Optimal. Leaf size=251 \[ \frac {1543648 \sqrt {x} (2+3 x)}{6567561 \sqrt {2+5 x+3 x^2}}-\frac {8 \sqrt {x} (397265+502911 x) \sqrt {2+5 x+3 x^2}}{2189187}+\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {1543648 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{6567561 \sqrt {2+5 x+3 x^2}}+\frac {349240 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2189187 \sqrt {2+5 x+3 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 251, 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 = {846, 828, 853,
1203, 1114, 1150} \begin {gather*} \frac {349240 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2189187 \sqrt {3 x^2+5 x+2}}-\frac {1543648 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{6567561 \sqrt {3 x^2+5 x+2}}+\frac {157160 \left (3 x^2+5 x+2\right )^{3/2} \sqrt {x}}{243243}-\frac {8 (502911 x+397265) \sqrt {3 x^2+5 x+2} \sqrt {x}}{2189187}+\frac {1543648 (3 x+2) \sqrt {x}}{6567561 \sqrt {3 x^2+5 x+2}}-\frac {10}{39} \left (3 x^2+5 x+2\right )^{3/2} x^{7/2}+\frac {656 \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}}{1287}-\frac {21620 \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}}{34749} \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^{7/2} \sqrt {2+5 x+3 x^2} \, dx &=-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2}{39} \int x^{5/2} (35+164 x) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {4 \int \left (-820-\frac {5405 x}{2}\right ) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx}{1287}\\ &=-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {8 \int \sqrt {x} \left (\frac {16215}{2}+\frac {58935 x}{2}\right ) \sqrt {2+5 x+3 x^2} \, dx}{34749}\\ &=\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {16 \int \frac {\left (-\frac {58935}{2}-\frac {838185 x}{4}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx}{729729}\\ &=-\frac {8 \sqrt {x} (397265+502911 x) \sqrt {2+5 x+3 x^2}}{2189187}+\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {32 \int \frac {-\frac {654825}{4}-\frac {723585 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{32837805}\\ &=-\frac {8 \sqrt {x} (397265+502911 x) \sqrt {2+5 x+3 x^2}}{2189187}+\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {64 \text {Subst}\left (\int \frac {-\frac {654825}{4}-\frac {723585 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{32837805}\\ &=-\frac {8 \sqrt {x} (397265+502911 x) \sqrt {2+5 x+3 x^2}}{2189187}+\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {698480 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{2189187}+\frac {1543648 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{2189187}\\ &=\frac {1543648 \sqrt {x} (2+3 x)}{6567561 \sqrt {2+5 x+3 x^2}}-\frac {8 \sqrt {x} (397265+502911 x) \sqrt {2+5 x+3 x^2}}{2189187}+\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {1543648 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{6567561 \sqrt {2+5 x+3 x^2}}+\frac {349240 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2189187 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 20.17, size = 178, normalized size = 0.71 \begin {gather*} \frac {2 \left (1543648+2811400 x+670548 x^2-141444 x^3+58374 x^4+2892348 x^5+671895 x^6-10195794 x^7-7577955 x^8\right )+1543648 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 )-495928 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 )}{6567561 \sqrt {x} \sqrt {2+5 x+3 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.76, size = 137, normalized size = 0.55
method | result | size |
default | \(-\frac {2 \left (22733865 x^{8}+30587382 x^{7}-2015685 x^{6}-8677044 x^{5}+633876 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-385912 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-175122 x^{4}+424332 x^{3}+4934772 x^{2}+3143160 x \right )}{19702683 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(137\) |
risch | \(-\frac {2 \left (841995 x^{5}-270459 x^{4}-185220 x^{3}+167634 x^{2}-162396 x +174620\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{2189187}-\frac {\left (-\frac {349240 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {771824 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(203\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 x^{5} \sqrt {3 x^{3}+5 x^{2}+2 x}}{13}+\frac {106 x^{4} \sqrt {3 x^{3}+5 x^{2}+2 x}}{429}+\frac {1960 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11583}-\frac {37252 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{243243}+\frac {2776 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{18711}-\frac {349240 \sqrt {3 x^{3}+5 x^{2}+2 x}}{2189187}+\frac {349240 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {771824 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(280\) |
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 = 0.45, size = 68, normalized size = 0.27 \begin {gather*} -\frac {2}{2189187} \, {\left (841995 \, x^{5} - 270459 \, x^{4} - 185220 \, x^{3} + 167634 \, x^{2} - 162396 \, x + 174620\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {204560}{8444007} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {1543648}{6567561} \, \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 {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 5 x^{\frac {9}{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^{7/2}\,\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]
________________________________________________________________________________________