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]

1/14*x*(7-5*I*7^(1/2))+1/28*x^2*(7-5*I*7^(1/2))+1/14*x*(7+5*I*7^(1/2))+1/28*x^2*(7+5*I*7^(1/2))-1/56*ln(4+4*x^
2+x*(1+I*7^(1/2)))*(35-9*I*7^(1/2))-1/56*ln(4+4*x^2+x*(1-I*7^(1/2)))*(35+9*I*7^(1/2))+1/2*arctan((1+8*x+I*7^(1
/2))/(70-2*I*7^(1/2))^(1/2))*(53*I-7^(1/2))/(490-14*I*7^(1/2))^(1/2)-1/2*arctan((1+8*x-I*7^(1/2))/(70+2*I*7^(1
/2))^(1/2))*(53*I+7^(1/2))/(490+14*I*7^(1/2))^(1/2)

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Rubi [A]  time = 0.39, antiderivative size = 269, 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, 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 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 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 {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*}

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Mathematica [C]  time = 0.02, 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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fricas [B]  time = 3.03, size = 1145, normalized size = 4.26 \[ \text {result too large to display} \]

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

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)

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maple [C]  time = 0.01, size = 67, normalized size = 0.25 \[ \frac {x^{2}}{2}+x +\frac {\left (-5 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}-3 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )-2\right ) \ln \left (-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+x \right )}{8 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}+3 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+10 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+1} \]

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(-_R+x),_R=RootOf(2*_Z^4+_Z^3+5*_Z^2+_Z+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} + x - \int \frac {5 \, x^{3} + x^{2} + 3 \, x + 2}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]

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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mupad [B]  time = 0.13, size = 188, normalized size = 0.70 \[ x+\frac {x^2}{2}+\left (\sum _{k=1}^4\ln \left (-\frac {179\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}{8}-7\,x-\frac {\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\,x\,459}{8}-\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2\,x\,665}{8}-\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3\,x\,147}{4}-\frac {35\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2}{32}+\frac {49\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3}{16}-15\right )\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + x^2/2 + symsum(log((49*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^3)/16 - 7*x - (459*root(z
^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)*x)/8 - (665*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128
/343, z, k)^2*x)/8 - (147*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^3*x)/4 - (35*root(z^4 + (5
*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^2)/32 - (179*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343,
z, k))/8 - 15)*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k), k, 1, 4)

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sympy [B]  time = 2.73, size = 3662, normalized size = 13.61 \[ \text {result too large to display} \]

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 + (-5/8 + sqrt(79/448 + sqrt(77)/49))*log(x**2 + x*(-1459*sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)
) + 21975 + 7648*sqrt(77))/536576 - 15*sqrt(77)*sqrt(553 + 64*sqrt(77))/2096 - 10391*sqrt(553 + 64*sqrt(77))/2
68288 + 1459*sqrt(77)/8384 + 522933/268288 + 45*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(
77)) + 21975 + 7648*sqrt(77))/536576) - 510895297*sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sq
rt(77))/71978450944 - 6009493*sqrt(22)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/1124663296 -
38714551*sqrt(77)*sqrt(553 + 64*sqrt(77))/2249326592 - 4417610843*sqrt(553 + 64*sqrt(77))/35989225472 + 15319
5*sqrt(22)*sqrt(553 + 64*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/2249326592 + 831
3499*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/71978450944 +
290832444193/35989225472 + 2303470247*sqrt(77)/2249326592) + (-5/8 - sqrt(79/448 + sqrt(77)/49))*log(x**2 + x
*(-45*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/536576 - 1459
*sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/536576 + 10391*sqrt(553 + 64*sqrt(77))/268
288 + 1459*sqrt(77)/8384 + 522933/268288 + 15*sqrt(77)*sqrt(553 + 64*sqrt(77))/2096) - 510895297*sqrt(14)*sqrt
(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/71978450944 - 6009493*sqrt(22)*sqrt(333*sqrt(553 + 64*sq
rt(77)) + 21975 + 7648*sqrt(77))/1124663296 - 8313499*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(333*sqrt(553 + 64*
sqrt(77)) + 21975 + 7648*sqrt(77))/71978450944 - 153195*sqrt(22)*sqrt(553 + 64*sqrt(77))*sqrt(333*sqrt(553 + 6
4*sqrt(77)) + 21975 + 7648*sqrt(77))/2249326592 + 4417610843*sqrt(553 + 64*sqrt(77))/35989225472 + 38714551*sq
rt(77)*sqrt(553 + 64*sqrt(77))/2249326592 + 290832444193/35989225472 + 2303470247*sqrt(77)/2249326592) + 2*sqr
t(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/1568 + 5/14 + 3*sqrt(77)/49)*atan(107315
2*x/(4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)
) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*s
qrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)
*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))) - 45*sqrt(14)*sqrt(553 + 64*s
qrt(77))*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/(4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(333*sqrt(
553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(333*sqrt(553
+ 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*
sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 764
8*sqrt(77)) + 560 + 96*sqrt(77))) - 1459*sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/(4
313*sqrt(2)*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30
*sqrt(7)*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(33
3*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(
333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))) + 20782*sqrt(553 + 64*sqrt(77))/(43
13*sqrt(2)*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*
sqrt(7)*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(333
*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(3
33*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))) + 93376*sqrt(77)/(4313*sqrt(2)*sqrt(
-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sq
rt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(333*sqrt(553 + 64*s
qrt(77)) + 21975 + 7648*sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64
*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))) + 1045866/(4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(333*sqrt
(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(333*sqrt(55
3 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648
*sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 76
48*sqrt(77)) + 560 + 96*sqrt(77))) + 3840*sqrt(77)*sqrt(553 + 64*sqrt(77))/(4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(3
33*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(333*
sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975
+ 7648*sqrt(77)) + 1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(333*sqrt(553 + 64*sqrt(77)) + 219
75 + 7648*sqrt(77)) + 560 + 96*sqrt(77)))) + 2*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648
*sqrt(77))/1568 + 5/14 + 3*sqrt(77)/49)*atan(1073152*x/(-1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*s
qrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*s
qrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqr
t(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77))
+ 21975 + 7648*sqrt(77))) + 45*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7
648*sqrt(77))/(-1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975
+ 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975
+ 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 +
7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))) + 1045866/(-14
59*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) +
560 + 96*sqrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) +
560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 5
60 + 96*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))) + 93376*sqrt(77)/(-1459*sqrt(2)*
sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*s
qrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*s
qrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqr
t(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))) - 20782*sqrt(553 + 64*sqrt(77))/(-1459*sqrt
(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 +
96*sqrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 +
96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96
*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))) - 3840*sqrt(77)*sqrt(553 + 64*sqrt(77))
/(-1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(7
7)) + 560 + 96*sqrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(7
7)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)
) + 560 + 96*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))) - 1459*sqrt(14)*sqrt(-333*s
qrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/(-1459*sqrt(2)*sqrt(553 + 64*sqrt(77))*sqrt(-sqrt(14)*sqrt(-33
3*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 4313*sqrt(2)*sqrt(-sqrt(14)*sqrt(-33
3*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77)) + 30*sqrt(7)*sqrt(-sqrt(14)*sqrt(-333*
sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77)) + 560 + 96*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 2197
5 + 7648*sqrt(77))))

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