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

Optimal. Leaf size=281 \[ \frac{3}{112} \left (7+11 i \sqrt{7}\right ) \log \left (4 i x^2+\left (-\sqrt{7}+i\right ) x+4 i\right )+\frac{3}{112} \left (7-11 i \sqrt{7}\right ) \log \left (4 i x^2+\left (\sqrt{7}+i\right ) x+4 i\right )-\frac{35+9 i \sqrt{7}}{28 x}-\frac{35-9 i \sqrt{7}}{28 x}-\frac{3}{56} \left (7+11 i \sqrt{7}\right ) \log (x)-\frac{3}{56} \left (7-11 i \sqrt{7}\right ) \log (x)+\frac{11 \left (9+5 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x-\sqrt{7}+i}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{11 \left (9-5 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x+\sqrt{7}+i}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35+i \sqrt{7}\right )}} \]

[Out]

-(35 - (9*I)*Sqrt[7])/(28*x) - (35 + (9*I)*Sqrt[7])/(28*x) + (11*(9 + (5*I)*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8
*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(4*Sqrt[14*(35 - I*Sqrt[7])]) - (11*(9 - (5*I)*Sqrt[7])*ArcTanh[(I + Sqrt[7]
+ (8*I)*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(4*Sqrt[14*(35 + I*Sqrt[7])]) - (3*(7 - (11*I)*Sqrt[7])*Log[x])/56 - (3
*(7 + (11*I)*Sqrt[7])*Log[x])/56 + (3*(7 + (11*I)*Sqrt[7])*Log[4*I + (I - Sqrt[7])*x + (4*I)*x^2])/112 + (3*(7
- (11*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7])*x + (4*I)*x^2])/112

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Rubi [A]  time = 0.467315, antiderivative size = 281, 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, 206, 628} \[ \frac{3}{112} \left (7+11 i \sqrt{7}\right ) \log \left (4 i x^2+\left (-\sqrt{7}+i\right ) x+4 i\right )+\frac{3}{112} \left (7-11 i \sqrt{7}\right ) \log \left (4 i x^2+\left (\sqrt{7}+i\right ) x+4 i\right )-\frac{35+9 i \sqrt{7}}{28 x}-\frac{35-9 i \sqrt{7}}{28 x}-\frac{3}{56} \left (7+11 i \sqrt{7}\right ) \log (x)-\frac{3}{56} \left (7-11 i \sqrt{7}\right ) \log (x)+\frac{11 \left (9+5 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x-\sqrt{7}+i}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{11 \left (9-5 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x+\sqrt{7}+i}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35+i \sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(35 - (9*I)*Sqrt[7])/(28*x) - (35 + (9*I)*Sqrt[7])/(28*x) + (11*(9 + (5*I)*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8
*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(4*Sqrt[14*(35 - I*Sqrt[7])]) - (11*(9 - (5*I)*Sqrt[7])*ArcTanh[(I + Sqrt[7]
+ (8*I)*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(4*Sqrt[14*(35 + I*Sqrt[7])]) - (3*(7 - (11*I)*Sqrt[7])*Log[x])/56 - (3
*(7 + (11*I)*Sqrt[7])*Log[x])/56 + (3*(7 + (11*I)*Sqrt[7])*Log[4*I + (I - Sqrt[7])*x + (4*I)*x^2])/112 + (3*(7
- (11*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7])*x + (4*I)*x^2])/112

