Optimal. Leaf size=234 \[ -\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}+\frac {\sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}}{19305}-\frac {\sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}}{162162}+\frac {(287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{14594580}+\frac {5021353 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5837832 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2742319 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{4169880 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.16, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}+\frac {\sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}}{19305}-\frac {\sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}}{162162}+\frac {(287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{14594580}+\frac {5021353 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5837832 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2742319 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{4169880 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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 )^{5/2} \, dx &=-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}+\frac {2}{45} \int \frac {\left (392+\frac {521 x}{2}\right ) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx\\ &=\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\int \frac {\left (\frac {16573}{2}+\frac {14847 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{3861}\\ &=-\frac {\sqrt {3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}+\frac {\int \frac {\left (-356538-\frac {889179 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{486486}\\ &=\frac {(287729-2667537 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{14594580}-\frac {\sqrt {3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\int \frac {\frac {48722901}{2}+\frac {57588699 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{43783740}\\ &=\frac {(287729-2667537 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{14594580}-\frac {\sqrt {3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2742319 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{8339760}+\frac {5021353 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{11675664}\\ &=\frac {(287729-2667537 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{14594580}-\frac {\sqrt {3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\left (2742319 \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 )}{4169880 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (5021353 \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 )}{5837832 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {(287729-2667537 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{14594580}-\frac {\sqrt {3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2742319 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{4169880 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {5021353 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5837832 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 218, normalized size = 0.93 \[ -\frac {2 \left (315242928 x^9+468822816 x^8-6333945660 x^7-30512259036 x^6-63978029658 x^5-76896556902 x^4-56607962679 x^3-25296672765 x^2-6298405666 x-666434848\right ) \sqrt {2 x+3}-4132174 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+19196233 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{87567480 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]
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 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {2 \, x + 3} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 166, normalized size = 0.71 \[ \frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (-6304858560 x^{9}-9376456320 x^{8}+126678913200 x^{7}+610245180720 x^{6}+1279560593160 x^{5}+1537931138040 x^{4}+1132159253580 x^{3}+507085229280 x^{2}+127887736620 x +19196233 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+5910532 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+14096546280\right )}{5254048800 x^{3}+16637821200 x^{2}+16637821200 x +5254048800} \]
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 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {2 \, x + 3} {\left (x - 5\right )}\,{d x} \]
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 \[ -\int \sqrt {2\,x+3}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]
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 \[ - \int \left (- 20 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 96 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 165 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 113 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 9 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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