3.30.51 \(\int \frac {x+\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}}{1-\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}} \, dx\)

Optimal. Leaf size=354 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3+\text {$\#$1}^2+\text {$\#$1}+4\& ,\frac {\text {$\#$1}^3 \left (-\log \left (x^2+x+1\right )\right )+\text {$\#$1}^3 \log \left (-\text {$\#$1} x^2-\text {$\#$1} x-\text {$\#$1}+\sqrt {2 x^5+5 x^4+8 x^3+7 x^2+4 x+1}\right )+2 \text {$\#$1}^2 \log \left (x^2+x+1\right )-2 \text {$\#$1}^2 \log \left (-\text {$\#$1} x^2-\text {$\#$1} x-\text {$\#$1}+\sqrt {2 x^5+5 x^4+8 x^3+7 x^2+4 x+1}\right )+\text {$\#$1} \log \left (x^2+x+1\right )-\text {$\#$1} \log \left (-\text {$\#$1} x^2-\text {$\#$1} x-\text {$\#$1}+\sqrt {2 x^5+5 x^4+8 x^3+7 x^2+4 x+1}\right )-4 \log \left (-\text {$\#$1} x^2-\text {$\#$1} x-\text {$\#$1}+\sqrt {2 x^5+5 x^4+8 x^3+7 x^2+4 x+1}\right )+4 \log \left (x^2+x+1\right )}{4 \text {$\#$1}^3+3 \text {$\#$1}^2+2 \text {$\#$1}+1}\& \right ]+\frac {1}{2} \log \left (x^2+x+1\right )-\frac {1}{2} \log \left (-x^2+\sqrt {2 x^5+5 x^4+8 x^3+7 x^2+4 x+1}-x-1\right )-x \]

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Rubi [F]  time = 9.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x+\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}}{1-\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x + Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5])/(1 - Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5]),x]

[Out]

-x + (Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*ArcTanh[Sqrt[1 + 2*x]])/(2*Sqrt[1 + 2*x]*(1 + x + x^2)) - Log[x]/4 + Log
[4 + 7*x + 8*x^2 + 5*x^3 + 2*x^4]/16 - (Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*Log[3 + 3*x + 2*x^2 - Sqrt[1 + 2*x] -
x*Sqrt[1 + 2*x]])/(16*Sqrt[1 + 2*x]*(1 + x + x^2)) + (Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*Log[3 + 3*x + 2*x^2 + Sq
rt[1 + 2*x] + x*Sqrt[1 + 2*x]])/(16*Sqrt[1 + 2*x]*(1 + x + x^2)) + (5*Defer[Int][(4 + 7*x + 8*x^2 + 5*x^3 + 2*
x^4)^(-1), x])/16 + Defer[Int][x/(4 + 7*x + 8*x^2 + 5*x^3 + 2*x^4), x] + (5*Defer[Int][x^2/(4 + 7*x + 8*x^2 +
5*x^3 + 2*x^4), x])/16 - (17*Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*Defer[Subst][Defer[Int][(4 - x + x^2 - x^3 + x^4)
^(-1), x], x, Sqrt[1 + 2*x]])/(16*Sqrt[1 + 2*x]*(1 + x + x^2)) + (3*Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*Defer[Subs
t][Defer[Int][x/(4 - x + x^2 - x^3 + x^4), x], x, Sqrt[1 + 2*x]])/(8*Sqrt[1 + 2*x]*(1 + x + x^2)) - (11*Sqrt[(
1 + 2*x)*(1 + x + x^2)^2]*Defer[Subst][Defer[Int][x^2/(4 - x + x^2 - x^3 + x^4), x], x, Sqrt[1 + 2*x]])/(16*Sq
rt[1 + 2*x]*(1 + x + x^2)) - (17*Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*Defer[Subst][Defer[Int][(4 + x + x^2 + x^3 +
x^4)^(-1), x], x, Sqrt[1 + 2*x]])/(16*Sqrt[1 + 2*x]*(1 + x + x^2)) - (3*Sqrt[(1 + 2*x)*(1 + x + x^2)^2]*Defer[
Subst][Defer[Int][x/(4 + x + x^2 + x^3 + x^4), x], x, Sqrt[1 + 2*x]])/(8*Sqrt[1 + 2*x]*(1 + x + x^2)) - (11*Sq
rt[(1 + 2*x)*(1 + x + x^2)^2]*Defer[Subst][Defer[Int][x^2/(4 + x + x^2 + x^3 + x^4), x], x, Sqrt[1 + 2*x]])/(1
6*Sqrt[1 + 2*x]*(1 + x + x^2))

