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

Optimal. Leaf size=307 \[ \frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4 x^2+\left (1-i \sqrt {7}\right ) x+4\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4 x^2+\left (1+i \sqrt {7}\right ) x+4\right )-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x+\frac {11 \left (5 \sqrt {7}+9 i\right ) \tan ^{-1}\left (\frac {8 x-i \sqrt {7}+1}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {11 \left (-5 \sqrt {7}+9 i\right ) \tan ^{-1}\left (\frac {8 x+i \sqrt {7}+1}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35-i \sqrt {7}\right )}} \]

[Out]

1/28*x^2*(7-5*I*7^(1/2))+1/42*x^3*(7-5*I*7^(1/2))+1/28*x^2*(7+5*I*7^(1/2))+1/42*x^3*(7+5*I*7^(1/2))-1/28*x*(35
-9*I*7^(1/2))-1/28*x*(35+9*I*7^(1/2))+3/112*ln(4+4*x^2+x*(1-I*7^(1/2)))*(7-11*I*7^(1/2))+3/112*ln(4+4*x^2+x*(1
+I*7^(1/2)))*(7+11*I*7^(1/2))-11/4*arctan((1+8*x+I*7^(1/2))/(70-2*I*7^(1/2))^(1/2))*(9*I-5*7^(1/2))/(490-14*I*
7^(1/2))^(1/2)+11/4*arctan((1+8*x-I*7^(1/2))/(70+2*I*7^(1/2))^(1/2))*(9*I+5*7^(1/2))/(490+14*I*7^(1/2))^(1/2)

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Rubi [A]  time = 0.58, antiderivative size = 307, 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} \[ \frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4 x^2+\left (1-i \sqrt {7}\right ) x+4\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4 x^2+\left (1+i \sqrt {7}\right ) x+4\right )-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x+\frac {11 \left (5 \sqrt {7}+9 i\right ) \tan ^{-1}\left (\frac {8 x-i \sqrt {7}+1}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {11 \left (-5 \sqrt {7}+9 i\right ) \tan ^{-1}\left (\frac {8 x+i \sqrt {7}+1}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35-i \sqrt {7}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((35 - (9*I)*Sqrt[7])*x)/28 - ((35 + (9*I)*Sqrt[7])*x)/28 + ((7 - (5*I)*Sqrt[7])*x^2)/28 + ((7 + (5*I)*Sqrt[7
])*x^2)/28 + ((7 - (5*I)*Sqrt[7])*x^3)/42 + ((7 + (5*I)*Sqrt[7])*x^3)/42 + (11*(9*I + 5*Sqrt[7])*ArcTan[(1 - I
*Sqrt[7] + 8*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(4*Sqrt[14*(35 + I*Sqrt[7])]) - (11*(9*I - 5*Sqrt[7])*ArcTan[(1 + I
*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(4*Sqrt[14*(35 - I*Sqrt[7])]) + (3*(7 - (11*I)*Sqrt[7])*Log[4 + (1
- I*Sqrt[7])*x + 4*x^2])/112 + (3*(7 + (11*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])*x + 4*x^2])/112

