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

Optimal. Leaf size=245 \[ -\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (-\sqrt {7}+i\right ) x+4 i\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (\sqrt {7}+i\right ) x+4 i\right )+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {\left (53+i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {8 i x-\sqrt {7}+i}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {\left (53-i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {8 i x+\sqrt {7}+i}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}} \]

[Out]

1/28*ln(x)*(35-9*I*7^(1/2))-1/56*ln(4*I+4*I*x^2+x*(I-7^(1/2)))*(35-9*I*7^(1/2))+1/28*ln(x)*(35+9*I*7^(1/2))-1/
56*ln(4*I+4*I*x^2+x*(I+7^(1/2)))*(35+9*I*7^(1/2))-1/2*arctanh((I+8*I*x-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*arctanh((I+8*I*x+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 = 0.47, antiderivative size = 245, 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} \[ -\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (-\sqrt {7}+i\right ) x+4 i\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (\sqrt {7}+i\right ) x+4 i\right )+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {\left (53+i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {8 i x-\sqrt {7}+i}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {\left (53-i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {8 i x+\sqrt {7}+i}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \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 \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 \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}+\frac {3 \left (11 i+\sqrt {7}\right )-2 \left (9 i-5 \sqrt {7}\right ) x}{2 \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}+\frac {3 \left (11 i-\sqrt {7}\right )-2 \left (9 i+5 \sqrt {7}\right ) x}{2 \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt {7}}\\ &=\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {i \int \frac {3 \left (11 i+\sqrt {7}\right )-2 \left (9 i-5 \sqrt {7}\right ) x}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx}{2 \sqrt {7}}+\frac {i \int \frac {3 \left (11 i-\sqrt {7}\right )-2 \left (9 i+5 \sqrt {7}\right ) x}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx}{2 \sqrt {7}}\\ &=\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\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 (35+9 i \sqrt {7}\right ) \int \frac {i+\sqrt {7}+8 i x}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx-\frac {1}{28} \left (-7 i+53 \sqrt {7}\right ) \int \frac {1}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx+\frac {1}{28} \left (7 i+53 \sqrt {7}\right ) \int \frac {1}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx\\ &=\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{14} \left (7 i-53 \sqrt {7}\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}{14} \left (7 i+53 \sqrt {7}\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 {\left (53+i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i-\sqrt {7}+8 i 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 ) \tanh ^{-1}\left (\frac {i+\sqrt {7}+8 i x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{56} \left (35+9 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 = 101, normalized size = 0.41 \[ \frac {5 \log (x)}{2}-\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^4+\text {$\#$1}^3+5 \text {$\#$1}^2+\text {$\#$1}+2\& ,\frac {10 \text {$\#$1}^3 \log (x-\text {$\#$1})+\text {$\#$1}^2 \log (x-\text {$\#$1})+19 \text {$\#$1} \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{8 \text {$\#$1}^3+3 \text {$\#$1}^2+10 \text {$\#$1}+1}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 3.24, size = 1143, normalized size = 4.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35)*log(49/4*(27*I*sqrt(7) + 84*sqrt(-37/392*I*sqrt(
7) + 79/56) + 1385)*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 2058*(-9/56*I*sqrt(7) - 1/
2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^3 - 5145*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8
)^2 + 1/64*(2352*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 945*I*sqrt(7) - 2940*sqrt(-
37/392*I*sqrt(7) + 79/56) - 28507)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 8384*x + 1323/2*I
*sqrt(7) + 2058*sqrt(-37/392*I*sqrt(7) + 79/56) + 16089/2) + 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*(27*I*sqrt(7) + 84*sqrt(-37/392*I*sqrt(7) + 79/56) + 1385)*(9/56*I*sqrt(
7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 15680*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56
) - 5/8)^2 - 1/64*(2352*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 945*I*sqrt(7) - 2940
*sqrt(-37/392*I*sqrt(7) + 79/56) - 28507)*(-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*s
qrt(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*s
qrt(37/392*I*sqrt(7) + 79/56) + 35) + 45/14*I*sqrt(7) + 10*sqrt(-37/392*I*sqrt(7) + 79/56) + 11/2)*((27*I*sqrt
(7) + 84*sqrt(-37/392*I*sqrt(7) + 79/56) + 1385)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 115
20*I*sqrt(7) + 35840*sqrt(-37/392*I*sqrt(7) + 79/56) - 35072) + 16768*x + 3492*I*sqrt(7) + 10864*sqrt(-37/392*
I*sqrt(7) + 79/56) + 5484) - 1/8*(2*sqrt(-12*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1
2*(-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*sqr
t(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(-37/3
92*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*(27*I*sqrt(7) + 84*sqrt(-37/392*I*sqrt(7) + 79/56) + 1385)*(9/56*I*sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) +
79/56) - 5/8)^2 - 15680*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1/64*(2352*(-9/56*I
*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 945*I*sqrt(7) - 2940*sqrt(-37/392*I*sqrt(7) + 79/56)
- 28507)*(-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(3
7/392*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)*((27*I*sqrt(7) + 84*sqrt(-37/392*I*sqrt(7)
+ 79/56) + 1385)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 11520*I*sqrt(7) + 35840*sqrt(-37/39
2*I*sqrt(7) + 79/56) - 35072) + 16768*x + 3492*I*sqrt(7) + 10864*sqrt(-37/392*I*sqrt(7) + 79/56) + 5484) - 1/5
6*(9*I*sqrt(7) + 28*sqrt(-37/392*I*sqrt(7) + 79/56) + 35)*log(2058*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(
7) + 79/56) - 5/8)^3 + 20825*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 + 8384*x - 8307/2
*I*sqrt(7) - 12922*sqrt(-37/392*I*sqrt(7) + 79/56) - 18673/2) + 5/2*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3*x^2+x+5)/x/(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), x)

