Optimal. Leaf size=233 \[ -\frac {16040}{243} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {33608 (3 x+2) \sqrt {x}}{729 \sqrt {3 x^2+5 x+2}}+\frac {16040 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {3 x^2+5 x+2}}-\frac {33608 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729 \sqrt {3 x^2+5 x+2}}+\frac {2 (95 x+74) x^{9/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {8 (905 x+773) x^{5/2}}{27 \sqrt {3 x^2+5 x+2}}+\frac {2348}{27} \sqrt {3 x^2+5 x+2} x^{3/2} \]
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Rubi [A] time = 0.16, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {818, 832, 839, 1189, 1100, 1136} \[ \frac {2 (95 x+74) x^{9/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {8 (905 x+773) x^{5/2}}{27 \sqrt {3 x^2+5 x+2}}+\frac {2348}{27} \sqrt {3 x^2+5 x+2} x^{3/2}-\frac {16040}{243} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {33608 (3 x+2) \sqrt {x}}{729 \sqrt {3 x^2+5 x+2}}+\frac {16040 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {3 x^2+5 x+2}}-\frac {33608 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 832
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(-333-245 x) x^{7/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {x^{3/2} \left (3865+\frac {8805 x}{2}\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {8}{405} \int \frac {\left (-\frac {26415}{2}-\frac {30075 x}{2}\right ) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {16 \int \frac {\frac {30075}{2}+\frac {63015 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{3645}\\ &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {32 \operatorname {Subst}\left (\int \frac {\frac {30075}{2}+\frac {63015 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{3645}\\ &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {32080}{243} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {33608}{243} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {33608 \sqrt {x} (2+3 x)}{729 \sqrt {2+5 x+3 x^2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {33608 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729 \sqrt {2+5 x+3 x^2}}+\frac {16040 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 179, normalized size = 0.77 \[ \frac {-486 x^6+2484 x^5-21276 x^4+161784 x^3+534680 x^2+14512 i \sqrt {\frac {2}{x}+2} \sqrt {\frac {2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+33608 i \sqrt {\frac {2}{x}+2} \sqrt {\frac {2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+479680 x+134432}{729 \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x^{6} - 2 \, x^{5}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]
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 \[ \int -\frac {{\left (5 \, x - 2\right )} x^{\frac {11}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 307, normalized size = 1.32 \[ -\frac {2 \left (729 x^{6}-3726 x^{5}+485622 x^{4}+1269684 x^{3}-25206 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+3438 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+1063224 x^{2}-42010 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+5730 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+288720 x -16804 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+2292 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right ) \sqrt {3 x^{2}+5 x +2}}{2187 \left (x +1\right )^{2} \left (3 x +2\right )^{2} \sqrt {x}} \]
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 \[ -\int \frac {{\left (5 \, x - 2\right )} x^{\frac {11}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {x^{11/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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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