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 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 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{5+x+3 x^2+2 x^3}{x^2 \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx &=\frac{i \int \frac{9-5 i \sqrt{7}+\left (10-2 i \sqrt{7}\right ) x}{x^2 \left (4+\left (1-i \sqrt{7}\right ) x+4 x^2\right )} \, dx}{\sqrt{7}}-\frac{i \int \frac{9+5 i \sqrt{7}+\left (10+2 i \sqrt{7}\right ) x}{x^2 \left (4+\left (1+i \sqrt{7}\right ) x+4 x^2\right )} \, dx}{\sqrt{7}}\\ &=-\frac{i \int \left (\frac{9+5 i \sqrt{7}}{4 x^2}+\frac{3 \left (11-i \sqrt{7}\right )}{8 x}+\frac{-7 \left (9 i-5 \sqrt{7}\right )-6 \left (11 i+\sqrt{7}\right ) x}{4 \left (4 i+\left (i-\sqrt{7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt{7}}+\frac{i \int \left (\frac{9-5 i \sqrt{7}}{4 x^2}+\frac{3 \left (11+i \sqrt{7}\right )}{8 x}+\frac{-7 \left (9 i+5 \sqrt{7}\right )-6 \left (11 i-\sqrt{7}\right ) x}{4 \left (4 i+\left (i+\sqrt{7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt{7}}\\ &=-\frac{35-9 i \sqrt{7}}{28 x}-\frac{35+9 i \sqrt{7}}{28 x}-\frac{3}{56} \left (7-11 i \sqrt{7}\right ) \log (x)-\frac{3}{56} \left (7+11 i \sqrt{7}\right ) \log (x)-\frac{i \int \frac{-7 \left (9 i-5 \sqrt{7}\right )-6 \left (11 i+\sqrt{7}\right ) x}{4 i+\left (i-\sqrt{7}\right ) x+4 i x^2} \, dx}{4 \sqrt{7}}+\frac{i \int \frac{-7 \left (9 i+5 \sqrt{7}\right )-6 \left (11 i-\sqrt{7}\right ) x}{4 i+\left (i+\sqrt{7}\right ) x+4 i x^2} \, dx}{4 \sqrt{7}}\\ &=-\frac{35-9 i \sqrt{7}}{28 x}-\frac{35+9 i \sqrt{7}}{28 x}-\frac{3}{56} \left (7-11 i \sqrt{7}\right ) \log (x)-\frac{3}{56} \left (7+11 i \sqrt{7}\right ) \log (x)-\frac{1}{56} \left (11 \left (35 i-9 \sqrt{7}\right )\right ) \int \frac{1}{4 i+\left (i+\sqrt{7}\right ) x+4 i x^2} \, dx+\frac{1}{112} \left (3 \left (7-11 i \sqrt{7}\right )\right ) \int \frac{i+\sqrt{7}+8 i x}{4 i+\left (i+\sqrt{7}\right ) x+4 i x^2} \, dx+\frac{1}{112} \left (3 \left (7+11 i \sqrt{7}\right )\right ) \int \frac{i-\sqrt{7}+8 i x}{4 i+\left (i-\sqrt{7}\right ) x+4 i x^2} \, dx-\frac{1}{56} \left (11 \left (35 i+9 \sqrt{7}\right )\right ) \int \frac{1}{4 i+\left (i-\sqrt{7}\right ) x+4 i x^2} \, dx\\ &=-\frac{35-9 i \sqrt{7}}{28 x}-\frac{35+9 i \sqrt{7}}{28 x}-\frac{3}{56} \left (7-11 i \sqrt{7}\right ) \log (x)-\frac{3}{56} \left (7+11 i \sqrt{7}\right ) \log (x)+\frac{3}{112} \left (7+11 i \sqrt{7}\right ) \log \left (4 i+\left (i-\sqrt{7}\right ) x+4 i x^2\right )+\frac{3}{112} \left (7-11 i \sqrt{7}\right ) \log \left (4 i+\left (i+\sqrt{7}\right ) x+4 i x^2\right )+\frac{1}{28} \left (11 \left (35 i-9 \sqrt{7}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (35+i \sqrt{7}\right )-x^2} \, dx,x,i+\sqrt{7}+8 i x\right )+\frac{1}{28} \left (11 \left (35 i+9 \sqrt{7}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (35-i \sqrt{7}\right )-x^2} \, dx,x,i-\sqrt{7}+8 i x\right )\\ &=-\frac{35-9 i \sqrt{7}}{28 x}-\frac{35+9 i \sqrt{7}}{28 x}+\frac{11 \left (9+5 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{i-\sqrt{7}+8 i x}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{11 \left (9-5 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{i+\sqrt{7}+8 i x}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35+i \sqrt{7}\right )}}-\frac{3}{56} \left (7-11 i \sqrt{7}\right ) \log (x)-\frac{3}{56} \left (7+11 i \sqrt{7}\right ) \log (x)+\frac{3}{112} \left (7+11 i \sqrt{7}\right ) \log \left (4 i+\left (i-\sqrt{7}\right ) x+4 i x^2\right )+\frac{3}{112} \left (7-11 i \sqrt{7}\right ) \log \left (4 i+\left (i+\sqrt{7}\right ) x+4 i x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0188368, size = 109, normalized size = 0.39 \[ \frac{1}{4} \text{RootSum}\left [2 \text{$\#$1}^4+\text{$\#$1}^3+5 \text{$\#$1}^2+\text{$\#$1}+2\& ,\frac{6 \text{$\#$1}^3 \log (x-\text{$\#$1})-17 \text{$\#$1}^2 \log (x-\text{$\#$1})+13 \text{$\#$1} \log (x-\text{$\#$1})-35 \log (x-\text{$\#$1})}{8 \text{$\#$1}^3+3 \text{$\#$1}^2+10 \text{$\#$1}+1}\& \right ]-\frac{5}{2 x}-\frac{3 \log (x)}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-5/(2*x) - (3*Log[x])/4 + RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (-35*Log[x - #1] + 13*Log[x - #1]*#1 - 1
7*Log[x - #1]*#1^2 + 6*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1^2 + 8*#1^3) & ]/4

