+5 x+3 x^2}}{8910}+\frac {1}{99} \sqrt {3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{33} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {\int \frac {-131691-\frac {320943 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{187110}\\ &=-\frac {\sqrt {3+2 x} (1246+3987 x) \sqrt {2+5 x+3 x^2}}{8910}+\frac {1}{99} \sqrt {3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{33} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {15283 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{35640}+\frac {4153 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{7128}\\ &=-\frac {\sqrt {3+2 x} (1246+3987 x) \sqrt {2+5 x+3 x^2}}{8910}+\frac {1}{99} \sqrt {3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{33} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\left (15283 \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 )}{17820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (4153 \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 )}{3564 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {\sqrt {3+2 x} (1246+3987 x) \sqrt {2+5 x+3 x^2}}{8910}+\frac {1}{99} \sqrt {3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{33} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {15283 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{17820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {4153 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{3564 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 208, normalized size = 1.03 \[ -\frac {2 \left (87480 x^7-48600 x^6-2001510 x^5-6002964 x^4-8112483 x^3-5740860 x^2-2059597 x-293686\right ) \sqrt {2 x+3}-2824 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+15283 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{53460 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]
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 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} \sqrt {2 \, x + 3} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 156, normalized size = 0.77 \[ \frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (-1749600 x^{7}+972000 x^{6}+40030200 x^{5}+120059280 x^{4}+162249660 x^{3}+115734180 x^{2}+42720240 x +15283 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+5482 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+6485040\right )}{3207600 x^{3}+10157400 x^{2}+10157400 x +3207600} \]
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 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} \sqrt {2 \, x + 3} {\left (x - 5\right )}\,{d x} \]
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 \[ -\int \sqrt {2\,x+3}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]
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 \[ - \int \left (- 10 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 23 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 3 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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