Optimal. Leaf size=228 \[ -\frac {261784 \sqrt {x} (2+3 x)}{841995 \sqrt {2+5 x+3 x^2}}+\frac {8 \sqrt {x} (57860+74313 x) \sqrt {2+5 x+3 x^2}}{280665}-\frac {4420 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {261784 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{841995 \sqrt {2+5 x+3 x^2}}-\frac {13016 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{56133 \sqrt {2+5 x+3 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 228, 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 = {846, 828, 853,
1203, 1114, 1150} \begin {gather*} -\frac {13016 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{56133 \sqrt {3 x^2+5 x+2}}+\frac {261784 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{841995 \sqrt {3 x^2+5 x+2}}-\frac {4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt {x}}{6237}+\frac {8 (74313 x+57860) \sqrt {3 x^2+5 x+2} \sqrt {x}}{280665}-\frac {261784 (3 x+2) \sqrt {x}}{841995 \sqrt {3 x^2+5 x+2}}-\frac {10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac {532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 828
Rule 846
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps
\begin {align*} \int (2-5 x) x^{5/2} \sqrt {2+5 x+3 x^2} \, dx &=-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2}{33} \int x^{3/2} (25+133 x) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {4}{891} \int \left (-399-\frac {3315 x}{2}\right ) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx\\ &=-\frac {4420 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {8 \int \frac {\left (\frac {3315}{2}+\frac {24771 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx}{18711}\\ &=\frac {8 \sqrt {x} (57860+74313 x) \sqrt {2+5 x+3 x^2}}{280665}-\frac {4420 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {16 \int \frac {\frac {24405}{2}+\frac {98169 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{841995}\\ &=\frac {8 \sqrt {x} (57860+74313 x) \sqrt {2+5 x+3 x^2}}{280665}-\frac {4420 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {32 \text {Subst}\left (\int \frac {\frac {24405}{2}+\frac {98169 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{841995}\\ &=\frac {8 \sqrt {x} (57860+74313 x) \sqrt {2+5 x+3 x^2}}{280665}-\frac {4420 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {26032 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{56133}-\frac {261784 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{280665}\\ &=-\frac {261784 \sqrt {x} (2+3 x)}{841995 \sqrt {2+5 x+3 x^2}}+\frac {8 \sqrt {x} (57860+74313 x) \sqrt {2+5 x+3 x^2}}{280665}-\frac {4420 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac {532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {261784 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{841995 \sqrt {2+5 x+3 x^2}}-\frac {13016 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{56133 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 20.13, size = 170, normalized size = 0.75 \begin {gather*} \frac {-523568-918440 x-198168 x^2+39780 x^3+947916 x^4+271350 x^5-3129840 x^6-2296350 x^7-261784 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 )+66544 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 )}{841995 \sqrt {x} \sqrt {2+5 x+3 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.84, size = 132, normalized size = 0.58
method | result | size |
default | \(\frac {-\frac {30 x^{7}}{11}-\frac {368 x^{6}}{99}+\frac {670 x^{5}}{2079}+\frac {65812 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{841995}-\frac {130892 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{2525985}+\frac {35108 x^{4}}{31185}+\frac {884 x^{3}}{18711}+\frac {195728 x^{2}}{280665}+\frac {26032 x}{56133}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(132\) |
risch | \(-\frac {2 \left (127575 x^{4}-38745 x^{3}-35550 x^{2}+32418 x -32540\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{280665}-\frac {\left (\frac {13016 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{168399 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {130892 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{841995 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 x^{4} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11}+\frac {82 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{297}+\frac {1580 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{6237}-\frac {7204 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{31185}+\frac {13016 \sqrt {3 x^{3}+5 x^{2}+2 x}}{56133}-\frac {13016 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{168399 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {130892 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{841995 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(259\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.64, size = 63, normalized size = 0.28 \begin {gather*} -\frac {2}{280665} \, {\left (127575 \, x^{4} - 38745 \, x^{3} - 35550 \, x^{2} + 32418 \, x - 32540\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} + \frac {3928}{216513} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {261784}{841995} \, \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 2 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 5 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int x^{5/2}\,\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.
[In]
[Out]
________________________________________________________________________________________