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Maple [C]  time = 0.008, size = 72, normalized size = 0.3 \begin{align*} -{\frac{5}{2\,x}}-{\frac{3\,\ln \left ( x \right ) }{4}}+{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+5\,{{\it \_Z}}^{2}+{\it \_Z}+2 \right ) }{\frac{ \left ( 6\,{{\it \_R}}^{3}-17\,{{\it \_R}}^{2}+13\,{\it \_R}-35 \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((2*x^3+3*x^2+x+5)/x^2/(2*x^4+x^3+5*x^2+x+2),x)

[Out]

-5/2/x-3/4*ln(x)+1/4*sum((6*_R^3-17*_R^2+13*_R-35)/(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{5}{2 \, x} + \frac{1}{4} \, \int \frac{6 \, x^{3} - 17 \, x^{2} + 13 \, x - 35}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} - \frac{3}{4} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2/x + 1/4*integrate((6*x^3 - 17*x^2 + 13*x - 35)/(2*x^4 + x^3 + 5*x^2 + x + 2), x) - 3/4*log(x)

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Fricas [B]  time = 9.74885, size = 5700, normalized size = 20.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/224*(2*x*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21)*log(91924*(33/112*I*sqrt(7) - 1/2*sqrt
(2101/1568*I*sqrt(7) - 55/32) + 3/16)^3 - 49/4*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3
/16)^2*(-2211*I*sqrt(7) + 3752*sqrt(2101/1568*I*sqrt(7) - 55/32) - 3839) - 1/256*(210112*(33/112*I*sqrt(7) - 1
/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 46431*I*sqrt(7) + 78792*sqrt(2101/1568*I*sqrt(7) - 55/32) - 1
17483)*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21) - 68943*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1
568*I*sqrt(7) - 55/32) + 3/16)^2 + 15488*x + 61908*I*sqrt(7) - 105056*sqrt(2101/1568*I*sqrt(7) - 55/32) + 1234
28) + 2*x*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21)*log(-91924*(33/112*I*sqrt(7) - 1/2*sqrt(
2101/1568*I*sqrt(7) - 55/32) + 3/16)^3 + 98735*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/1
6)^2 + 15488*x - 146487/2*I*sqrt(7) + 124292*sqrt(2101/1568*I*sqrt(7) - 55/32) - 285347/2) + (4*sqrt(7)*sqrt(-
336*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 336*(-33/112*I*sqrt(7) - 1/2*sqrt(-2
101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 1/56*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21)*(-33*
I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) + 63) + 99/2*I*sqrt(7) - 84*sqrt(2101/1568*I*sqrt(7) - 55/32)
- 1859/2)*x - x*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21) - x*(-33*I*sqrt(7) + 56*sqrt(2101
/1568*I*sqrt(7) - 55/32) - 21) - 84*x)*log(49/4*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) +
3/16)^2*(-2211*I*sqrt(7) + 3752*sqrt(2101/1568*I*sqrt(7) - 55/32) - 3839) + 1/256*(210112*(33/112*I*sqrt(7) -
1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 46431*I*sqrt(7) + 78792*sqrt(2101/1568*I*sqrt(7) - 55/32) -
117483)*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21) - 29792*(33/112*I*sqrt(7) - 1/2*sqrt(2101/
1568*I*sqrt(7) - 55/32) + 3/16)^2 + 1/256*((67*sqrt(7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) -
