Optimal. Leaf size=205 \[ \frac {2360 \sqrt {x} (2+3 x)}{5103 \sqrt {2+5 x+3 x^2}}-\frac {4 \sqrt {x} (779+1035 x) \sqrt {2+5 x+3 x^2}}{1701}+\frac {136}{189} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2360 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {2+5 x+3 x^2}}+\frac {668 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1701 \sqrt {2+5 x+3 x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {846, 828, 853,
1203, 1114, 1150} \begin {gather*} \frac {668 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1701 \sqrt {3 x^2+5 x+2}}-\frac {2360 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {3 x^2+5 x+2}}+\frac {136}{189} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}-\frac {4 \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}}{1701}+\frac {2360 \sqrt {x} (3 x+2)}{5103 \sqrt {3 x^2+5 x+2}}-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 828
Rule 846
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps
\begin {align*} \int (2-5 x) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx &=-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2}{27} \int \sqrt {x} (15+102 x) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {136}{189} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {4}{567} \int \frac {\left (-102-\frac {1725 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx\\ &=-\frac {4 \sqrt {x} (779+1035 x) \sqrt {2+5 x+3 x^2}}{1701}+\frac {136}{189} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {8 \int \frac {-\frac {2505}{2}-\frac {4425 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{25515}\\ &=-\frac {4 \sqrt {x} (779+1035 x) \sqrt {2+5 x+3 x^2}}{1701}+\frac {136}{189} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {16 \text {Subst}\left (\int \frac {-\frac {2505}{2}-\frac {4425 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{25515}\\ &=-\frac {4 \sqrt {x} (779+1035 x) \sqrt {2+5 x+3 x^2}}{1701}+\frac {136}{189} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {1336 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{1701}+\frac {2360 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{1701}\\ &=\frac {2360 \sqrt {x} (2+3 x)}{5103 \sqrt {2+5 x+3 x^2}}-\frac {4 \sqrt {x} (779+1035 x) \sqrt {2+5 x+3 x^2}}{1701}+\frac {136}{189} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{27} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2360 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {2+5 x+3 x^2}}+\frac {668 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1701 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.15, size = 165, normalized size = 0.80 \begin {gather*} \frac {4720+7792 x+1380 x^2+7920 x^3+2970 x^4-23652 x^5-17010 x^6+2360 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-356 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{5103 \sqrt {x} \sqrt {2+5 x+3 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 127, normalized size = 0.62
method | result | size |
default | \(-\frac {2 \left (25515 x^{6}+35478 x^{5}+768 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-590 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-4455 x^{4}-11880 x^{3}+8550 x^{2}+6012 x \right )}{15309 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(127\) |
risch | \(-\frac {2 \left (945 x^{3}-261 x^{2}-360 x +334\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{1701}-\frac {\left (-\frac {668 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {1180 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(193\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{9}+\frac {58 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}+\frac {80 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}-\frac {668 \sqrt {3 x^{3}+5 x^{2}+2 x}}{1701}+\frac {668 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {1180 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(238\) |
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 = 0.71, size = 58, normalized size = 0.28 \begin {gather*} -\frac {2}{1701} \, {\left (945 \, x^{3} - 261 \, x^{2} - 360 \, x + 334\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} + \frac {32}{6561} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {2360}{5103} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \end {gather*}
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 \begin {gather*} - \int \left (- 2 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 5 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \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 x^{3/2}\,\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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