Rubi steps

\begin {align*} \int \frac {x+\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}}{1-\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}} \, dx &=\int \frac {x+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{1-\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx\\ &=\int \left (-\frac {x}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}\right ) \, dx\\ &=-\int \frac {x}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx-\int \frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x} \left (1+x+x^2\right )}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \left (\frac {1}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \left (\frac {\sqrt {1+2 x}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}+\frac {x \sqrt {1+2 x}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}+\frac {x^2 \sqrt {1+2 x}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}\right ) \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\int \frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x} \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {x \sqrt {1+2 x}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {x^2 \sqrt {1+2 x}}{-1+\sqrt {(1+2 x) \left (1+x+x^2\right )^2}} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \left (\frac {\sqrt {1+2 x}}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {\sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \left (\frac {x \sqrt {1+2 x}}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {x \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \left (\frac {\sqrt {1+2 x}}{4 x}+\frac {\sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4 x}-\frac {7 \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4 \left (4+7 x+8 x^2+5 x^3+2 x^4\right )}-\frac {2 x \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4}-\frac {5 x^2 \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4 \left (4+7 x+8 x^2+5 x^3+2 x^4\right )}-\frac {x^3 \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{2 \left (4+7 x+8 x^2+5 x^3+2 x^4\right )}+\frac {\sqrt {1+2 x} \left (-7-8 x-5 x^2-2 x^3\right )}{4 \left (4+7 x+8 x^2+5 x^3+2 x^4\right )}\right ) \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (2 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3+x^4\right )}{16+7 x^2+7 x^4+x^6+x^8} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x}}{x} \, dx}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{x} \, dx}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x} \left (-7-8 x-5 x^2-2 x^3\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {x^3 \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {x \sqrt {1+2 x}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {\sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {x \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (5 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \int \frac {x^2 \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (7 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \int \frac {\sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (2 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \int \frac {x \sqrt {1+2 x} \sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (2 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1+x) x}{2 \left (4-x+x^2-x^3+x^4\right )}+\frac {x (1+x)}{2 \left (4+x+x^2+x^3+x^4\right )}\right ) \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{2 \left (1+x+x^2\right )}-\frac {1}{4} \int \frac {(1+2 x) \left (1+x+x^2\right )}{x} \, dx+\frac {1}{2} \int \frac {x^3 (1+2 x) \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{4} \int \frac {x^2 (1+2 x) \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {7}{4} \int \frac {(1+2 x) \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+2 \int \frac {x (1+2 x) \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \int \frac {1}{x \sqrt {1+2 x}} \, dx}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2 \left (16+9 x^2+2 x^4+x^6\right )}{16+7 x^2+7 x^4+x^6+x^8} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {(-1+x) x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x (1+x)}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (4 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )}{16+7 x^2+7 x^4+x^6+x^8} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (8 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{16+7 x^2+7 x^4+x^6+x^8} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\int \frac {(1+2 x) \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\int \frac {x (1+2 x) \left (1+x+x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2}}{2 \left (1+x+x^2\right )}+\frac {3}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )-\frac {1}{8} \int \frac {-6-8 x-6 x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {7}{32} \int \frac {-6-8 x-6 x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {1}{4} \int \left (3+\frac {1}{x}+3 x+2 x^2\right ) \, dx+\frac {1}{2} \int \left (-x+x^2+\frac {x \left (4+3 x+2 x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx+\frac {5}{4} \int \left (-1+x+\frac {4+3 x+2 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx+2 \int \left (1-\frac {4+6 x+5 x^2+2 x^3}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {x^2}{2}} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (1-\frac {16-9 x^2-2 x^4-x^6}{16+7 x^2+7 x^4+x^6+x^8}\right ) \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (-\frac {x}{4-x+x^2-x^3+x^4}+\frac {x^2}{4-x+x^2-x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {x}{4+x+x^2+x^3+x^4}+\frac {x^2}{4+x+x^2+x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (4 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \left (\frac {x \left (2-x+x^2\right )}{4 \left (4-x+x^2-x^3+x^4\right )}-\frac {x \left (2+x+x^2\right )}{4 \left (4+x+x^2+x^3+x^4\right )}\right ) \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (8 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1+x) x^2}{8 \left (4-x+x^2-x^3+x^4\right )}+\frac {x^2 (1+x)}{8 \left (4+x+x^2+x^3+x^4\right )}\right ) \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\int \left (1-\frac {4+6 x+5 x^2+2 x^3}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx\\ &=-x+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \tanh ^{-1}\left (\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\log (x)}{4}+\frac {3}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )-\frac {1}{8} \int \left (-\frac {6}{4+7 x+8 x^2+5 x^3+2 x^4}-\frac {8 x}{4+7 x+8 x^2+5 x^3+2 x^4}-\frac {6 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx+\frac {7}{32} \int \left (-\frac {6}{4+7 x+8 x^2+5 x^3+2 x^4}-\frac {8 x}{4+7 x+8 x^2+5 x^3+2 x^4}-\frac {6 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx+\frac {1}{2} \int \frac {x \left (4+3 x+2 x^2\right )}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{4} \int \frac {4+3 x+2 x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-2 \int \frac {4+6 x+5 x^2+2 x^3}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {16-9 x^2-2 x^4-x^6}{16+7 x^2+7 x^4+x^6+x^8} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {(-1+x) x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x \left (2-x+x^2\right )}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2 (1+x)}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x \left (2+x+x^2\right )}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\int \frac {4+6 x+5 x^2+2 