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^3 \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^3 \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^3 \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}{4} \left (-9+5 i \sqrt {7}\right )+\frac {1}{2} \left (5-i \sqrt {7}\right ) x+\frac {1}{2} \left (5-i \sqrt {7}\right ) x^2+\frac {2 \left (9-5 i \sqrt {7}\right )-3 \left (11+i \sqrt {7}\right ) x}{2 \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )}\right ) \, dx}{\sqrt {7}}-\frac {i \int \left (\frac {1}{4} \left (-9-5 i \sqrt {7}\right )+\frac {1}{2} \left (5+i \sqrt {7}\right ) x+\frac {1}{2} \left (5+i \sqrt {7}\right ) x^2+\frac {2 \left (9+5 i \sqrt {7}\right )-3 \left (11-i \sqrt {7}\right ) x}{2 \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )}\right ) \, dx}{\sqrt {7}}\\ &=-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 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}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {i \int \frac {2 \left (9-5 i \sqrt {7}\right )-3 \left (11+i \sqrt {7}\right ) x}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx}{2 \sqrt {7}}-\frac {i \int \frac {2 \left (9+5 i \sqrt {7}\right )-3 \left (11-i \sqrt {7}\right ) x}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx}{2 \sqrt {7}}\\ &=-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 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}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {1}{56} \left (11 \left (35-9 i \sqrt {7}\right )\right ) \int \frac {1}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{56} \left (11 \left (35+9 i \sqrt {7}\right )\right ) \int \frac {1}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{112} \left (3 \left (7-11 i \sqrt {7}\right )\right ) \int \frac {1-i \sqrt {7}+8 x}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{112} \left (3 \left (7+11 i \sqrt {7}\right )\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 (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 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}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )-\frac {1}{28} \left (11 \left (35-9 i \sqrt {7}\right )\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}{28} \left (11 \left (35+9 i \sqrt {7}\right )\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}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 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}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {11 \left (9 i+5 \sqrt {7}\right ) \tan ^{-1}\left (\frac {1-i \sqrt {7}+8 x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {11 \left (9 i-5 \sqrt {7}\right ) \tan ^{-1}\left (\frac {1+i \sqrt {7}+8 x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {3}{112} \left (7+11 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 = 109, normalized size = 0.36 \[ \frac {1}{6} \left (3 \text {RootSum}\left [2 \text {$\#$1}^4+\text {$\#$1}^3+5 \text {$\#$1}^2+\text {$\#$1}+2\& ,\frac {3 \text {$\#$1}^3 \log (x-\text {$\#$1})+19 \text {$\#$1}^2 \log (x-\text {$\#$1})+\text {$\#$1} \log (x-\text {$\#$1})+10 \log (x-\text {$\#$1})}{8 \text {$\#$1}^3+3 \text {$\#$1}^2+10 \text {$\#$1}+1}\& \right ]+x \left (2 x^2+3 x-15\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-15 + 3*x + 2*x^2) + 3*RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (10*Log[x - #1] + Log[x - #1]*#1 + 19*L
og[x - #1]*#1^2 + 3*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1^2 + 8*#1^3) & ])/6

