Optimal. Leaf size=187 \[ \frac {1450 \sqrt {x} (3 x+2)}{9 \sqrt {3 x^2+5 x+2}}-\frac {2 \sqrt {x} (2175 x+1831)}{9 \sqrt {3 x^2+5 x+2}}+\frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {598 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {1450 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.11, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {818, 822, 839, 1189, 1100, 1136} \[ \frac {1450 \sqrt {x} (3 x+2)}{9 \sqrt {3 x^2+5 x+2}}-\frac {2 \sqrt {x} (2175 x+1831)}{9 \sqrt {3 x^2+5 x+2}}+\frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {598 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {1450 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 822
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac {2 \sqrt {x} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {-37+135 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2 \sqrt {x} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2 \sqrt {x} (1831+2175 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {2}{9} \int \frac {-897-\frac {2175 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 \sqrt {x} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2 \sqrt {x} (1831+2175 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {4}{9} \operatorname {Subst}\left (\int \frac {-897-\frac {2175 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2 \sqrt {x} (1831+2175 x)}{9 \sqrt {2+5 x+3 x^2}}+\frac {1196}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {1450}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {1450 \sqrt {x} (2+3 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {2 \sqrt {x} (1831+2175 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {1450 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}}+\frac {598 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.25, size = 164, normalized size = 0.88 \[ \frac {10764 x^3+26830 x^2+344 i \sqrt {\frac {2}{x}+2} \sqrt {\frac {2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+1450 i \sqrt {\frac {2}{x}+2} \sqrt {\frac {2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+21824 x+5800}{9 \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x^{2} - 2 \, x\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]
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 \[ \int -\frac {{\left (5 \, x - 2\right )} x^{\frac {3}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 297, normalized size = 1.59 \[ -\frac {\left (39150 x^{4}+98208 x^{3}-2175 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+1143 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+80460 x^{2}-3625 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+1905 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+21528 x -1450 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+762 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right ) \sqrt {3 x^{2}+5 x +2}}{27 \left (x +1\right )^{2} \left (3 x +2\right )^{2} \sqrt {x}} \]
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 \[ -\int \frac {{\left (5 \, x - 2\right )} x^{\frac {3}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ -\int \frac {x^{3/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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