3.331 \(\int \frac {\sqrt {3-x+2 x^2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^6} \, dx\)

Optimal. Leaf size=165 \[ -\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{179159040 (2 x+5)^3}+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{829440 (2 x+5)^4}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}-\frac {(3174439702 x+4583087983) \sqrt {2 x^2-x+3}}{6879707136 (2 x+5)^2}+\frac {12895597463 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{82556485632 \sqrt {2}}-\frac {5 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{32 \sqrt {2}} \]

[Out]

-3667/2880*(2*x^2-x+3)^(3/2)/(5+2*x)^5+711961/829440*(2*x^2-x+3)^(3/2)/(5+2*x)^4-38732321/179159040*(2*x^2-x+3
)^(3/2)/(5+2*x)^3-5/64*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)[phone]/165112971264*arctanh(1/24*(17-22*x)*
2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-1/6879707136*(4583087983+3174439702*x)*(2*x^2-x+3)^(1/2)/(5+2*x)^2

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1650, 810, 843, 619, 215, 724, 206} \[ -\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{179159040 (2 x+5)^3}+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{829440 (2 x+5)^4}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}-\frac {(3174439702 x+4583087983) \sqrt {2 x^2-x+3}}{6879707136 (2 x+5)^2}+\frac {12895597463 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{82556485632 \sqrt {2}}-\frac {5 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{32 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^6,x]

[Out]

-((4583087983 + 3174439702*x)*Sqrt[3 - x + 2*x^2])/(6879707136*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(3/2))/(28
80*(5 + 2*x)^5) + (711961*(3 - x + 2*x^2)^(3/2))/(829440*(5 + 2*x)^4) - (38732321*(3 - x + 2*x^2)^(3/2))/(1791
59040*(5 + 2*x)^3) - (5*ArcSinh[(1 - 4*x)/Sqrt[23]])/(32*Sqrt[2]) + (12895597463*ArcTanh[(17 - 22*x)/(12*Sqrt[
2]*Sqrt[3 - x + 2*x^2])])/(82556485632*Sqrt[2])

Rule 206

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

Rule 215

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

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}-\frac {1}{360} \int \frac {\sqrt {3-x+2 x^2} \left (\frac {52701}{16}-\frac {9563 x}{2}+2430 x^2-900 x^3\right )}{(5+2 x)^5} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}+\frac {\int \frac {\sqrt {3-x+2 x^2} \left (\frac {5935131}{16}-\frac {1983719 x}{4}+129600 x^2\right )}{(5+2 x)^4} \, dx}{103680}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}-\frac {38732321 \left (3-x+2 x^2\right )^{3/2}}{179159040 (5+2 x)^3}-\frac {\int \frac {\left (\frac {138672015}{16}-13996800 x\right ) \sqrt {3-x+2 x^2}}{(5+2 x)^3} \, dx}{22394880}\\ &=-\frac {(4583087983+3174439702 x) \sqrt {3-x+2 x^2}}{6879707136 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}-\frac {38732321 \left (3-x+2 x^2\right )^{3/2}}{179159040 (5+2 x)^3}+\frac {\int \frac {-\frac {32190825945}{8}[phone] x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{25798901760}\\ &=-\frac {(4583087983+3174439702 x) \sqrt {3-x+2 x^2}}{6879707136 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}-\frac {38732321 \left (3-x+2 x^2\right )^{3/2}}{179159040 (5+2 x)^3}+\frac {5}{32} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx-\frac {12895597463 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{13759414272}\\ &=-\frac {(4583087983+3174439702 x) \sqrt {3-x+2 x^2}}{6879707136 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}-\frac {38732321 \left (3-x+2 x^2\right )^{3/2}}{179159040 (5+2 x)^3}+\frac {12895597463 \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{6879707136}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{32 \sqrt {46}}\\ &=-\frac {(4583087983+3174439702 x) \sqrt {3-x+2 x^2}}{6879707136 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}-\frac {38732321 \left (3-x+2 x^2\right )^{3/2}}{179159040 (5+2 x)^3}-\frac {5 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{32 \sqrt {2}}+\frac {12895597463 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{82556485632 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 98, normalized size = 0.59 \[ \frac {64477987315 \sqrt {2} \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )-\frac {24 \sqrt {2 x^2-x+3} \left (186470433136 x^4+1285267446304 x^3+3919478861832 x^2+5608297138216 x+3110673952831\right )}{(2 x+5)^5}-64497254400 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{825564856320} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^6,x]

[Out]

((-24*Sqrt[3 - x + 2*x^2]*(3110673952831 + 5608297138216*x + 3919478861832*x^2 + 1285267446304*x^3 + 186470433
136*x^4))/(5 + 2*x)^5 - 64497254400*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 64477987315*Sqrt[2]*ArcTanh[(17 - 22
*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/825564856320

________________________________________________________________________________________

fricas [A]  time = 0.89, size = 203, normalized size = 1.23 \[ \frac {64497254400 \, \sqrt {2} {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 64477987315 \, \sqrt {2} {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \, {\left (186470433136 \, x^{4} + 1285267446304 \, x^{3} + 3919478861832 \, x^{2} + 5608297138216 \, x + 3110673952831\right )} \sqrt {2 \, x^{2} - x + 3}}{1651129712640 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^6,x, algorithm="fricas")

