Optimal. Leaf size=205 \[ \frac {62 \sqrt {x} (2+3 x)}{21 \sqrt {2+5 x+3 x^2}}-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {62 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}}+\frac {43 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{7 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]
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Rubi [A]
time = 0.09, 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 = {824, 848, 853,
1203, 1114, 1150} \begin {gather*} \frac {43 (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{7 \sqrt {2} \sqrt {3 x^2+5 x+2}}-\frac {62 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {3 x^2+5 x+2}}-\frac {62 \sqrt {3 x^2+5 x+2}}{21 \sqrt {x}}+\frac {62 \sqrt {x} (3 x+2)}{21 \sqrt {3 x^2+5 x+2}}-\frac {4 \sqrt {3 x^2+5 x+2} (1-3 x)}{7 x^{7/2}}+\frac {43 \sqrt {3 x^2+5 x+2}}{21 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 824
Rule 848
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps
\begin {align*} \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx &=-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}-\frac {1}{35} \int \frac {215+255 x}{x^{5/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}+\frac {1}{105} \int \frac {310+\frac {645 x}{2}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {1}{105} \int \frac {-\frac {645}{2}-465 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {2}{105} \text {Subst}\left (\int \frac {-\frac {645}{2}-465 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}+\frac {43}{7} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {62}{7} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {62 \sqrt {x} (2+3 x)}{21 \sqrt {2+5 x+3 x^2}}-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {62 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}}+\frac {43 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{7 \sqrt {2} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.14, size = 155, normalized size = 0.76 \begin {gather*} \frac {-48+24 x+460 x^2+646 x^3+258 x^4+124 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{9/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+5 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{9/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{42 x^{7/2} \sqrt {2+5 x+3 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 129, normalized size = 0.63
method | result | size |
default | \(-\frac {57 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}-62 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}+1116 x^{5}+1086 x^{4}-1194 x^{3}-1380 x^{2}-72 x +144}{126 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {7}{2}}}\) | \(129\) |
risch | \(-\frac {186 x^{5}+181 x^{4}-199 x^{3}-230 x^{2}-12 x +24}{21 x^{\frac {7}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (-\frac {31 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{21 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {43 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{42 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(203\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {4 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{4}}+\frac {12 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{3}}+\frac {43 \sqrt {3 x^{3}+5 x^{2}+2 x}}{21 x^{2}}-\frac {62 \left (3 x^{2}+5 x +2\right )}{21 \sqrt {x \left (3 x^{2}+5 x +2\right )}}+\frac {43 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{42 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {31 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{21 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(248\) |
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 = 0.45, size = 69, normalized size = 0.34 \begin {gather*} \frac {77 \, \sqrt {3} x^{4} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 558 \, \sqrt {3} x^{4} {\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 (62 \, x^{3} - 43 \, x^{2} - 36 \, x + 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{189 \, x^{4}} \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 {2 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {9}{2}}}\right )\, dx - \int \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {7}{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 -\frac {\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2}}{x^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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