3.251 \(\int \frac{x^2 (5+x+3 x^2+2 x^3)}{2+x+5 x^2+x^3+2 x^4} \, dx\)

Optimal. Leaf size=269 \[ \frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1-i \sqrt{7}\right ) x+4\right )-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1+i \sqrt{7}\right ) x+4\right )+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x-\frac{\left (\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x-i \sqrt{7}+1}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}}+\frac{\left (-\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x+i \sqrt{7}+1}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}} \]

[Out]

((7 - (5*I)*Sqrt[7])*x)/14 + ((7 + (5*I)*Sqrt[7])*x)/14 + ((7 - (5*I)*Sqrt[7])*x^2)/28 + ((7 + (5*I)*Sqrt[7])*
x^2)/28 - ((53*I + Sqrt[7])*ArcTan[(1 - I*Sqrt[7] + 8*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(2*Sqrt[14*(35 + I*Sqrt[7]
)]) + ((53*I - Sqrt[7])*ArcTan[(1 + I*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(2*Sqrt[14*(35 - I*Sqrt[7])])
- ((35 + (9*I)*Sqrt[7])*Log[4 + (1 - I*Sqrt[7])*x + 4*x^2])/56 - ((35 - (9*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])
*x + 4*x^2])/56

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Rubi [A]  time = 0.392077, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2087, 800, 634, 618, 204, 628} \[ \frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1-i \sqrt{7}\right ) x+4\right )-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1+i \sqrt{7}\right ) x+4\right )+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x-\frac{\left (\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x-i \sqrt{7}+1}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}}+\frac{\left (-\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x+i \sqrt{7}+1}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7 - (5*I)*Sqrt[7])*x)/14 + ((7 + (5*I)*Sqrt[7])*x)/14 + ((7 - (5*I)*Sqrt[7])*x^2)/28 + ((7 + (5*I)*Sqrt[7])*
x^2)/28 - ((53*I + Sqrt[7])*ArcTan[(1 - I*Sqrt[7] + 8*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(2*Sqrt[14*(35 + I*Sqrt[7]
)]) + ((53*I - Sqrt[7])*ArcTan[(1 + I*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(2*Sqrt[14*(35 - I*Sqrt[7])])
- ((35 + (9*I)*Sqrt[7])*Log[4 + (1 - I*Sqrt[7])*x + 4*x^2])/56 - ((35 - (9*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])
*x + 4*x^2])/56

Rule 2087

Int[((P3_)*(x_)^(m_.))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4), x_Symbol] :> With[{q
= Sqrt[8*a^2 + b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3
]}, Dist[1/q, Int[(x^m*(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x))/(2*a + (b + q)*x + 2*a*x^2
), x], x] - Dist[1/q, Int[(x^m*(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x))/(2*a + (b - q)*x +
2*a*x^2), x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), 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] && IntegerQ[m]

