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

Optimal. Leaf size=317 \[ -\frac{35+9 i \sqrt{7}}{56 x^2}-\frac{35-9 i \sqrt{7}}{56 x^2}+\frac{1}{32} \left (35-9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (-\sqrt{7}+i\right ) x+4 i\right )+\frac{1}{32} \left (35+9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (\sqrt{7}+i\right ) x+4 i\right )+\frac{3 \left (7+11 i \sqrt{7}\right )}{56 x}+\frac{3 \left (7-11 i \sqrt{7}\right )}{56 x}-\frac{1}{16} \left (35+9 i \sqrt{7}\right ) \log (x)-\frac{1}{16} \left (35-9 i \sqrt{7}\right ) \log (x)+\frac{\left (355-73 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x-\sqrt{7}+i}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{8 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{\left (355+73 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x+\sqrt{7}+i}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{8 \sqrt{14 \left (35+i \sqrt{7}\right )}} \]

[Out]

-(35 - (9*I)*Sqrt[7])/(56*x^2) - (35 + (9*I)*Sqrt[7])/(56*x^2) + (3*(7 - (11*I)*Sqrt[7]))/(56*x) + (3*(7 + (11
*I)*Sqrt[7]))/(56*x) + ((355 - (73*I)*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(8*S
qrt[14*(35 - I*Sqrt[7])]) - ((355 + (73*I)*Sqrt[7])*ArcTanh[(I + Sqrt[7] + (8*I)*x)/Sqrt[2*(35 + I*Sqrt[7])]])
/(8*Sqrt[14*(35 + I*Sqrt[7])]) - ((35 - (9*I)*Sqrt[7])*Log[x])/16 - ((35 + (9*I)*Sqrt[7])*Log[x])/16 + ((35 -
(9*I)*Sqrt[7])*Log[4*I + (I - Sqrt[7])*x + (4*I)*x^2])/32 + ((35 + (9*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7])*x +
(4*I)*x^2])/32

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Rubi [A]  time = 0.539418, antiderivative size = 317, 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} \[ -\frac{35+9 i \sqrt{7}}{56 x^2}-\frac{35-9 i \sqrt{7}}{56 x^2}+\frac{1}{32} \left (35-9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (-\sqrt{7}+i\right ) x+4 i\right )+\frac{1}{32} \left (35+9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (\sqrt{7}+i\right ) x+4 i\right )+\frac{3 \left (7+11 i \sqrt{7}\right )}{56 x}+\frac{3 \left (7-11 i \sqrt{7}\right )}{56 x}-\frac{1}{16} \left (35+9 i \sqrt{7}\right ) \log (x)-\frac{1}{16} \left (35-9 i \sqrt{7}\right ) \log (x)+\frac{\left (355-73 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x-\sqrt{7}+i}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{8 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{\left (355+73 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x+\sqrt{7}+i}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{8 \sqrt{14 \left (35+i \sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(35 - (9*I)*Sqrt[7])/(56*x^2) - (35 + (9*I)*Sqrt[7])/(56*x^2) + (3*(7 - (11*I)*Sqrt[7]))/(56*x) + (3*(7 + (11
*I)*Sqrt[7]))/(56*x) + ((355 - (73*I)*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(8*S
qrt[14*(35 - I*Sqrt[7])]) - ((355 + (73*I)*Sqrt[7])*ArcTanh[(I + Sqrt[7] + (8*I)*x)/Sqrt[2*(35 + I*Sqrt[7])]])
/(8*Sqrt[14*(35 + I*Sqrt[7])]) - ((35 - (9*I)*Sqrt[7])*Log[x])/16 - ((35 + (9*I)*Sqrt[7])*Log[x])/16 + ((35 -
(9*I)*Sqrt[7])*Log[4*I + (I - Sqrt[7])*x + (4*I)*x^2])/32 + ((35 + (9*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7])*x +
(4*I)*x^2])/32

