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]

-((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])

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Rubi [A]  time = 0.22853, antiderivative size = 165, normalized size of antiderivative = 1., 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 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]

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 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 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 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 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])

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 = 0.212094, size = 98, normalized size = 0.59 \[ \frac{-\frac{24 \sqrt{2 x^2-x+3} \left (186470433136 x^4+1285267446304 x^3+3919478861832 x^2+5608297138216 x+3110673952831\right )}{(2 x+5)^5}[phone] \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-64497254400 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{825564856320} \]

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

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Maple [A]  time = 0.074, size = 188, normalized size = 1.1 \begin{align*} -{\frac{562688629}{247669456896} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{-562688629+2250754516\,x}{495338913792}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{46569601}{6879707136} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{12895597463\,\sqrt{2}}{165112971264}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }+{\frac{5\,\sqrt{2}}{64}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{12895597463}{495338913792}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{711961}{13271040} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}}-{\frac{38732321}{1433272320} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{3667}{92160} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}} \end{align*}

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]

-562688629/247669456896/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(3/2)[phone]/495338913792*(-1+4*x)*(2*(x+5/2)^2-11*
x-19/2)^(1/2)[phone]/6879707136/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(3/2)[phone]/165112971264*2^(1/2)*arc
tanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))+5/64*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-12895
597463/495338913792*(2*(x+5/2)^2-11*x-19/2)^(1/2)+711961/13271040/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(3/2)-3873
2321/1433272320/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(3/2)-3667/92160/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(3/2)

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Maxima [A]  time = 1.61647, size = 300, normalized size = 1.82 \begin{align*} \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 )}} \end{align*}

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)

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Fricas [A]  time = 1.40667, size = 694, normalized size = 4.21 \begin{align*} \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 )}} \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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)

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Giac [B]  time = 1.29892, size = 522, normalized size = 3.16 \begin{align*} -\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}} \end{align*}

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