3.49 \(\int \frac{d+e x^4}{x^2 \left (a+b x^4+c x^8\right )} \, dx\)

Optimal. Leaf size=392 \[ -\frac{\sqrt [4]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{d}{a x} \]

[Out]

-(d/(a*x)) - (c^(1/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1
/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1
/4)) - (c^(1/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)
/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) +
(c^(1/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b
- Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(
1/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqr
t[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 1.45766, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt [4]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{d}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-(d/(a*x)) - (c^(1/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1
/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1
/4)) - (c^(1/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)
/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) +
(c^(1/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b
- Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(
1/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqr
t[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 160.037, size = 391, normalized size = 1. \[ \frac{\sqrt [4]{2} \sqrt [4]{c} \left (2 a e - b d - d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [4]{2} \sqrt [4]{c} \left (2 a e - b d - d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [4]{2} \sqrt [4]{c} \left (2 a e - b d + d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt [4]{2} \sqrt [4]{c} \left (2 a e - b d + d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{d}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*c**(1/4)*(2*a*e - b*d - d*sqrt(-4*a*c + b**2))*atan(2**(1/4)*c**(1/4)*x
/(-b + sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b + sqrt(-4*a*c + b**2))**(1/4)*sqrt(
-4*a*c + b**2)) - 2**(1/4)*c**(1/4)*(2*a*e - b*d - d*sqrt(-4*a*c + b**2))*atanh(
2**(1/4)*c**(1/4)*x/(-b + sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b + sqrt(-4*a*c +
b**2))**(1/4)*sqrt(-4*a*c + b**2)) - 2**(1/4)*c**(1/4)*(2*a*e - b*d + d*sqrt(-4*
a*c + b**2))*atan(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-
b - sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*a*c + b**2)) + 2**(1/4)*c**(1/4)*(2*a*e
- b*d + d*sqrt(-4*a*c + b**2))*atanh(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c + b**
2))**(1/4))/(4*a*(-b - sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*a*c + b**2)) - d/(a*x
)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0889316, size = 85, normalized size = 0.22 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right ]}{4 a}-\frac{d}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-(d/(a*x)) - RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1]
+ c*d*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(4*a)

_______________________________________________________________________________________

Maple [C]  time = 0.01, size = 72, normalized size = 0.2 \[ -{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ({{\it \_R}}^{6}cd+ \left ( -ae+bd \right ){{\it \_R}}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}}-{\frac{d}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/a*sum((_R^6*c*d+(-a*e+b*d)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*
c+_Z^4*b+a))-d/a/x

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{c d x^{6} +{\left (b d - a e\right )} x^{2}}{c x^{8} + b x^{4} + a}\,{d x}}{a} - \frac{d}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((c*d*x^6 + (b*d - a*e)*x^2)/(c*x^8 + b*x^4 + a), x)/a - d/(a*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{4} + d}{{\left (c x^{8} + b x^{4} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^4 + d)/((c*x^8 + b*x^4 + a)*x^2), x)