3.1.83 \(\int \frac {1}{\sqrt {3-x+2 x^2} (2+3 x+5 x^2)} \, dx\) [83]

Optimal. Leaf size=148 \[ \sqrt {\frac {1}{682} \left (13+10 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (13+10 \sqrt {2}\right )}} \left (7+3 \sqrt {2}+\left (13+10 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\sqrt {\frac {1}{682} \left (-13+10 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-13+10 \sqrt {2}\right )}} \left (7-3 \sqrt {2}+\left (13-10 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]

[Out]

-1/682*arctanh(1/31*(7+x*(13-10*2^(1/2))-3*2^(1/2))*341^(1/2)/(-13+10*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-8866
+6820*2^(1/2))^(1/2)+1/682*arctan(1/31*(7+3*2^(1/2)+x*(13+10*2^(1/2)))*341^(1/2)/(13+10*2^(1/2))^(1/2)/(2*x^2-
x+3)^(1/2))*(8866+6820*2^(1/2))^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1000, 1043, 210, 212} \begin {gather*} \sqrt {\frac {1}{682} \left (13+10 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (13+10 \sqrt {2}\right )}} \left (\left (13+10 \sqrt {2}\right ) x+3 \sqrt {2}+7\right )}{\sqrt {2 x^2-x+3}}\right )-\sqrt {\frac {1}{682} \left (10 \sqrt {2}-13\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (10 \sqrt {2}-13\right )}} \left (\left (13-10 \sqrt {2}\right ) x-3 \sqrt {2}+7\right )}{\sqrt {2 x^2-x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)),x]

[Out]

Sqrt[(13 + 10*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(13 + 10*Sqrt[2]))]*(7 + 3*Sqrt[2] + (13 + 10*Sqrt[2])*x))/Sqr
t[3 - x + 2*x^2]] - Sqrt[(-13 + 10*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-13 + 10*Sqrt[2]))]*(7 - 3*Sqrt[2] + (1
3 - 10*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1000

Int[1/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt
[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c
*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sq
rt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] &&
NeQ[c*e - b*f, 0] && NegQ[b^2 - 4*a*c]

Rule 1043

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx &=-\frac {\int \frac {11-11 \sqrt {2}-11 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{22 \sqrt {2}}+\frac {\int \frac {11+11 \sqrt {2}-11 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{22 \sqrt {2}}\\ &=-\left (\frac {1}{2} \left (11 \left (20-13 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-3751 \left (13-10 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {11 \left (7-3 \sqrt {2}\right )+11 \left (13-10 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )\right )-\frac {1}{2} \left (11 \left (20+13 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-3751 \left (13+10 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {11 \left (7+3 \sqrt {2}\right )+11 \left (13+10 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )\\ &=\sqrt {\frac {1}{682} \left (13+10 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (13+10 \sqrt {2}\right )}} \left (7+3 \sqrt {2}+\left (13+10 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\sqrt {\frac {1}{682} \left (-13+10 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-13+10 \sqrt {2}\right )}} \left (7-3 \sqrt {2}+\left (13-10 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.19, size = 135, normalized size = 0.91 \begin {gather*} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)),x]

[Out]

RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] -
#1] + 2*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1
^3) & ]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(683\) vs. \(2(109)=218\).
time = 0.57, size = 684, normalized size = 4.62

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+465124 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) \ln \left (\frac {-18721241 \RootOf \left (\textit {\_Z}^{2}+465124 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{4} x +280302 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} \RootOf \left (\textit {\_Z}^{2}+465124 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) x [phone] \sqrt {2 x^{2}-x +3}\, \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}-238700 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} \RootOf \left (\textit {\_Z}^{2}+465124 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right )+1739 \RootOf \left (\textit {\_Z}^{2}+465124 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) x +649946 \sqrt {2 x^{2}-x +3}+4700 \RootOf \left (\textit {\_Z}^{2}+465124 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right )}{341 x \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+4 x +1}\right )}{682}+\RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right ) \ln \left (-\frac {74884964 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{5} x +3976060 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{3} x +391468 \sqrt {2 x^{2}-x +3}\, \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}-954800 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{3}+41625 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right ) x +3650 \sqrt {2 x^{2}-x +3}-37000 \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )}{682 x \RootOf \left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+5 x -2}\right )\)	\(441\)
default	\(\frac {\sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (369 \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+520 \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+465124 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-866822 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{21142 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\)	\(684\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/21142*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)*2^(1/2
)*(369*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*
(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+10368
*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(
2^(1/2)+1-x)^2+23)*(2^(1/2)-1+x)/(2^(1/2)+1-x)*(8+3*2^(1/2)))*(-775687+549362*2^(1/2))^(1/2)+520*(-8866+6820*2
^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2
)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-
x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(2^(1/2)-
1+x)/(2^(1/2)+1-x)*(8+3*2^(1/2)))*(-775687+549362*2^(1/2))^(1/2)+465124*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/
2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2))*2^(1/2)-866
822*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/
2)/(-8866+6820*2^(1/2))^(1/2)))/((8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+
8-3*2^(1/2))/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))^2)^(1/2)/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))/(8+3*2^(1/2))/(-8866+682
0*2^(1/2))^(1/2)

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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.

