3.911 \(\int \frac{x^4}{a+b+2 a x^2+a x^4} \, dx\)

Optimal. Leaf size=432 \[ \frac{\left (-2 \sqrt{a} \sqrt{a+b}+a+b\right ) \log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\left (-2 \sqrt{a} \sqrt{a+b}+a+b\right ) \log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\left (2 \sqrt{a} \sqrt{a+b}+a+b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\left (2 \sqrt{a} \sqrt{a+b}+a+b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{x}{a} \]

[Out]

x/a + ((a + b + 2*Sqrt[a]*Sqrt[a + b])*ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sq
rt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]])/(2*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sq
rt[Sqrt[a] + Sqrt[a + b]]) - ((a + b + 2*Sqrt[a]*Sqrt[a + b])*ArcTan[(Sqrt[-Sqrt
[a] + Sqrt[a + b]] + Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]])/(2*Sqrt[2]
*a^(5/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) + ((a + b - 2*Sqrt[a]*Sqrt[a +
b])*Log[Sqrt[a + b] - Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*
x^2])/(4*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) - ((a + b - 2
*Sqrt[a]*Sqrt[a + b])*Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a +
b]]*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b
]])

_______________________________________________________________________________________

Rubi [A]  time = 2.05947, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\left (-2 \sqrt{a} \sqrt{a+b}+a+b\right ) \log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\left (-2 \sqrt{a} \sqrt{a+b}+a+b\right ) \log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\left (2 \sqrt{a} \sqrt{a+b}+a+b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\left (2 \sqrt{a} \sqrt{a+b}+a+b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{x}{a} \]

Antiderivative was successfully verified.

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

[Out]

x/a + ((a + b + 2*Sqrt[a]*Sqrt[a + b])*ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sq
rt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]])/(2*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sq
rt[Sqrt[a] + Sqrt[a + b]]) - ((a + b + 2*Sqrt[a]*Sqrt[a + b])*ArcTan[(Sqrt[-Sqrt
[a] + Sqrt[a + b]] + Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]])/(2*Sqrt[2]
*a^(5/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) + ((a + b - 2*Sqrt[a]*Sqrt[a +
b])*Log[Sqrt[a + b] - Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*
x^2])/(4*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) - ((a + b - 2
*Sqrt[a]*Sqrt[a + b])*Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a +
b]]*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b
]])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 158.426, size = 393, normalized size = 0.91 \[ \frac{x}{a} - \frac{\sqrt{2} \left (2 \sqrt{a} \sqrt{a + b} + a + b\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt [4]{a} x - \frac{\sqrt{- 2 \sqrt{a} + 2 \sqrt{a + b}}}{2}\right )}{\sqrt{\sqrt{a} + \sqrt{a + b}}} \right )}}{4 a^{\frac{5}{4}} \sqrt{\sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} - \frac{\sqrt{2} \left (2 \sqrt{a} \sqrt{a + b} + a + b\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt [4]{a} x + \frac{\sqrt{- 2 \sqrt{a} + 2 \sqrt{a + b}}}{2}\right )}{\sqrt{\sqrt{a} + \sqrt{a + b}}} \right )}}{4 a^{\frac{5}{4}} \sqrt{\sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} + \frac{\sqrt{2} \left (- 2 \sqrt{a} \sqrt{a + b} + a + b\right ) \log{\left (x^{2} + \frac{\sqrt{a + b}}{\sqrt{a}} - \frac{\sqrt{2} x \sqrt{- \sqrt{a} + \sqrt{a + b}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} \sqrt{- \sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} - \frac{\sqrt{2} \left (- 2 \sqrt{a} \sqrt{a + b} + a + b\right ) \log{\left (x^{2} + \frac{\sqrt{a + b}}{\sqrt{a}} + \frac{\sqrt{2} x \sqrt{- \sqrt{a} + \sqrt{a + b}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} \sqrt{- \sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a*x**4+2*a*x**2+a+b),x)

[Out]

x/a - sqrt(2)*(2*sqrt(a)*sqrt(a + b) + a + b)*atan(sqrt(2)*(a**(1/4)*x - sqrt(-2
*sqrt(a) + 2*sqrt(a + b))/2)/sqrt(sqrt(a) + sqrt(a + b)))/(4*a**(5/4)*sqrt(sqrt(
a) + sqrt(a + b))*sqrt(a + b)) - sqrt(2)*(2*sqrt(a)*sqrt(a + b) + a + b)*atan(sq
rt(2)*(a**(1/4)*x + sqrt(-2*sqrt(a) + 2*sqrt(a + b))/2)/sqrt(sqrt(a) + sqrt(a +
b)))/(4*a**(5/4)*sqrt(sqrt(a) + sqrt(a + b))*sqrt(a + b)) + sqrt(2)*(-2*sqrt(a)*
sqrt(a + b) + a + b)*log(x**2 + sqrt(a + b)/sqrt(a) - sqrt(2)*x*sqrt(-sqrt(a) +
sqrt(a + b))/a**(1/4))/(8*a**(5/4)*sqrt(-sqrt(a) + sqrt(a + b))*sqrt(a + b)) - s
qrt(2)*(-2*sqrt(a)*sqrt(a + b) + a + b)*log(x**2 + sqrt(a + b)/sqrt(a) + sqrt(2)
*x*sqrt(-sqrt(a) + sqrt(a + b))/a**(1/4))/(8*a**(5/4)*sqrt(-sqrt(a) + sqrt(a + b
))*sqrt(a + b))

_______________________________________________________________________________________

Mathematica [C]  time = 0.162432, size = 164, normalized size = 0.38 \[ -\frac{i \left (\sqrt{a}-i \sqrt{b}\right )^2 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right )}{2 a \sqrt{b} \sqrt{a-i \sqrt{a} \sqrt{b}}}+\frac{i \left (\sqrt{a}+i \sqrt{b}\right )^2 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right )}{2 a \sqrt{b} \sqrt{a+i \sqrt{a} \sqrt{b}}}+\frac{x}{a} \]

Antiderivative was successfully verified.

