3.278 \(\int \frac{-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{(-1+2 x^2)^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx\)

Optimal. Leaf size=94 \[ \frac{(2 x+1) \sqrt{x^4+4 x^3+2 x^2+1}}{2 \left (2 x^2-1\right )}-\tanh ^{-1}\left (\frac{x (x+2) \left (33 x^3+27 x^2-x+7\right )}{\left (31 x^3+37 x^2+2\right ) \sqrt{x^4+4 x^3+2 x^2+1}}\right ) \]

[Out]

((1 + 2*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4])/(2*(-1 + 2*x^2)) - ArcTanh[(x*(2 + x)*(7 - x + 27*x^2 + 33*x^3))/((2
+ 37*x^2 + 31*x^3)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4])]

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Rubi [F]  time = 1.83995, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(-8 - 8*x - x^2 - 3*x^3 + 7*x^4 + 4*x^5 + 2*x^6)/((-1 + 2*x^2)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]),x]

[Out]

(9*Defer[Int][1/Sqrt[1 + 2*x^2 + 4*x^3 + x^4], x])/4 - (13*Defer[Int][1/((Sqrt[2] - 2*x)^2*Sqrt[1 + 2*x^2 + 4*
x^3 + x^4]), x])/4 + Defer[Int][x/Sqrt[1 + 2*x^2 + 4*x^3 + x^4], x] + Defer[Int][x^2/Sqrt[1 + 2*x^2 + 4*x^3 +
x^4], x]/2 - (13*Defer[Int][1/((Sqrt[2] + 2*x)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/4 - (13*Defer[Int][1/((1
- Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - ((15 + Sqrt[2])*Defer[Int][1/((1 - Sqrt[2]*x)*Sqrt[1 + 2*
x^2 + 4*x^3 + x^4]), x])/8 - (13*Defer[Int][1/((1 + Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - ((15 -
Sqrt[2])*Defer[Int][1/((1 + Sqrt[2]*x)*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/8 - (17*Defer[Int][x/((-1 + 2*x^2)^
2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]), x])/2

Rubi steps

\begin{align*} \int \frac{-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx &=\int \left (\frac{9}{4 \sqrt{1+2 x^2+4 x^3+x^4}}+\frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}}+\frac{x^2}{2 \sqrt{1+2 x^2+4 x^3+x^4}}+\frac{-13-17 x}{2 \left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}}+\frac{15+2 x}{4 \left (-1+2 x^2\right ) \sqrt{1+2 x^2+4 x^3+x^4}}\right ) \, dx\\ &=\frac{1}{4} \int \frac{15+2 x}{\left (-1+2 x^2\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{2} \int \frac{-13-17 x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac{1}{4} \int \left (-\frac{15+\sqrt{2}}{2 \left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}}-\frac{15-\sqrt{2}}{2 \left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}}\right ) \, dx+\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{2} \int \left (-\frac{13}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}}-\frac{17 x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}}\right ) \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{2} \int \frac{1}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{17}{2} \int \frac{x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15-\sqrt{2}\right ) \int \frac{1}{\left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15+\sqrt{2}\right ) \int \frac{1}{\left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{2} \int \left (\frac{1}{2 \left (\sqrt{2}-2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}}+\frac{1}{2 \left (\sqrt{2}+2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}}+\frac{1}{\left (2-4 x^2\right ) \sqrt{1+2 x^2+4 x^3+x^4}}\right ) \, dx-\frac{17}{2} \int \frac{x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15-\sqrt{2}\right ) \int \frac{1}{\left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15+\sqrt{2}\right ) \int \frac{1}{\left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{4} \int \frac{1}{\left (\sqrt{2}-2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{4} \int \frac{1}{\left (\sqrt{2}+2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{2} \int \frac{1}{\left (2-4 x^2\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{17}{2} \int \frac{x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15-\sqrt{2}\right ) \int \frac{1}{\left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15+\sqrt{2}\right ) \int \frac{1}{\left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{4} \int \frac{1}{\left (\sqrt{2}-2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{4} \int \frac{1}{\left (\sqrt{2}+2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{2} \int \left (\frac{1}{4 \left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}}+\frac{1}{4 \left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}}\right ) \, dx-\frac{17}{2} \int \frac{x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15-\sqrt{2}\right ) \int \frac{1}{\left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15+\sqrt{2}\right ) \int \frac{1}{\left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ &=\frac{1}{2} \int \frac{x^2}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{8} \int \frac{1}{\left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{8} \int \frac{1}{\left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{9}{4} \int \frac{1}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{4} \int \frac{1}{\left (\sqrt{2}-2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{13}{4} \int \frac{1}{\left (\sqrt{2}+2 x\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx-\frac{17}{2} \int \frac{x}{\left (-1+2 x^2\right )^2 \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15-\sqrt{2}\right ) \int \frac{1}{\left (1-\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\frac{1}{8} \left (-15+\sqrt{2}\right ) \int \frac{1}{\left (1+\sqrt{2} x\right ) \sqrt{1+2 x^2+4 x^3+x^4}} \, dx+\int \frac{x}{\sqrt{1+2 x^2+4 x^3+x^4}} \, dx\\ \end{align*}

