Optimal. Leaf size=170 \[ -\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac {1}{945} \sqrt {2 x+3} (2169 x+2327) \sqrt {3 x^2+5 x+2}+\frac {1039 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{378 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {697 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{270 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.12, antiderivative size = 170, 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 = {832, 814, 843, 718, 424, 419} \[ -\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac {1}{945} \sqrt {2 x+3} (2169 x+2327) \sqrt {3 x^2+5 x+2}+\frac {1039 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{378 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {697 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{270 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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} \sqrt {2+5 x+3 x^2} \, dx &=-\frac {2}{21} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2}{21} \int \frac {\left (182+\frac {241 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx\\ &=\frac {1}{945} \sqrt {3+2 x} (2327+2169 x) \sqrt {2+5 x+3 x^2}-\frac {2}{21} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{945} \int \frac {\frac {4721}{2}+\frac {4879 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{945} \sqrt {3+2 x} (2327+2169 x) \sqrt {2+5 x+3 x^2}-\frac {2}{21} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {697}{540} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {1039}{756} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{945} \sqrt {3+2 x} (2327+2169 x) \sqrt {2+5 x+3 x^2}-\frac {2}{21} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {\left (697 \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 )}{270 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (1039 \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 )}{378 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {1}{945} \sqrt {3+2 x} (2327+2169 x) \sqrt {2+5 x+3 x^2}-\frac {2}{21} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {697 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{270 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1039 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{378 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 198, normalized size = 1.16 \[ -\frac {2 \left (4860 x^5-15552 x^4-121239 x^3-200865 x^2-128926 x-28888\right ) \sqrt {2 x+3}-1762 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+4879 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{5670 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}, x\right ) \]
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 \[ \int -\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 146, normalized size = 0.86 \[ \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-97200 x^{5}+311040 x^{4}+2424780 x^{3}+4310040 x^{2}+3066420 x +4879 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+316 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+772920\right )}{340200 x^{3}+1077300 x^{2}+1077300 x +340200} \]
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 \[ -\int \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \sqrt {2\,x+3}\,\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2} \,d x \]
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 \[ - \int \left (- 5 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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