Rule 634

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

Rule 618

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

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx &=\frac{i \int \frac{x^2 \left (9-5 i \sqrt{7}+\left (10-2 i \sqrt{7}\right ) x\right )}{4+\left (1-i \sqrt{7}\right ) x+4 x^2} \, dx}{\sqrt{7}}-\frac{i \int \frac{x^2 \left (9+5 i \sqrt{7}+\left (10+2 i \sqrt{7}\right ) x\right )}{4+\left (1+i \sqrt{7}\right ) x+4 x^2} \, dx}{\sqrt{7}}\\ &=\frac{i \int \left (\frac{1}{2} \left (5-i \sqrt{7}\right )+\frac{1}{2} \left (5-i \sqrt{7}\right ) x+\frac{i \left (2 \left (5 i+\sqrt{7}\right )+\left (9 i+5 \sqrt{7}\right ) x\right )}{4+\left (1-i \sqrt{7}\right ) x+4 x^2}\right ) \, dx}{\sqrt{7}}-\frac{i \int \left (\frac{1}{2} \left (5+i \sqrt{7}\right )+\frac{1}{2} \left (5+i \sqrt{7}\right ) x-\frac{i \left (-2 \left (5 i-\sqrt{7}\right )-\left (9 i-5 \sqrt{7}\right ) x\right )}{4+\left (1+i \sqrt{7}\right ) x+4 x^2}\right ) \, dx}{\sqrt{7}}\\ &=\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2-\frac{\int \frac{2 \left (5 i+\sqrt{7}\right )+\left (9 i+5 \sqrt{7}\right ) x}{4+\left (1-i \sqrt{7}\right ) x+4 x^2} \, dx}{\sqrt{7}}-\frac{\int \frac{-2 \left (5 i-\sqrt{7}\right )-\left (9 i-5 \sqrt{7}\right ) x}{4+\left (1+i \sqrt{7}\right ) x+4 x^2} \, dx}{\sqrt{7}}\\ &=\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \int \frac{1+i \sqrt{7}+8 x}{4+\left (1+i \sqrt{7}\right ) x+4 x^2} \, dx-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \int \frac{1-i \sqrt{7}+8 x}{4+\left (1-i \sqrt{7}\right ) x+4 x^2} \, dx-\frac{1}{28} \left (7-53 i \sqrt{7}\right ) \int \frac{1}{4+\left (1+i \sqrt{7}\right ) x+4 x^2} \, dx-\frac{1}{28} \left (7+53 i \sqrt{7}\right ) \int \frac{1}{4+\left (1-i \sqrt{7}\right ) x+4 x^2} \, dx\\ &=\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4+\left (1-i \sqrt{7}\right ) x+4 x^2\right )-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4+\left (1+i \sqrt{7}\right ) x+4 x^2\right )-\frac{1}{14} \left (-7+53 i \sqrt{7}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (35-i \sqrt{7}\right )-x^2} \, dx,x,1+i \sqrt{7}+8 x\right )+\frac{1}{14} \left (7+53 i \sqrt{7}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (35+i \sqrt{7}\right )-x^2} \, dx,x,1-i \sqrt{7}+8 x\right )\\ &=\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2-\frac{\left (53 i+\sqrt{7}\right ) \tan ^{-1}\left (\frac{1-i \sqrt{7}+8 x}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}}+\frac{\left (53 i-\sqrt{7}\right ) \tan ^{-1}\left (\frac{1+i \sqrt{7}+8 x}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4+\left (1-i \sqrt{7}\right ) x+4 x^2\right )-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4+\left (1+i \sqrt{7}\right ) x+4 x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0163307, size = 101, normalized size = 0.38 \[ -\text{RootSum}\left [2 \text{$\#$1}^4+\text{$\#$1}^3+5 \text{$\#$1}^2+\text{$\#$1}+2\& ,\frac{5 \text{$\#$1}^3 \log (x-\text{$\#$1})+\text{$\#$1}^2 \log (x-\text{$\#$1})+3 \text{$\#$1} \log (x-\text{$\#$1})+2 \log (x-\text{$\#$1})}{8 \text{$\#$1}^3+3 \text{$\#$1}^2+10 \text{$\#$1}+1}\& \right ]+\frac{x^2}{2}+x \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + x^2/2 - RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (2*Log[x - #1] + 3*Log[x - #1]*#1 + Log[x - #1]*#1^2 +
5*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1^2 + 8*#1^3) & ]