21) - 2432*sqrt(7))*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21) - 2432*sqrt(7)*(-33*I*sqrt(7)
+ 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21) + 147456*sqrt(7))*sqrt(-336*(33/112*I*sqrt(7) - 1/2*sqrt(2101/15
68*I*sqrt(7) - 55/32) + 3/16)^2 - 336*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 -
1/56*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7)
- 55/32) + 63) + 99/2*I*sqrt(7) - 84*sqrt(2101/1568*I*sqrt(7) - 55/32) - 1859/2) + 30976*x + 22671/2*I*sqrt(7)
- 19236*sqrt(2101/1568*I*sqrt(7) - 55/32) + 53979/2) - (4*sqrt(7)*sqrt(-336*(33/112*I*sqrt(7) - 1/2*sqrt(2101
/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 336*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3/16)^2
- 1/56*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(
7) - 55/32) + 63) + 99/2*I*sqrt(7) - 84*sqrt(2101/1568*I*sqrt(7) - 55/32) - 1859/2)*x + x*(33*I*sqrt(7) + 56*s
qrt(-2101/1568*I*sqrt(7) - 55/32) - 21) + x*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21) + 84*x
)*log(49/4*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3/16)^2*(-2211*I*sqrt(7) + 3752*sqrt(
2101/1568*I*sqrt(7) - 55/32) - 3839) + 1/256*(210112*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32)
+ 3/16)^2 - 46431*I*sqrt(7) + 78792*sqrt(2101/1568*I*sqrt(7) - 55/32) - 117483)*(33*I*sqrt(7) + 56*sqrt(-2101
/1568*I*sqrt(7) - 55/32) - 21) - 29792*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 1
/256*((67*sqrt(7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21) - 2432*sqrt(7))*(33*I*sqrt(7) +
56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21) - 2432*sqrt(7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/3
2) - 21) + 147456*sqrt(7))*sqrt(-336*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 336
*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 1/56*(33*I*sqrt(7) + 56*sqrt(-2101/15
68*I*sqrt(7) - 55/32) - 21)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) + 63) + 99/2*I*sqrt(7) - 84*
sqrt(2101/1568*I*sqrt(7) - 55/32) - 1859/2) + 30976*x + 22671/2*I*sqrt(7) - 19236*sqrt(2101/1568*I*sqrt(7) - 5
5/32) + 53979/2) + 168*x*log(x) + 560)/x

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Sympy [A]  time = 2.41469, size = 65, normalized size = 0.23 \begin{align*} - \frac{3 \log{\left (x \right )}}{4} + \operatorname{RootSum}{\left (1372 t^{4} - 1029 t^{3} + 3136 t^{2} + 2688 t + 512, \left ( t \mapsto t \log{\left (- \frac{506797249 t^{4}}{34947704} + \frac{21584647 t^{3}}{4368463} - \frac{14969669687 t^{2}}{559163264} - \frac{282513301 t}{6354128} + x - \frac{101471979}{8736926} \right )} \right )\right )} - \frac{5}{2 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*log(x)/4 + RootSum(1372*_t**4 - 1029*_t**3 + 3136*_t**2 + 2688*_t + 512, Lambda(_t, _t*log(-506797249*_t**4
/34947704 + 21584647*_t**3/4368463 - 14969669687*_t**2/559163264 - 282513301*_t/6354128 + x - 101471979/873692
6))) - 5/(2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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