[In]

integrate(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2002 vs. \(2 (109) = 218\).
time = 2.37, size = 2002, normalized size = 13.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

-1/845680*sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(13*sqrt(2) + 20)*(13*sqrt(2) - 20)*log(1240*(sqrt(341)*200
^(1/4)*sqrt(31)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*x - 1) - 3*x - 5)*sqrt(13*sqrt(2) + 20) + 7595*x^2 + 6
820*sqrt(2)*(2*x^2 - x + 3) - 23405*x + 31000)/x^2) + 1/845680*sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(13*sq
rt(2) + 20)*(13*sqrt(2) - 20)*log(-1240*(sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*
x - 1) - 3*x - 5)*sqrt(13*sqrt(2) + 20) - 7595*x^2 - 6820*sqrt(2)*(2*x^2 - x + 3) + 23405*x - 31000)/x^2) - 1/
6820*sqrt(341)*200^(1/4)*sqrt(5)*sqrt(2)*sqrt(13*sqrt(2) + 20)*arctan(1/2762875*(14260*sqrt(341)*sqrt(5)*sqrt(
2*x^2 - x + 3)*(11*200^(3/4)*(8056*x^7 - 28976*x^6 + 61838*x^5 - 93342*x^4 + 45376*x^3 - 18288*x^2 - sqrt(2)*(
4702*x^7 - 19541*x^6 + 40352*x^5 - 68777*x^4 + 35480*x^3 - 19080*x^2 - 34560*x + 27648) - 55296*x + 34560) + 5
*200^(1/4)*(18463*x^7 - 280047*x^6 + 1453472*x^5 - 3238500*x^4 + 4140576*x^3 - 2378592*x^2 - sqrt(2)*(11418*x^
7 - 177633*x^6 + 957180*x^5 - 2237548*x^4 + 2920320*x^3 - 2005920*x^2 - 1990656*x + 1534464) - 3068928*x + 199
0656))*sqrt(13*sqrt(2) + 20) + 7843000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1
549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 -
752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(310)*(sqrt(341)*sqrt(5)*sqrt(2*x^
2 - x + 3)*(11*200^(3/4)*(30876*x^7 - 44014*x^6 + 139674*x^5 - 42464*x^4 + 38736*x^3 + 89856*x^2 - sqrt(2)*(15
454*x^7 - 22399*x^6 + 73509*x^5 - 37360*x^4 + 52200*x^3 + 13824*x^2 - 13824*x) - 89856*x) + 5*200^(1/4)*(69479
*x^7 - 898236*x^6 + 3454740*x^5 - 4394304*x^4 + 5347296*x^3 + 4478976*x^2 - sqrt(2)*(38627*x^7 - 500012*x^6 +
1934180*x^5 - 2560320*x^4 + 3506400*x^3 + 1202688*x^2 - 1202688*x) - 4478976*x))*sqrt(13*sqrt(2) + 20) + 550*s
qrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 -
sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800
*x) + 3276288*x) + 25*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219
328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2
- 1944*x) + 144820224*x))*sqrt(-(sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*x - 1) -
3*x - 5)*sqrt(13*sqrt(2) + 20) - 7595*x^2 - 6820*sqrt(2)*(2*x^2 - x + 3) + 23405*x - 31000)/x^2) + 89125*sqrt
(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2
- 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184)
+ 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^
3 - 34615296*x^2 - 24772608*x + 18579456)) - 1/6820*sqrt(341)*200^(1/4)*sqrt(5)*sqrt(2)*sqrt(13*sqrt(2) + 20)*
arctan(1/2762875*(14260*sqrt(341)*sqrt(5)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(8056*x^7 - 28976*x^6 + 61838*x^5
- 93342*x^4 + 45376*x^3 - 18288*x^2 - sqrt(2)*(4702*x^7 - 19541*x^6 + 40352*x^5 - 68777*x^4 + 35480*x^3 - 1908
0*x^2 - 34560*x + 27648) - 55296*x + 34560) + 5*200^(1/4)*(18463*x^7 - 280047*x^6 + 1453472*x^5 - 3238500*x^4
+ 4140576*x^3 - 2378592*x^2 - sqrt(2)*(11418*x^7 - 177633*x^6 + 957180*x^5 - 2237548*x^4 + 2920320*x^3 - 20059
20*x^2 - 1990656*x + 1534464) - 3068928*x + 1990656))*sqrt(13*sqrt(2) + 20) - 7843000*sqrt(31)*sqrt(2)*(28180*
x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 10233
5*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456
192) - 2*sqrt(310)*(sqrt(341)*sqrt(5)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(30876*x^7 - 44014*x^6 + 139674*x^5 -
42464*x^4 + 38736*x^3 + 89856*x^2 - sqrt(2)*(15454*x^7 - 22399*x^6 + 73509*x^5 - 37360*x^4 + 52200*x^3 + 13824
*x^2 - 13824*x) - 89856*x) + 5*200^(1/4)*(69479*x^7 - 898236*x^6 + 3454740*x^5 - 4394304*x^4 + 5347296*x^3 + 4
478976*x^2 - sqrt(2)*(38627*x^7 - 500012*x^6 + 1934180*x^5 - 2560320*x^4 + 3506400*x^3 + 1202688*x^2 - 1202688
*x) - 4478976*x))*sqrt(13*sqrt(2) + 20) - 550*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 329307
2*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 10
53960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 25*sqrt(31)*(254591*x^8 - 4815126*x^7 + 3230
3580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x
^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((sqrt(341)*200^(1/4)*sqrt(31)*sq
rt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*x - 1) - 3*x - 5)*sqrt(13*sqrt(2) + 20) + 7595*x^2 + 6820*sqrt(2)*(2*x^2
- x + 3) - 23405*x + 31000)/x^2) - 89125*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 +
254146592*x^4 - 249300096*x^3 + 37981440*x^2 -...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 x^{2} - x + 3} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x**2+3*x+2)/(2*x**2-x+3)**(1/2),x)

[Out]

Integral(1/(sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,i
nfinity,inf

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {2\,x^2-x+3}\,\left (5\,x^2+3\,x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x^2 - x + 3)^(1/2)*(3*x + 5*x^2 + 2)),x)

[Out]

int(1/((2*x^2 - x + 3)^(1/2)*(3*x + 5*x^2 + 2)), x)

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