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^3 \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^3 \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^3 \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^3}+\frac{3 \left (11-i \sqrt{7}\right )}{8 x^2}-\frac{7 i \left (-9 i+5 \sqrt{7}\right )}{16 x}+\frac{-223 i-61 \sqrt{7}+14 \left (9 i-5 \sqrt{7}\right ) x}{8 \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^3}+\frac{3 \left (11+i \sqrt{7}\right )}{8 x^2}+\frac{7 i \left (9 i+5 \sqrt{7}\right )}{16 x}+\frac{-223 i+61 \sqrt{7}+14 \left (9 i+5 \sqrt{7}\right ) x}{8 \left (4 i+\left (i+\sqrt{7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt{7}}\\ &=-\frac{35-9 i \sqrt{7}}{56 x^2}-\frac{35+9 i \sqrt{7}}{56 x^2}+\frac{3 \left (7-11 i \sqrt{7}\right )}{56 x}+\frac{3 \left (7+11 i \sqrt{7}\right )}{56 x}-\frac{1}{16} \left (35-9 i \sqrt{7}\right ) \log (x)-\frac{1}{16} \left (35+9 i \sqrt{7}\right ) \log (x)-\frac{i \int \frac{-223 i-61 \sqrt{7}+14 \left (9 i-5 \sqrt{7}\right ) x}{4 i+\left (i-\sqrt{7}\right ) x+4 i x^2} \, dx}{8 \sqrt{7}}+\frac{i \int \frac{-223 i+61 \sqrt{7}+14 \left (9 i+5 \sqrt{7}\right ) x}{4 i+\left (i+\sqrt{7}\right ) x+4 i x^2} \, dx}{8 \sqrt{7}}\\ &=-\frac{35-9 i \sqrt{7}}{56 x^2}-\frac{35+9 i \sqrt{7}}{56 x^2}+\frac{3 \left (7-11 i \sqrt{7}\right )}{56 x}+\frac{3 \left (7+11 i \sqrt{7}\right )}{56 x}-\frac{1}{16} \left (35-9 i \sqrt{7}\right ) \log (x)-\frac{1}{16} \left (35+9 i \sqrt{7}\right ) \log (x)+\frac{1}{112} \left (511 i-355 \sqrt{7}\right ) \int \frac{1}{4 i+\left (i-\sqrt{7}\right ) x+4 i x^2} \, dx-\frac{1}{32} \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}{32} \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}{112} \left (511 i+355 \sqrt{7}\right ) \int \frac{1}{4 i+\left (i+\sqrt{7}\right ) x+4 i x^2} \, dx\\ &=-\frac{35-9 i \sqrt{7}}{56 x^2}-\frac{35+9 i \sqrt{7}}{56 x^2}+\frac{3 \left (7-11 i \sqrt{7}\right )}{56 x}+\frac{3 \left (7+11 i \sqrt{7}\right )}{56 x}-\frac{1}{16} \left (35-9 i \sqrt{7}\right ) \log (x)-\frac{1}{16} \left (35+9 i \sqrt{7}\right ) \log (x)+\frac{1}{32} \left (35-9 i \sqrt{7}\right ) \log \left (4 i+\left (i-\sqrt{7}\right ) x+4 i x^2\right )+\frac{1}{32} \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 (-511 i+355 \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}{56} \left (511 i+355 \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{35-9 i \sqrt{7}}{56 x^2}-\frac{35+9 i \sqrt{7}}{56 x^2}+\frac{3 \left (7-11 i \sqrt{7}\right )}{56 x}+\frac{3 \left (7+11 i \sqrt{7}\right )}{56 x}+\frac{\left (355-73 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{i-\sqrt{7}+8 i x}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{8 \sqrt{14 \left (35-i \sqrt{7}\right )}}-\frac{\left (355+73 i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{i+\sqrt{7}+8 i x}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{8 \sqrt{14 \left (35+i \sqrt{7}\right )}}-\frac{1}{16} \left (35-9 i \sqrt{7}\right ) \log (x)-\frac{1}{16} \left (35+9 i \sqrt{7}\right ) \log (x)+\frac{1}{32} \left (35-9 i \sqrt{7}\right ) \log \left (4 i+\left (i-\sqrt{7}\right ) x+4 i x^2\right )+\frac{1}{32} \left (35+9 i \sqrt{7}\right ) \log \left (4 i+\left (i+\sqrt{7}\right ) x+4 i x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0179647, size = 116, normalized size = 0.37 \[ \frac{1}{8} \text{RootSum}\left [2 \text{$\#$1}^4+\text{$\#$1}^3+5 \text{$\#$1}^2+\text{$\#$1}+2\& ,\frac{70 \text{$\#$1}^3 \log (x-\text{$\#$1})+47 \text{$\#$1}^2 \log (x-\text{$\#$1})+141 \text{$\#$1} \log (x-\text{$\#$1})+61 \log (x-\text{$\#$1})}{8 \text{$\#$1}^3+3 \text{$\#$1}^2+10 \text{$\#$1}+1}\& \right ]-\frac{5}{4 x^2}+\frac{3}{4 x}-\frac{35 \log (x)}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-5/(4*x^2) + 3/(4*x) - (35*Log[x])/8 + RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (61*Log[x - #1] + 141*Log[x
- #1]*#1 + 47*Log[x - #1]*#1^2 + 70*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1^2 + 8*#1^3) & ]/8