Mathematica [C]  time = 6.41877, size = 5137, normalized size = 54.65 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 8*x - x^2 - 3*x^3 + 7*x^4 + 4*x^5 + 2*x^6)/((-1 + 2*x^2)^2*Sqrt[1 + 2*x^2 + 4*x^3 + x^4]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.914, size = 1197351, normalized size = 12737.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2+1)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1}{\left (2 \, x^{2} - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.69053, size = 464, normalized size = 4.94 \begin{align*} \frac{{\left (2 \, x^{2} - 1\right )} \log \left (\frac{1025 \, x^{10} + 6138 \, x^{9} + 12307 \, x^{8} + 10188 \, x^{7} + 4503 \, x^{6} + 3134 \, x^{5} + 1589 \, x^{4} + 140 \, x^{3} + 176 \, x^{2} -{\left (1023 \, x^{8} + 4104 \, x^{7} + 5084 \, x^{6} + 2182 \, x^{5} + 805 \, x^{4} + 624 \, x^{3} + 10 \, x^{2} + 28 \, x\right )} \sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} + 2}{32 \, x^{10} - 80 \, x^{8} + 80 \, x^{6} - 40 \, x^{4} + 10 \, x^{2} - 1}\right ) + \sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1}{\left (2 \, x + 1\right )}}{2 \,{\left (2 \, x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*((2*x^2 - 1)*log((1025*x^10 + 6138*x^9 + 12307*x^8 + 10188*x^7 + 4503*x^6 + 3134*x^5 + 1589*x^4 + 140*x^3
+ 176*x^2 - (1023*x^8 + 4104*x^7 + 5084*x^6 + 2182*x^5 + 805*x^4 + 624*x^3 + 10*x^2 + 28*x)*sqrt(x^4 + 4*x^3 +
2*x^2 + 1) + 2)/(32*x^10 - 80*x^8 + 80*x^6 - 40*x^4 + 10*x^2 - 1)) + sqrt(x^4 + 4*x^3 + 2*x^2 + 1)*(2*x + 1))
/(2*x^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 x^{6} + 4 x^{5} + 7 x^{4} - 3 x^{3} - x^{2} - 8 x - 8}{\sqrt{\left (x + 1\right ) \left (x^{3} + 3 x^{2} - x + 1\right )} \left (2 x^{2} - 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**6+4*x**5+7*x**4-3*x**3-x**2-8*x-8)/(2*x**2-1)**2/(x**4+4*x**3+2*x**2+1)**(1/2),x)

[Out]

Integral((2*x**6 + 4*x**5 + 7*x**4 - 3*x**3 - x**2 - 8*x - 8)/(sqrt((x + 1)*(x**3 + 3*x**2 - x + 1))*(2*x**2 -
1)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt{x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1}{\left (2 \, x^{2} - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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