x^3}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-x+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \tanh ^{-1}\left (\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\log (x)}{4}+\frac {1}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )+\frac {1}{16} \int \frac {-14-6 x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {1}{8} \int \frac {18+16 x+10 x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {1}{4} \int \frac {18+16 x+10 x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{4} \int \left (\frac {4}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {3 x}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {2 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx-\frac {21}{16} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {1-2 x-x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {1+6 x-x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {-1-2 x+x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {-1+6 x+x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {4-5 x-2 x^2+x^3}{2 \left (4-x+x^2-x^3+x^4\right )}+\frac {4+5 x-2 x^2-x^3}{2 \left (4+x+x^2+x^3+x^4\right )}\right ) \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-x+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \tanh ^{-1}\left (\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\log (x)}{4}+\frac {1}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )+\frac {1}{16} \int \left (-\frac {14}{4+7 x+8 x^2+5 x^3+2 x^4}-\frac {6 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx+\frac {1}{8} \int \left (\frac {18}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {16 x}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {10 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx-\frac {1}{4} \int \left (\frac {18}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {16 x}{4+7 x+8 x^2+5 x^3+2 x^4}+\frac {10 x^2}{4+7 x+8 x^2+5 x^3+2 x^4}\right ) \, dx+\frac {3}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{2} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {15}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+5 \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {4-5 x-2 x^2+x^3}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {4+5 x-2 x^2-x^3}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {1}{4-x+x^2-x^3+x^4}-\frac {2 x}{4-x+x^2-x^3+x^4}-\frac {x^2}{4-x+x^2-x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {1}{4-x+x^2-x^3+x^4}+\frac {6 x}{4-x+x^2-x^3+x^4}-\frac {x^2}{4-x+x^2-x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {1}{-4-x-x^2-x^3-x^4}-\frac {2 x}{4+x+x^2+x^3+x^4}+\frac {x^2}{4+x+x^2+x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {1}{-4-x-x^2-x^3-x^4}+\frac {6 x}{4+x+x^2+x^3+x^4}+\frac {x^2}{4+x+x^2+x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{4 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-x+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \tanh ^{-1}\left (\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\log (x)}{4}+\frac {1}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \log \left (3+3 x+2 x^2-\sqrt {1+2 x}-x \sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \log \left (3+3 x+2 x^2+\sqrt {1+2 x}+x \sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {3}{8} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{8} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+2 \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {9}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {15}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-4 \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {9}{2} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+5 \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {17-22 x-5 x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {17+22 x-5 x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (3 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (3 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-x+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \tanh ^{-1}\left (\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\log (x)}{4}+\frac {1}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \log \left (3+3 x+2 x^2-\sqrt {1+2 x}-x \sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \log \left (3+3 x+2 x^2+\sqrt {1+2 x}+x \sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {3}{8} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{8} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+2 \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {9}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {15}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-4 \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {9}{2} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+5 \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {17}{4-x+x^2-x^3+x^4}-\frac {22 x}{4-x+x^2-x^3+x^4}-\frac {5 x^2}{4-x+x^2-x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \left (\frac {17}{4+x+x^2+x^3+x^4}+\frac {22 x}{4+x+x^2+x^3+x^4}-\frac {5 x^2}{4+x+x^2+x^3+x^4}\right ) \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (3 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (3 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ &=-x+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \tanh ^{-1}\left (\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\log (x)}{4}+\frac {1}{16} \log \left (4+7 x+8 x^2+5 x^3+2 x^4\right )-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \log \left (3+3 x+2 x^2-\sqrt {1+2 x}-x \sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \log \left (3+3 x+2 x^2+\sqrt {1+2 x}+x \sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {3}{8} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {3}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{8} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {5}{4} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {21}{16} \int \frac {x^2}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {7}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+2 \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {9}{4} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {15}{4} \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-4 \int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx-\frac {9}{2} \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+5 \int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\frac {\left (5 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (5 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\sqrt {(1+2 x) \left (1+x+x^2\right )^2} \operatorname {Subst}\left (\int \frac {x^2}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (17 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (17 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{16 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (11 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{8 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (11 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{8 \sqrt {1+2 x} \left (1+x+x^2\right )}-\frac {\left (3 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4-x+x^2-x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}+\frac {\left (3 \sqrt {(1+2 x) \left (1+x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{4+x+x^2+x^3+x^4} \, dx,x,\sqrt {1+2 x}\right )}{2 \sqrt {1+2 x} \left (1+x+x^2\right )}-\int \frac {1}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx+\int \frac {x}{4+7 x+8 x^2+5 x^3+2 x^4} \, dx\\ \end {align*}