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

[Out]

x/a - ((I/2)*(Sqrt[a] - I*Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[
b]]])/(a*Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((I/2)*(Sqrt[a] + I*Sqrt[b])^2*A
rcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + I*Sqrt[a]*Sqrt[b]]*S
qrt[b])

_______________________________________________________________________________________

Maple [B]  time = 0.128, size = 1658, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a*x^4+2*a*x^2+a+b),x)

[Out]

x/a+1/8/a/b*ln(-a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2))*(a+b)^(
1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/8/a^2/b*ln(-a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-
2*a)^(1/2)*x-(a+b)^(1/2))*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1
/2)-1/4/a^(3/2)/b*ln(-a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2))*(
2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/4/a^(1/2)/b*ln(-a^(1/2)*x^2+(2*(a
*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/a/(4*a^(
1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*x+(2*(a*(a+b))^
(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(a+b)^(1/
2)-1/4/a/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2
)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)
^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-
1/4/a^2/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)
*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^
(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(
a^2+a*b)^(1/2)+1/2/a^(3/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)
*arctan((-2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a
*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^
(1/2)*(a^2+a*b)^(1/2)+1/2/a^(1/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a
)^(1/2)*arctan((-2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/
2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2
)-2*a)^(1/2)-1/8/a/b*ln(a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x+(a+b)^(1/2))
*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8/a^2/b*ln(a^(1/2)*x^2+(2*(a*(a+b))
^(1/2)-2*a)^(1/2)*x+(a+b)^(1/2))*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+
a*b)^(1/2)+1/4/a^(3/2)/b*ln(a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x+(a+b)^(1
/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/4/a^(1/2)/b*ln(a^(1/2)*x^2+
(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/a/(
4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*x+(2*(a*(a+
b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(a+b)
^(1/2)+1/4/a/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((2*a^(
1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2
*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/
2)+1/4/a^2/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((2*a^(1/
2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a
)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(a+b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)
*(a^2+a*b)^(1/2)-1/2/a^(3/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/
2)*arctan((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(
a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)
^(1/2)*(a^2+a*b)^(1/2)-1/2/a^(1/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*
a)^(1/2)*arctan((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/
2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2
)-2*a)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x}{a} - \frac{\int \frac{2 \, a x^{2} + a + b}{a x^{4} + 2 \, a x^{2} + a + b}\,{d x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/a - integrate((2*a*x^2 + a + b)/(a*x^4 + 2*a*x^2 + a + b), x)/a

_______________________________________________________________________________________

Fricas [A]  time = 0.285478, size = 830, normalized size = 1.92 \[ \frac{a \sqrt{\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x +{\left (a^{4} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b - a b^{2}\right )} \sqrt{\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}}\right ) - a \sqrt{\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x -{\left (a^{4} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b - a b^{2}\right )} \sqrt{\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}}\right ) - a \sqrt{-\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x +{\left (a^{4} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b + a b^{2}\right )} \sqrt{-\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}}\right ) + a \sqrt{-\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x -{\left (a^{4} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b + a b^{2}\right )} \sqrt{-\frac{a^{2} b \sqrt{-\frac{9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}}\right ) + 4 \, x}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(a*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))*log(
-(3*a^2 + 2*a*b - b^2)*x + (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + 3*a^2*b
- a*b^2)*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b)))
- a*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))*log(-(3
*a^2 + 2*a*b - b^2)*x - (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + 3*a^2*b -
a*b^2)*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))) - a
*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))*log(-(3*a
^2 + 2*a*b - b^2)*x + (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - 3*a^2*b + a*
b^2)*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))) + a*
sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))*log(-(3*a^
2 + 2*a*b - b^2)*x - (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - 3*a^2*b + a*b
^2)*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))) + 4*x
)/a

_______________________________________________________________________________________

Sympy [A]  time = 3.71654, size = 105, normalized size = 0.24 \[ \operatorname{RootSum}{\left (256 t^{4} a^{5} b^{2} + t^{2} \left (- 32 a^{4} b + 96 a^{3} b^{2}\right ) + a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} b + 4 t a^{3} - 24 t a^{2} b + 4 t a b^{2}}{3 a^{2} + 2 a b - b^{2}} \right )} \right )\right )} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a*x**4+2*a*x**2+a+b),x)

[Out]

RootSum(256*_t**4*a**5*b**2 + _t**2*(-32*a**4*b + 96*a**3*b**2) + a**3 + 3*a**2*
b + 3*a*b**2 + b**3, Lambda(_t, _t*log(x + (-64*_t**3*a**4*b + 4*_t*a**3 - 24*_t
*a**2*b + 4*_t*a*b**2)/(3*a**2 + 2*a*b - b**2)))) + x/a

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError