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]

1/28*(-35+9*I*7^(1/2))/x+1/28*(-35-9*I*7^(1/2))/x-3/56*ln(x)*(7-11*I*7^(1/2))+3/112*ln(4*I+4*I*x^2+x*(I+7^(1/2
)))*(7-11*I*7^(1/2))-3/56*ln(x)*(7+11*I*7^(1/2))+3/112*ln(4*I+4*I*x^2+x*(I-7^(1/2)))*(7+11*I*7^(1/2))+11/4*arc
tanh((I+8*I*x-7^(1/2))/(70-2*I*7^(1/2))^(1/2))*(9+5*I*7^(1/2))/(490-14*I*7^(1/2))^(1/2)-11/4*arctanh((I+8*I*x+
7^(1/2))/(70+2*I*7^(1/2))^(1/2))*(9-5*I*7^(1/2))/(490+14*I*7^(1/2))^(1/2)

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Rubi [A]  time = 0.47, antiderivative size = 281, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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 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]

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*}

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Mathematica [C]  time = 0.02, 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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fricas [B]  time = 3.27, size = 1245, normalized size = 4.43 \[ \text {result too large to display} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, x^{3} + 3 \, x^{2} + x + 5}{{\left (2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2\right )} x^{2}}\,{d x} \]

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)

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maple [C]  time = 0.01, size = 72, normalized size = 0.26 \[ -\frac {3 \ln \relax (x )}{4}+\frac {\left (6 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}-17 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+13 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )-35\right ) \ln \left (-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+x \right )}{32 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}+12 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+40 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+4}-\frac {5}{2 x} \]

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]

1/4*sum((6*_R^3-17*_R^2+13*_R-35)/(8*_R^3+3*_R^2+10*_R+1)*ln(-_R+x),_R=RootOf(2*_Z^4+_Z^3+5*_Z^2+_Z+2))-5/2/x-
3/4*ln(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {5}{2 \, x} + \frac {1}{4} \, \int \frac {6 \, x^{3} - 17 \, x^{2} + 13 \, x - 35}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} - \frac {3}{4} \, \log \relax (x) \]

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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mupad [B]  time = 2.30, size = 242, normalized size = 0.86 \[ \left (\sum _{k=1}^4\ln \left (\frac {1199\,\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}{32}+25\,x+\frac {\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,x\,4169}{32}+\frac {{\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}^2\,x\,43993}{256}+{\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}^3\,x\,28+\frac {{\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}^4\,x\,3675}{32}+\frac {11647\,{\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}^2}{128}+\frac {7273\,{\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}^3}{128}-\frac {441\,{\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )}^4}{32}+\frac {21}{4}\right )\,\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\right )-\frac {3\,\ln \relax (x)}{4}-\frac {5}{2\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((1199*root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k))/32 + 25*x + (4169*root(z^4 -
(3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)*x)/32 + (43993*root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/
49 + 128/343, z, k)^2*x)/256 + 28*root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)^3*x + (3675*r
oot(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)^4*x)/32 + (11647*root(z^4 - (3*z^3)/4 + (16*z^2)
/7 + (96*z)/49 + 128/343, z, k)^2)/128 + (7273*root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)^
3)/128 - (441*root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)^4)/32 + 21/4)*root(z^4 - (3*z^3)/
4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k), k, 1, 4) - (3*log(x))/4 - 5/(2*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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