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Mathematica [A]  time = 2.34, size = 418, normalized size = 1.18 \begin {gather*} \frac {\sqrt {2 x+1} \left (x^2+x+1\right ) \left (\text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3+\text {$\#$1}^2+\text {$\#$1}+4\&,\frac {\text {$\#$1}^3 \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )-2 \text {$\#$1}^2 \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )-\text {$\#$1} \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )-4 \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )}{4 \text {$\#$1}^3+3 \text {$\#$1}^2+2 \text {$\#$1}+1}\&\right ]-\text {RootSum}\left [\text {$\#$1}^4-\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}+4\&,\frac {\text {$\#$1}^3 \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )+2 \text {$\#$1}^2 \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )-\text {$\#$1} \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )+4 \log \left (\sqrt {2 x+1}-\text {$\#$1}\right )}{4 \text {$\#$1}^3-3 \text {$\#$1}^2+2 \text {$\#$1}-1}\&\right ]\right )}{4 \sqrt {(2 x+1) \left (x^2+x+1\right )^2}}+\frac {1}{4} \left (\text {RootSum}\left [2 \text {$\#$1}^4+5 \text {$\#$1}^3+8 \text {$\#$1}^2+7 \text {$\#$1}+4\&,\frac {2 \text {$\#$1}^3 \log (x-\text {$\#$1})+5 \text {$\#$1}^2 \log (x-\text {$\#$1})+8 \text {$\#$1} \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{8 \text {$\#$1}^3+15 \text {$\#$1}^2+16 \text {$\#$1}+7}\&\right ]+2 \tanh ^{-1}\left (\frac {\sqrt {(2 x+1) \left (x^2+x+1\right )^2}}{x^2+x+1}\right )-4 x-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5])/(1 - Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^
5]),x]

[Out]

