Optimal. Leaf size=256 \[ \frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {1521056 \sqrt {x} (2+3 x)}{76545 \sqrt {2+5 x+3 x^2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {1521056 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {2+5 x+3 x^2}}-\frac {211144 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {2+5 x+3 x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {832, 846, 853,
1203, 1114, 1150} \begin {gather*} -\frac {211144 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {3 x^2+5 x+2}}+\frac {1521056 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {3 x^2+5 x+2}}+\frac {211144 \sqrt {3 x^2+5 x+2} \sqrt {x}}{5103}-\frac {1521056 (3 x+2) \sqrt {x}}{76545 \sqrt {3 x^2+5 x+2}}+\frac {2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {4 (1685 x+1484) x^{7/2}}{27 \sqrt {3 x^2+5 x+2}}+\frac {45820}{567} \sqrt {3 x^2+5 x+2} x^{5/2}-\frac {167336 \sqrt {3 x^2+5 x+2} x^{3/2}}{2835} \end {gather*}
Antiderivative was successfully verified.
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Rule 832
Rule 846
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(-407-340 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {x^{5/2} \left (5194+\frac {11455 x}{2}\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {8}{567} \int \frac {\left (-\frac {57275}{2}-\frac {62751 x}{2}\right ) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {16 \int \frac {\sqrt {x} \left (\frac {188253}{2}+\frac {395895 x}{4}\right )}{\sqrt {2+5 x+3 x^2}} \, dx}{8505}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {32 \int \frac {-\frac {395895}{4}-\frac {142599 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{76545}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {64 \text {Subst}\left (\int \frac {-\frac {395895}{4}-\frac {142599 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{76545}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {1521056 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{25515}-\frac {422288 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{5103}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {1521056 \sqrt {x} (2+3 x)}{76545 \sqrt {2+5 x+3 x^2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {1521056 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {2+5 x+3 x^2}}-\frac {211144 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.22, size = 187, normalized size = 0.73 \begin {gather*} \frac {-2 \left (3042112+8876240 x+5504080 x^2-2967300 x^3-2106756 x^4+262710 x^5-70956 x^6+18225 x^7\right )-1521056 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-1646104 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{76545 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 312, normalized size = 1.22
method | result | size |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {22460}{19683}-\frac {32666 x}{19683}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (-\frac {100085}{6561}-\frac {33460 x}{2187}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {10 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}+\frac {1084 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{2835}-\frac {9286 \sqrt {3 x^{3}+5 x^{2}+2 x}}{5103}-\frac {211144 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{15309 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {760528 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{76545 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(272\) |
default | \(-\frac {2 \left (1328364 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+1140792 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+54675 x^{7}+2213940 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +1901320 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -212868 x^{6}+885576 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+760528 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+788130 x^{5}-26854524 x^{4}-77349420 x^{3}-67906368 x^{2}-19002960 x \right ) \sqrt {3 x^{2}+5 x +2}}{229635 \sqrt {x}\, \left (2+3 x \right )^{2} \left (x +1\right )^{2}}\) | \(312\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.88, size = 137, normalized size = 0.54 \begin {gather*} -\frac {2 \, {\left (5698840 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 6844752 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 27 \, {\left (6075 \, x^{6} - 23652 \, x^{5} + 87570 \, x^{4} - 2983836 \, x^{3} - 8594380 \, x^{2} - 7545152 \, x - 2111440\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{688905 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {x^{13/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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