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maple [C]  time = 0.01, size = 67, normalized size = 0.27 \[ \frac {5 \ln \relax (x )}{2}+\frac {\left (-10 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}-19 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )-3\right ) \ln \left (-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+x \right )}{16 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}+6 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+20 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sum((-10*_R^3-_R^2-19*_R-3)/(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*ln(x
)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, \int \frac {10 \, x^{3} + x^{2} + 19 \, x + 3}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} + \frac {5}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.34, size = 237, normalized size = 0.97 \[ \frac {5\,\ln \relax (x)}{2}+\left (\sum _{k=1}^4\ln \left (\frac {223\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}{8}-\frac {31\,x}{2}+\frac {\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\,x\,71}{16}-\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2\,x\,4463}{64}+\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3\,x\,1449}{16}+\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^4\,x\,3675}{32}+\frac {257\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2}{32}+\frac {1673\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3}{64}-\frac {441\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^4}{32}+10\right )\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*log(x))/2 + symsum(log((223*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k))/8 - (31*x)/2 + (71*r
oot(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)*x)/16 - (4463*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/4
9 + 128/343, z, k)^2*x)/64 + (1449*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^3*x)/16 + (3675*r
oot(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^4*x)/32 + (257*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/
49 + 128/343, z, k)^2)/32 + (1673*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^3)/64 - (441*root(
z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k)^4)/32 + 10)*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 12
8/343, z, k), k, 1, 4)

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sympy [A]  time = 12.66, size = 60, normalized size = 0.24 \[ \frac {5 \log {\relax (x )}}{2} + \operatorname {RootSum} {\left (686 t^{4} + 1715 t^{3} + 1372 t^{2} + 448 t + 256, \left (t \mapsto t \log {\left (- \frac {160344611 t^{4}}{532759184} - \frac {16880402 t^{3}}{33297449} + \frac {4010520787 t^{2}}{2131036736} + \frac {1537535671 t}{532759184} + x + \frac {46660495}{66594898} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*log(x)/2 + RootSum(686*_t**4 + 1715*_t**3 + 1372*_t**2 + 448*_t + 256, Lambda(_t, _t*log(-160344611*_t**4/53
2759184 - 16880402*_t**3/33297449 + 4010520787*_t**2/2131036736 + 1537535671*_t/532759184 + x + 46660495/66594
898)))

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