(Sqrt[1 + 2*x]*(1 + x + x^2)*(-RootSum[4 - #1 + #1^2 - #1^3 + #1^4 & , (4*Log[Sqrt[1 + 2*x] - #1] - Log[Sqrt[1
+ 2*x] - #1]*#1 + 2*Log[Sqrt[1 + 2*x] - #1]*#1^2 + Log[Sqrt[1 + 2*x] - #1]*#1^3)/(-1 + 2*#1 - 3*#1^2 + 4*#1^3
) & ] + RootSum[4 + #1 + #1^2 + #1^3 + #1^4 & , (-4*Log[Sqrt[1 + 2*x] - #1] - Log[Sqrt[1 + 2*x] - #1]*#1 - 2*L
og[Sqrt[1 + 2*x] - #1]*#1^2 + Log[Sqrt[1 + 2*x] - #1]*#1^3)/(1 + 2*#1 + 3*#1^2 + 4*#1^3) & ]))/(4*Sqrt[(1 + 2*
x)*(1 + x + x^2)^2]) + (-4*x + 2*ArcTanh[Sqrt[(1 + 2*x)*(1 + x + x^2)^2]/(1 + x + x^2)] - Log[x] + RootSum[4 +
7*#1 + 8*#1^2 + 5*#1^3 + 2*#1^4 & , (3*Log[x - #1] + 8*Log[x - #1]*#1 + 5*Log[x - #1]*#1^2 + 2*Log[x - #1]*#1
^3)/(7 + 16*#1 + 15*#1^2 + 8*#1^3) & ])/4

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IntegrateAlgebraic [A]  time = 0.24, size = 354, normalized size = 1.00 \begin {gather*} -x+\frac {1}{2} \log \left (1+x+x^2\right )-\frac {1}{2} \log \left (-1-x-x^2+\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}\right )+\frac {1}{2} \text {RootSum}\left [4+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {4 \log \left (1+x+x^2\right )-4 \log \left (\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text {$\#$1}-x \text {$\#$1}-x^2 \text {$\#$1}\right )+\log \left (1+x+x^2\right ) \text {$\#$1}-\log \left (\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text {$\#$1}-x \text {$\#$1}-x^2 \text {$\#$1}\right ) \text {$\#$1}+2 \log \left (1+x+x^2\right ) \text {$\#$1}^2-2 \log \left (\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text {$\#$1}-x \text {$\#$1}-x^2 \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (1+x+x^2\right ) \text {$\#$1}^3+\log \left (\sqrt {1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text {$\#$1}-x \text {$\#$1}-x^2 \text {$\#$1}\right ) \text {$\#$1}^3}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x + Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5])/(1 - Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x
^4 + 2*x^5]),x]

[Out]

-x + Log[1 + x + x^2]/2 - Log[-1 - x - x^2 + Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5]]/2 + RootSum[4 + #1
+ #1^2 + #1^3 + #1^4 & , (4*Log[1 + x + x^2] - 4*Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#
1 - x^2*#1] + Log[1 + x + x^2]*#1 - Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1]*#1
+ 2*Log[1 + x + x^2]*#1^2 - 2*Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1]*#1^2 -
Log[1 + x + x^2]*#1^3 + Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1]*#1^3)/(1 + 2*#
1 + 3*#1^2 + 4*#1^3) & ]/2

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2))/(1-(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)),x, algorithm="fr
icas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x + \sqrt {2 \, x^{5} + 5 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 4 \, x + 1}}{\sqrt {2 \, x^{5} + 5 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 4 \, x + 1} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2))/(1-(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)),x, algorithm="gi
ac")

[Out]

integrate(-(x + sqrt(2*x^5 + 5*x^4 + 8*x^3 + 7*x^2 + 4*x + 1))/(sqrt(2*x^5 + 5*x^4 + 8*x^3 + 7*x^2 + 4*x + 1)
- 1), x)

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maple [B]  time = 0.03, size = 494, normalized size = 1.40 \[-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}\, \textit {\_Z}^{5}-4 \sqrt {1+2 x}\, x^{2}+3 \sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}\, \textit {\_Z} -4 \sqrt {1+2 x}\, x -4 \sqrt {1+2 x}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+1\right ) \ln \left (\sqrt {1+2 x}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+3}\right ) \sqrt {1+2 x}\, x^{2}+4 \left (\munderset {\textit {\_R} =\RootOf \left (\sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}\, \textit {\_Z}^{5}-4 \sqrt {1+2 x}\, x^{2}+3 \sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}\, \textit {\_Z} -4 \sqrt {1+2 x}\, x -4 \sqrt {1+2 x}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+1\right ) \ln \left (\sqrt {1+2 x}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+3}\right ) \sqrt {1+2 x}\, x +2 x \sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}+4 \sqrt {1+2 x}\, \left (\munderset {\textit {\_R} =\RootOf \left (\sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}\, \textit {\_Z}^{5}-4 \sqrt {1+2 x}\, x^{2}+3 \sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}\, \textit {\_Z} -4 \sqrt {1+2 x}\, x -4 \sqrt {1+2 x}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+1\right ) \ln \left (\sqrt {1+2 x}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+3}\right )+\sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}}{2 \sqrt {2 x^{5}+5 x^{4}+8 x^{3}+7 x^{2}+4 x +1}}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2))/(1-(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)),x)