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fricas [B]  time = 2.94, size = 1202, normalized size = 3.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 1/2*x^2 - 1/112*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21)*log(23324*(33/112*I*sqrt
(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^3 - 23765*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7)
- 55/32) + 3/16)^2 + 7744*x + 19470*I*sqrt(7) - 33040*sqrt(2101/1568*I*sqrt(7) - 55/32) + 38950) - 1/112*(33*
I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21)*log(-23324*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sq
rt(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*(-561*I*
sqrt(7) + 952*sqrt(2101/1568*I*sqrt(7) - 55/32) - 869) + 1/256*(53312*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I
*sqrt(7) - 55/32) + 3/16)^2 - 11781*I*sqrt(7) + 19992*sqrt(2101/1568*I*sqrt(7) - 55/32) - 36681)*(33*I*sqrt(7)
+ 56*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 21) + 17493*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/3
2) + 3/16)^2 + 7744*x - 15708*I*sqrt(7) + 26656*sqrt(2101/1568*I*sqrt(7) - 55/32) - 29132) + 1/112*(2*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*s
qrt(-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) + 28*sqrt(2101/1568*I*sqrt(7) - 55/32) + 28*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 21)*log(-49/
4*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3/16)^2*(-561*I*sqrt(7) + 952*sqrt(2101/1568*I
*sqrt(7) - 55/32) - 869) - 1/256*(53312*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 -
11781*I*sqrt(7) + 19992*sqrt(2101/1568*I*sqrt(7) - 55/32) - 36681)*(33*I*sqrt(7) + 56*sqrt(-2101/1568*I*sqrt(7
) - 55/32) - 21) + 6272*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 + 1/256*((17*sqrt(
7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21) - 512*sqrt(7))*(33*I*sqrt(7) + 56*sqrt(-2101/15
68*I*sqrt(7) - 55/32) - 21) - 512*sqrt(7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21) + 73728*
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*s
qrt(7) - 55/32) - 1859/2) + 15488*x - 3762*I*sqrt(7) + 6384*sqrt(2101/1568*I*sqrt(7) - 55/32) - 5946) - 1/112*
(2*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) - 5
5/32) - 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) - 28*sqrt(2101/1568*I*sqrt(7) - 55/32) - 28*sqrt(-2101/1568*I*sqrt(7) - 55/32) - 2
1)*log(-49/4*(-33/112*I*sqrt(7) - 1/2*sqrt(-2101/1568*I*sqrt(7) - 55/32) + 3/16)^2*(-561*I*sqrt(7) + 952*sqrt(
2101/1568*I*sqrt(7) - 55/32) - 869) - 1/256*(53312*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) +
3/16)^2 - 11781*I*sqrt(7) + 19992*sqrt(2101/1568*I*sqrt(7) - 55/32) - 36681)*(33*I*sqrt(7) + 56*sqrt(-2101/15
68*I*sqrt(7) - 55/32) - 21) + 6272*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 1/256
*((17*sqrt(7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 21) - 512*sqrt(7))*(33*I*sqrt(7) + 56*sq
rt(-2101/1568*I*sqrt(7) - 55/32) - 21) - 512*sqrt(7)*(-33*I*sqrt(7) + 56*sqrt(2101/1568*I*sqrt(7) - 55/32) - 2
1) + 73728*sqrt(7))*sqrt(-336*(33/112*I*sqrt(7) - 1/2*sqrt(2101/1568*I*sqrt(7) - 55/32) + 3/16)^2 - 336*(-33/1
12*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*sq
rt(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(21
01/1568*I*sqrt(7) - 55/32) - 1859/2) + 15488*x - 3762*I*sqrt(7) + 6384*sqrt(2101/1568*I*sqrt(7) - 55/32) - 594
6) - 5/2*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [C]  time = 0.01, size = 74, normalized size = 0.24 \[ \frac {x^{3}}{3}+\frac {x^{2}}{2}-\frac {5 x}{2}+\frac {\left (3 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}+19 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+10\right ) \ln \left (-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+x \right )}{16 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}+6 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+20 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3+1/2*x^2-5/2*x+1/2*sum((3*_R^3+19*_R^2+_R+10)/(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 \[ \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} - \frac {5}{2} \, x + \frac {1}{2} \, \int \frac {3 \, x^{3} + 19 \, x^{2} + x + 10}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 1/2*x^2 - 5/2*x + 1/2*integrate((3*x^3 + 19*x^2 + x + 10)/(2*x^4 + x^3 + 5*x^2 + x + 2), x)

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mupad [B]  time = 2.17, size = 128, normalized size = 0.42 \[ \left (\sum _{k=1}^4\ln \left (-29\,x+\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,\left (-\frac {289\,x}{4}+\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,\left (\frac {581\,x}{16}-\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,\left (\frac {147\,x}{4}-\frac {49}{16}\right )+\frac {1141}{64}\right )+\frac {47}{4}\right )+7\right )\,\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\right )-\frac {5\,x}{2}+\frac {x^2}{2}+\frac {x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)*(root(z^4 - (3*z^3)/4 + (16*z^2)/7 +
(96*z)/49 + 128/343, z, k)*((581*x)/16 - root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/343, z, k)*((147
*x)/4 - 49/16) + 1141/64) - (289*x)/4 + 47/4) - 29*x + 7)*root(z^4 - (3*z^3)/4 + (16*z^2)/7 + (96*z)/49 + 128/
343, z, k), k, 1, 4) - (5*x)/2 + x^2/2 + x^3/3

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sympy [A]  time = 0.99, size = 61, normalized size = 0.20 \[ \frac {x^{3}}{3} + \frac {x^{2}}{2} - \frac {5 x}{2} + \operatorname {RootSum} {\left (1372 t^{4} - 1029 t^{3} + 3136 t^{2} + 2688 t + 512, \left (t \mapsto t \log {\left (\frac {5831 t^{3}}{1936} - \frac {23765 t^{2}}{7744} + \frac {2065 t}{242} + x + \frac {415}{121} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 + x**2/2 - 5*x/2 + RootSum(1372*_t**4 - 1029*_t**3 + 3136*_t**2 + 2688*_t + 512, Lambda(_t, _t*log(5831
*_t**3/1936 - 23765*_t**2/7744 + 2065*_t/242 + x + 415/121)))

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