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Maple [C]  time = 0.008, size = 77, normalized size = 0.2 \begin{align*} -{\frac{5}{4\,{x}^{2}}}+{\frac{3}{4\,x}}-{\frac{35\,\ln \left ( x \right ) }{8}}+{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+5\,{{\it \_Z}}^{2}+{\it \_Z}+2 \right ) }{\frac{ \left ( 70\,{{\it \_R}}^{3}+47\,{{\it \_R}}^{2}+141\,{\it \_R}+61 \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^3/(2*x^4+x^3+5*x^2+x+2),x)

[Out]

-5/4/x^2+3/4/x-35/8*ln(x)+1/8*sum((70*_R^3+47*_R^2+141*_R+61)/(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{3 \, x - 5}{4 \, x^{2}} + \frac{1}{8} \, \int \frac{70 \, x^{3} + 47 \, x^{2} + 141 \, x + 61}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} - \frac{35}{8} \, \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^3/(2*x^4+x^3+5*x^2+x+2),x, algorithm="maxima")

[Out]

1/4*(3*x - 5)/x^2 + 1/8*integrate((70*x^3 + 47*x^2 + 141*x + 61)/(2*x^4 + x^3 + 5*x^2 + x + 2), x) - 35/8*log(
x)

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Fricas [B]  time = 9.77968, size = 6164, normalized size = 19.44 \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^3/(2*x^4+x^3+5*x^2+x+2),x, algorithm="fricas")

[Out]