[Out]

1/1651129712640*(64497254400*sqrt(2)*(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125)*log(-4*sqrt(2)*s
qrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 64477987315*sqrt(2)*(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x
^2 + 6250*x + 3125)*log((24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 1153)/(4*x^2 + 20*x
+ 25)) - 48*(186470433136*x^4 + 1285267446304*x^3 + 3919478861832*x^2 + 5608297138216*x + 3110673952831)*sqrt(
2*x^2 - x + 3))/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125)

________________________________________________________________________________________

giac [B]  time = 0.28, size = 387, normalized size = 2.35 \[ -\frac {5}{64} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {12895597463}{165112971264} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {12895597463}{165112971264} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (4368922304720 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 124570969998480 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 637804348664160 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} + 1828845222532320 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 3763189300187016 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} - 10794416351958120 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 25049834283305880 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 34708488692384520 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10654664764755165 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 2507056315485767\right )}}{68797071360 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^6,x, algorithm="giac")

[Out]

-5/64*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 12895597463/165112971264*sqrt(2)*log(abs
(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 12895597463/165112971264*sqrt(2)*log(abs(-2*sqrt(2)*x - 11
*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/68797071360*sqrt(2)*(4368922304720*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x
+ 3))^9 + 124570969998480*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^8 + 637804348664160*sqrt(2)*(sqrt(2)*x - sqrt(2*x^
2 - x + 3))^7 + 1828845222532320*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^6 - 3763189300187016*sqrt(2)*(sqrt(2)*x - s
qrt(2*x^2 - x + 3))^5 - 10794416351958120*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 + 25049834283305880*sqrt(2)*(sqr
t(2)*x - sqrt(2*x^2 - x + 3))^3 - 34708488692384520*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10654664764755165*sq
rt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 2507056315485767)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2
)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^5

________________________________________________________________________________________

maple [A]  time = 0.01, size = 188, normalized size = 1.14 \[ \frac {5 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{64}+\frac {12895597463 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{165112971264}-\frac {12895597463 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{495338913792}-\frac {3667 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{92160 \left (x +\frac {5}{2}\right )^{5}}-\frac {38732321 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{1433272320 \left (x +\frac {5}{2}\right )^{3}}+\frac {711961 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{13271040 \left (x +\frac {5}{2}\right )^{4}}+\frac {46569601 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{6879707136 \left (x +\frac {5}{2}\right )^{2}}+\frac {562688629 \left (4 x -1\right ) \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{495338913792}-\frac {562688629 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{247669456896 \left (x +\frac {5}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^6,x)

[Out]

-12895597463/495338913792*(-11*x+2*(x+5/2)^2-19/2)^(1/2)-3667/92160/(x+5/2)^5*(-11*x+2*(x+5/2)^2-19/2)^(3/2)-3
8732321/1433272320/(x+5/2)^3*(-11*x+2*(x+5/2)^2-19/2)^(3/2)+711961/13271040/(x+5/2)^4*(-11*x+2*(x+5/2)^2-19/2)
^(3/2)[phone]/6879707136/(x+5/2)^2*(-11*x+2*(x+5/2)^2-19/2)^(3/2)[phone]/495338913792*(4*x-1)*(-11*x+2*(x
+5/2)^2-19/2)^(1/2)-562688629/247669456896/(x+5/2)*(-11*x+2*(x+5/2)^2-19/2)^(3/2)[phone]/165112971264*2^(
1/2)*arctanh(1/12*(-11*x+17/2)*2^(1/2)/(-11*x+2*(x+5/2)^2-19/2)^(1/2))+5/64*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1
/4))

________________________________________________________________________________________

maxima [A]  time = 1.04, size = 222, normalized size = 1.35 \[ \frac {5}{64} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {12895597463}{165112971264} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {46569601}{3439853568} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2880 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac {711961 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{829440 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac {38732321 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{179159040 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac {46569601 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1719926784 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {562688629 \, \sqrt {2 \, x^{2} - x + 3}}{6879707136 \, {\left (2 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^6,x, algorithm="maxima")

[Out]

5/64*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 12895597463/165112971264*sqrt(2)*arcsinh(22/23*sqrt(23
)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 46569601/3439853568*sqrt(2*x^2 - x + 3) - 3667/2880*(2*x^2 -
x + 3)^(3/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125) + 711961/829440*(2*x^2 - x + 3)^(3/2)/(
16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) - 38732321/179159040*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2 + 150*x
+ 125) + 46569601/1719926784*(2*x^2 - x + 3)^(3/2)/(4*x^2 + 20*x + 25) - 562688629/6879707136*sqrt(2*x^2 - x +
3)/(2*x + 5)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2 - x + 3)^(1/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^6,x)

[Out]

int(((2*x^2 - x + 3)^(1/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^6, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**4-x**3+3*x**2+x+2)*(2*x**2-x+3)**(1/2)/(5+2*x)**6,x)

[Out]

Integral(sqrt(2*x**2 - x + 3)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**6, x)

________________________________________________________________________________________