[Out]

-1/2*(4*sum(_R*(_R^2+1)*ln((1+2*x)^(1/2)-_R)/(5*_R^4+3),_R=RootOf((2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)*_Z^5-4
*(1+2*x)^(1/2)*x^2+3*(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)*_Z-4*(1+2*x)^(1/2)*x-4*(1+2*x)^(1/2)))*(1+2*x)^(1/2
)*x^2+4*sum(_R*(_R^2+1)*ln((1+2*x)^(1/2)-_R)/(5*_R^4+3),_R=RootOf((2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)*_Z^5-4
*(1+2*x)^(1/2)*x^2+3*(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)*_Z-4*(1+2*x)^(1/2)*x-4*(1+2*x)^(1/2)))*(1+2*x)^(1/2
)*x+2*x*(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)+4*(1+2*x)^(1/2)*sum(_R*(_R^2+1)*ln((1+2*x)^(1/2)-_R)/(5*_R^4+3),
_R=RootOf((2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)*_Z^5-4*(1+2*x)^(1/2)*x^2+3*(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/
2)*_Z-4*(1+2*x)^(1/2)*x-4*(1+2*x)^(1/2)))+(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2))/(2*x^5+5*x^4+8*x^3+7*x^2+4*x+
1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{4} \, x^{2} - \frac {1}{2} \, x + \int -\frac {2 \, x^{6} + 7 \, x^{5} + 13 \, x^{4} + 15 \, x^{3} + 11 \, x^{2} + 4 \, x}{2 \, {\left (2 \, x^{5} + 5 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} - 2 \, {\left (x^{2} + x + 1\right )} \sqrt {2 \, x + 1} + 4 \, x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2))/(1-(2*x^5+5*x^4+8*x^3+7*x^2+4*x+1)^(1/2)),x, algorithm="ma
xima")

[Out]

1/4*x^2 - 1/2*x + integrate(-1/2*(2*x^6 + 7*x^5 + 13*x^4 + 15*x^3 + 11*x^2 + 4*x)/(2*x^5 + 5*x^4 + 8*x^3 + 7*x
^2 - 2*(x^2 + x + 1)*sqrt(2*x + 1) + 4*x + 2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x+\sqrt {2\,x^5+5\,x^4+8\,x^3+7\,x^2+4\,x+1}}{\sqrt {2\,x^5+5\,x^4+8\,x^3+7\,x^2+4\,x+1}-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + (4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5 + 1)^(1/2))/((4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5 + 1)^(1/2) -
1),x)

[Out]

int(-(x + (4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5 + 1)^(1/2))/((4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5 + 1)^(1/2) -
1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{\sqrt {2 x^{5} + 5 x^{4} + 8 x^{3} + 7 x^{2} + 4 x + 1} - 1}\, dx - \int \frac {\sqrt {2 x^{5} + 5 x^{4} + 8 x^{3} + 7 x^{2} + 4 x + 1}}{\sqrt {2 x^{5} + 5 x^{4} + 8 x^{3} + 7 x^{2} + 4 x + 1} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(2*x**5+5*x**4+8*x**3+7*x**2+4*x+1)**(1/2))/(1-(2*x**5+5*x**4+8*x**3+7*x**2+4*x+1)**(1/2)),x)

[Out]

-Integral(x/(sqrt(2*x**5 + 5*x**4 + 8*x**3 + 7*x**2 + 4*x + 1) - 1), x) - Integral(sqrt(2*x**5 + 5*x**4 + 8*x*
*3 + 7*x**2 + 4*x + 1)/(sqrt(2*x**5 + 5*x**4 + 8*x**3 + 7*x**2 + 4*x + 1) - 1), x)

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