3.6 \(\int \frac{d+e x^4}{d^2+f x^4+e^2 x^8} \, dx\)

Optimal. Leaf size=791 \[ -\frac{\log \left (-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\log \left (x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}-\frac{\log \left (-x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left (x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}} \]

[Out]

-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]
*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]
]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqr
t[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e
- f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sqrt[2
*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*
d*e - f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sq
rt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqr
t[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqr
t[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) + Log[Sqrt[d] +
Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqr
t[d]*Sqrt[e] - Sqrt[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2
*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]
) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*
Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]])

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Rubi [A]  time = 1.75234, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\log \left (-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\log \left (x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}-\frac{\log \left (-x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left (x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8),x]

[Out]

-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]
*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]
]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqr
t[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e
- f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sqrt[2
*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*
d*e - f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sq
rt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqr
t[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqr
t[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) + Log[Sqrt[d] +
Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqr
t[d]*Sqrt[e] - Sqrt[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2
*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]
) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*
Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**4+d)/(e**2*x**8+f*x**4+d**2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0476941, size = 67, normalized size = 0.08 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8 e^2+\text{$\#$1}^4 f+d^2\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2+\text{$\#$1}^3 f}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8),x]

[Out]

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

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Maple [C]  time = 0.06, size = 53, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({e}^{2}{{\it \_Z}}^{8}+f{{\it \_Z}}^{4}+{d}^{2} \right ) }{\frac{ \left ({{\it \_R}}^{4}e+d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}{e}^{2}+{{\it \_R}}^{3}f}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^4+d)/(e^2*x^8+f*x^4+d^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{4} + d}{e^{2} x^{8} + f x^{4} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/(e^2*x^8 + f*x^4 + d^2),x, algorithm="maxima")

[Out]

integrate((e*x^4 + d)/(e^2*x^8 + f*x^4 + d^2), x)

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Fricas [A]  time = 0.313566, size = 3082, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/(e^2*x^8 + f*x^4 + d^2),x, algorithm="fricas")

[Out]

sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7
*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*
f^2)))*arctan(-1/2*(2*d*e - (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/
(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)*sqrt(-((
4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6
*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))/(e*x + sqrt(1/2)
*e*sqrt((2*e^2*x^2 + sqrt(1/2)*(2*d*e*f + f^2 - (8*d^5*e^3 + 12*d^4*e^2*f + 6*d^
3*e*f^2 + d^2*f^3)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d
^4*f^3)))*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3
+ 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2))
)/e^2))) - sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e -
f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*
e*f + d^2*f^2)))*arctan(1/2*(2*d*e + (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*
d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)
*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e
^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))/(e*x + s
qrt(1/2)*e*sqrt((2*e^2*x^2 + sqrt(1/2)*(2*d*e*f + f^2 + (8*d^5*e^3 + 12*d^4*e^2*
f + 6*d^3*e*f^2 + d^2*f^3)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e
*f^2 + d^4*f^3)))*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d
^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^
2*f^2)))/e^2))) + 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sq
rt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*
e^2 + 4*d^3*e*f + d^2*f^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 + 4*d^3*e*f + d^2
*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)
*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^
7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2
*f^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*
d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4
*d^3*e*f + d^2*f^2)))*log(e*x - 1/2*(2*d*e - (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*s
qrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(s
qrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 +
12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))
) + 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)
/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f
+ d^2*f^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*
d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)
*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e
^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))) - 1/4*s
qrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e
^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^
2)))*log(e*x - 1/2*(2*d*e + (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/
(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)*sqrt(((4
*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*
d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2))))

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Sympy [A]  time = 18.9669, size = 136, normalized size = 0.17 \[ \operatorname{RootSum}{\left (t^{8} \left (1048576 d^{6} e^{4} + 2097152 d^{5} e^{3} f + 1572864 d^{4} e^{2} f^{2} + 524288 d^{3} e f^{3} + 65536 d^{2} f^{4}\right ) + t^{4} \left (1024 d^{2} e^{2} f + 1024 d e f^{2} + 256 f^{3}\right ) + e^{2}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{5} d^{4} e^{2} + 4096 t^{5} d^{3} e f + 1024 t^{5} d^{2} f^{2} + 4 t d e + 4 t f}{e} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**4+d)/(e**2*x**8+f*x**4+d**2),x)

[Out]

RootSum(_t**8*(1048576*d**6*e**4 + 2097152*d**5*e**3*f + 1572864*d**4*e**2*f**2
+ 524288*d**3*e*f**3 + 65536*d**2*f**4) + _t**4*(1024*d**2*e**2*f + 1024*d*e*f**
2 + 256*f**3) + e**2, Lambda(_t, _t*log(x + (4096*_t**5*d**4*e**2 + 4096*_t**5*d
**3*e*f + 1024*_t**5*d**2*f**2 + 4*_t*d*e + 4*_t*f)/e)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{4} + d}{e^{2} x^{8} + f x^{4} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/(e^2*x^8 + f*x^4 + d^2),x, algorithm="giac")

[Out]

integrate((e*x^4 + d)/(e^2*x^8 + f*x^4 + d^2), x)