Optimal. Leaf size=183 \[ -\frac {34 \sqrt {x} (2+3 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2 (2-x) \sqrt {2+5 x+3 x^2}}{\sqrt {x}}-\frac {2 (2+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{3 x^{3/2}}+\frac {34 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}-\frac {14 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {826, 853, 1203,
1114, 1150} \begin {gather*} -\frac {14 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {34 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}+\frac {2 (2-x) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-\frac {34 \sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (3 x+2) \left (3 x^2+5 x+2\right )^{3/2}}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 826
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps
\begin {align*} \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{5/2}} \, dx &=-\frac {2 (2+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{3 x^{3/2}}-\frac {2}{5} \int \frac {\left (5+\frac {15 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx\\ &=\frac {2 (2-x) \sqrt {2+5 x+3 x^2}}{\sqrt {x}}-\frac {2 (2+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{3 x^{3/2}}+\frac {4}{15} \int \frac {-\frac {105}{2}-\frac {255 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 (2-x) \sqrt {2+5 x+3 x^2}}{\sqrt {x}}-\frac {2 (2+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{3 x^{3/2}}+\frac {8}{15} \text {Subst}\left (\int \frac {-\frac {105}{2}-\frac {255 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 (2-x) \sqrt {2+5 x+3 x^2}}{\sqrt {x}}-\frac {2 (2+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{3 x^{3/2}}-28 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-34 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {34 \sqrt {x} (2+3 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2 (2-x) \sqrt {2+5 x+3 x^2}}{\sqrt {x}}-\frac {2 (2+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{3 x^{3/2}}+\frac {34 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}-\frac {14 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.17, size = 163, normalized size = 0.89 \begin {gather*} \frac {-34 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{5/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (8+74 x+195 x^2+219 x^3+117 x^4+27 x^5+4 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{5/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )\right )}{3 x^{3/2} \sqrt {2+5 x+3 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 125, normalized size = 0.68
method | result | size |
default | \(\frac {9 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -17 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -162 x^{5}-702 x^{4}-1008 x^{3}-660 x^{2}-240 x -48}{9 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {3}{2}}}\) | \(125\) |
risch | \(-\frac {2 \left (27 x^{5}+117 x^{4}+168 x^{3}+110 x^{2}+40 x +8\right )}{3 x^{\frac {3}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (\frac {14 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {17 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(203\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{3 x^{2}}-\frac {20 \left (3 x^{2}+5 x +2\right )}{3 \sqrt {x \left (3 x^{2}+5 x +2\right )}}-6 x \sqrt {3 x^{3}+5 x^{2}+2 x}-16 \sqrt {3 x^{3}+5 x^{2}+2 x}-\frac {14 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {17 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(243\) |
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 = 0.41, size = 69, normalized size = 0.38 \begin {gather*} -\frac {2 \, {\left (41 \, \sqrt {3} x^{2} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 153 \, \sqrt {3} x^{2} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 9 \, {\left (9 \, x^{3} + 24 \, x^{2} + 10 \, x + 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{27 \, x^{2}} \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 (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {5}{2}}}\right )\, dx - \int \frac {19 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {x}}\, dx - \int 15 \sqrt {x} \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 = -0.01 \begin {gather*} \int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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