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Maple [C]  time = 0.006, size = 67, normalized size = 0.3 \begin{align*}{\frac{{x}^{2}}{2}}+x+\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+5\,{{\it \_Z}}^{2}+{\it \_Z}+2 \right ) }{\frac{ \left ( -5\,{{\it \_R}}^{3}-{{\it \_R}}^{2}-3\,{\it \_R}-2 \right ) \ln \left ( x-{\it \_R} \right ) }{8\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+10\,{\it \_R}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2+x+sum((-5*_R^3-_R^2-3*_R-2)/(8*_R^3+3*_R^2+10*_R+1)*ln(x-_R),_R=RootOf(2*_Z^4+_Z^3+5*_Z^2+_Z+2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} + x - \int \frac{5 \, x^{3} + x^{2} + 3 \, x + 2}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 9.30163, size = 5019, normalized size = 18.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/56*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35)*log(49/4*(135*I*sqrt(7) + 420*sqrt(-37/
392*I*sqrt(7) + 79/56) - 1459)*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 10290*(-9/56*I*
sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^3 - 25725*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) +
79/56) - 5/8)^2 + 3/64*(3920*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1575*I*sqrt(7)
- 4900*sqrt(-37/392*I*sqrt(7) + 79/56) + 5587)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 8384
*x + 6615/2*I*sqrt(7) + 10290*sqrt(-37/392*I*sqrt(7) + 79/56) + 13373/2) + 1/8*(2*sqrt(-12*(9/56*I*sqrt(7) - 1
/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 12*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^
2 - 1/392*(9*I*sqrt(7) + 28*sqrt(-37/392*I*sqrt(7) + 79/56) - 105)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) +
79/56) + 35) + 45/14*I*sqrt(7) + 10*sqrt(-37/392*I*sqrt(7) + 79/56) + 11/2) + 2*sqrt(37/392*I*sqrt(7) + 79/56)
+ 2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5)*log(-49/4*(135*I*sqrt(7) + 420*sqrt(-37/392*I*sqrt(7) + 79/56) - 145
9)*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 + 24304*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I
*sqrt(7) + 79/56) - 5/8)^2 - 3/64*(3920*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1575
*I*sqrt(7) - 4900*sqrt(-37/392*I*sqrt(7) + 79/56) + 5587)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) +
35) + 7/64*sqrt(-12*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 12*(-9/56*I*sqrt(7) - 1/2*
sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1/392*(9*I*sqrt(7) + 28*sqrt(-37/392*I*sqrt(7) + 79/56) - 105)*(-9*
I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 45/14*I*sqrt(7) + 10*sqrt(-37/392*I*sqrt(7) + 79/56) + 1
1/2)*((135*I*sqrt(7) + 420*sqrt(-37/392*I*sqrt(7) + 79/56) - 1459)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) +
79/56) + 35) - 17856*I*sqrt(7) - 55552*sqrt(-37/392*I*sqrt(7) + 79/56) + 67776) + 16768*x - 4941*I*sqrt(7) - 1
5372*sqrt(-37/392*I*sqrt(7) + 79/56) - 9391) - 1/8*(2*sqrt(-12*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 7
9/56) - 5/8)^2 - 12*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1/392*(9*I*sqrt(7) + 28*
sqrt(-37/392*I*sqrt(7) + 79/56) - 105)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 45/14*I*sqrt(
7) + 10*sqrt(-37/392*I*sqrt(7) + 79/56) + 11/2) - 2*sqrt(37/392*I*sqrt(7) + 79/56) - 2*sqrt(-37/392*I*sqrt(7)
+ 79/56) + 5)*log(-49/4*(135*I*sqrt(7) + 420*sqrt(-37/392*I*sqrt(7) + 79/56) - 1459)*(9/56*I*sqrt(7) - 1/2*sqr
t(37/392*I*sqrt(7) + 79/56) - 5/8)^2 + 24304*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 -
3/64*(3920*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1575*I*sqrt(7) - 4900*sqrt(-37/3
92*I*sqrt(7) + 79/56) + 5587)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) - 7/64*sqrt(-12*(9/56*I*
sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 12*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/
56) - 5/8)^2 - 1/392*(9*I*sqrt(7) + 28*sqrt(-37/392*I*sqrt(7) + 79/56) - 105)*(-9*I*sqrt(7) + 28*sqrt(37/392*I
*sqrt(7) + 79/56) + 35) + 45/14*I*sqrt(7) + 10*sqrt(-37/392*I*sqrt(7) + 79/56) + 11/2)*((135*I*sqrt(7) + 420*s
qrt(-37/392*I*sqrt(7) + 79/56) - 1459)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) - 17856*I*sqrt(
7) - 55552*sqrt(-37/392*I*sqrt(7) + 79/56) + 67776) + 16768*x - 4941*I*sqrt(7) - 15372*sqrt(-37/392*I*sqrt(7)
+ 79/56) - 9391) - 1/56*(9*I*sqrt(7) + 28*sqrt(-37/392*I*sqrt(7) + 79/56) + 35)*log(10290*(-9/56*I*sqrt(7) - 1
/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^3 + 1421*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/
8)^2 + 8384*x + 3267/2*I*sqrt(7) + 5082*sqrt(-37/392*I*sqrt(7) + 79/56) + 13793/2) + x

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Sympy [A]  time = 0.72476, size = 53, normalized size = 0.2 \begin{align*} \frac{x^{2}}{2} + x + \operatorname{RootSum}{\left (686 t^{4} + 1715 t^{3} + 1372 t^{2} + 448 t + 256, \left ( t \mapsto t \log{\left (\frac{5145 t^{3}}{4192} + \frac{1421 t^{2}}{8384} - \frac{2541 t}{2096} + x + \frac{17}{262} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 + x + RootSum(686*_t**4 + 1715*_t**3 + 1372*_t**2 + 448*_t + 256, Lambda(_t, _t*log(5145*_t**3/4192 + 1
421*_t**2/8384 - 2541*_t/2096 + x + 17/262)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x^{2}}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^3 + 3*x^2 + x + 5)*x^2/(2*x^4 + x^3 + 5*x^2 + x + 2), x)