-1/448*(14*x^2*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35)*log(-49/4*(207711*I*sqrt(7) + 369
264*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 957269)*(9/32*I*sqrt(7) - 1/2*sqrt(9803/6272*I*sqrt(7) + 2815/896)
+ 35/32)^2 + 9046968*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^3 - 39580485*(-9/3
2*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 21/1024*(13785856*(-9/32*I*sqrt(7) - 1/2*
sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 + 16963065*I*sqrt(7) + 30156560*sqrt(-9803/6272*I*sqrt(7) + 2
815/896) - 68488563)*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) + 9662336*x - 68336919/4*I*
sqrt(7) - 30371964*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 257023549/4) + 14*x^2*(9*I*sqrt(7) + 16*sqrt(-9803/
6272*I*sqrt(7) + 2815/896) - 35)*log(-9046968*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 3
5/32)^3 + 41411909*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 + 9662336*x + 70198
191/4*I*sqrt(7) + 31199196*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 240366533/4) + 1960*x^2*log(x) + (4*sqrt(7)
*sqrt(-1344*(9/32*I*sqrt(7) - 1/2*sqrt(9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 1344*(-9/32*I*sqrt(7) - 1/
2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 7/8*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/89
6) + 105)*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 2205/2*I*sqrt(7) - 1960*sqrt(-9803/6
272*I*sqrt(7) + 2815/896) + 1661/2)*x^2 - 7*x^2*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 35)
- 7*x^2*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 980*x^2)*log(49/4*(207711*I*sqrt(7) +
369264*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 957269)*(9/32*I*sqrt(7) - 1/2*sqrt(9803/6272*I*sqrt(7) + 2815/8
96) + 35/32)^2 - 1831424*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 + 21/1024*(13
785856*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 + 16963065*I*sqrt(7) + 30156560
*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 68488563)*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 3
5) + 1/1024*sqrt(-1344*(9/32*I*sqrt(7) - 1/2*sqrt(9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 1344*(-9/32*I*s
qrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 7/8*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7
) + 2815/896) + 105)*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 2205/2*I*sqrt(7) - 1960*s
qrt(-9803/6272*I*sqrt(7) + 2815/896) + 1661/2)*(7*(23079*sqrt(7)*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) +
2815/896) - 35) - 149504*sqrt(7))*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 1046528*sqr
t(7)*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 35) - 116260864*sqrt(7)) + 19324672*x - 465318*
I*sqrt(7) - 827232*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 666914) - (4*sqrt(7)*sqrt(-1344*(9/32*I*sqrt(7) - 1
/2*sqrt(9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 1344*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2
815/896) + 35/32)^2 - 7/8*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 105)*(-9*I*sqrt(7) + 16*sq
rt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 2205/2*I*sqrt(7) - 1960*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 166
1/2)*x^2 + 7*x^2*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 35) + 7*x^2*(-9*I*sqrt(7) + 16*sqrt
(9803/6272*I*sqrt(7) + 2815/896) - 35) + 980*x^2)*log(49/4*(207711*I*sqrt(7) + 369264*sqrt(-9803/6272*I*sqrt(7
) + 2815/896) - 957269)*(9/32*I*sqrt(7) - 1/2*sqrt(9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 1831424*(-9/32
*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 + 21/1024*(13785856*(-9/32*I*sqrt(7) - 1/2*s
qrt(-9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 + 16963065*I*sqrt(7) + 30156560*sqrt(-9803/6272*I*sqrt(7) + 28
15/896) - 68488563)*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 1/1024*sqrt(-1344*(9/32*I*
sqrt(7) - 1/2*sqrt(9803/6272*I*sqrt(7) + 2815/896) + 35/32)^2 - 1344*(-9/32*I*sqrt(7) - 1/2*sqrt(-9803/6272*I*
sqrt(7) + 2815/896) + 35/32)^2 - 7/8*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/896) + 105)*(-9*I*sqrt
(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 2205/2*I*sqrt(7) - 1960*sqrt(-9803/6272*I*sqrt(7) + 2815
/896) + 1661/2)*(7*(23079*sqrt(7)*(9*I*sqrt(7) + 16*sqrt(-9803/6272*I*sqrt(7) + 2815/896) - 35) - 149504*sqrt(
7))*(-9*I*sqrt(7) + 16*sqrt(9803/6272*I*sqrt(7) + 2815/896) - 35) - 1046528*sqrt(7)*(9*I*sqrt(7) + 16*sqrt(-98
03/6272*I*sqrt(7) + 2815/896) - 35) - 116260864*sqrt(7)) + 19324672*x - 465318*I*sqrt(7) - 827232*sqrt(-9803/6
272*I*sqrt(7) + 2815/896) + 666914) - 336*x + 560)/x^2

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Sympy [A]  time = 1.67274, size = 70, normalized size = 0.22 \begin{align*} - \frac{35 \log{\left (x \right )}}{8} + \operatorname{RootSum}{\left (2744 t^{4} - 12005 t^{3} + 18424 t^{2} - 3136 t + 1024, \left ( t \mapsto t \log{\left (- \frac{20101387287723 t^{4}}{91907904361586} + \frac{944515214496 t^{3}}{45953952180793} + \frac{16572327093911939 t^{2}}{5882105879141504} - \frac{4564471749800865 t}{735263234892688} + x + \frac{70084064010625}{91907904361586} \right )} \right )\right )} + \frac{3 x - 5}{4 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35*log(x)/8 + RootSum(2744*_t**4 - 12005*_t**3 + 18424*_t**2 - 3136*_t + 1024, Lambda(_t, _t*log(-20101387287
723*_t**4/91907904361586 + 944515214496*_t**3/45953952180793 + 16572327093911939*_t**2/5882105879141504 - 4564
471749800865*_t/735263234892688 + x + 70084064010625/91907904361586))) + (3*x - 5)/(4*x**2)

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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^{3}}\,{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^3/(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^3), x)