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]] - 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]
)*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)*S
qrt[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)*S
qrt[-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]])

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Rubi [A]  time = 0.890587, 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, Rules used = {1122, 1169, 634, 618, 204, 628} \[ \frac{\left (-2 \sqrt{a} \sqrt{a+b}+a+b\right ) \log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\left (-2 \sqrt{a} \sqrt{a+b}+a+b\right ) \log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\left (2 \sqrt{a} \sqrt{a+b}+a+b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\left (2 \sqrt{a} \sqrt{a+b}+a+b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{x}{a} \]

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]] - 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]
)*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)*S
qrt[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)*S
qrt[-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]])

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{a+b+2 a x^2+a x^4} \, dx &=\frac{x}{a}-\frac{\int \frac{a+b+2 a x^2}{a+b+2 a x^2+a x^4} \, dx}{a}\\ &=\frac{x}{a}-\frac{\int \frac{\frac{\sqrt{2} (a+b) \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}-\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) x}{\frac{\sqrt{a+b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\int \frac{\frac{\sqrt{2} (a+b) \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) x}{\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ &=\frac{x}{a}+\frac{\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) \int \frac{-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x}{\frac{\sqrt{a+b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) \int \frac{\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x}{\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\left (a+b+2 \sqrt{a} \sqrt{a+b}\right ) \int \frac{1}{\frac{\sqrt{a+b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a^{3/2} \sqrt{a+b}}-\frac{\left (a+b+2 \sqrt{a} \sqrt{a+b}\right ) \int \frac{1}{\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a^{3/2} \sqrt{a+b}}\\ &=\frac{x}{a}+\frac{\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) \log \left (\sqrt{a+b}-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) \log \left (\sqrt{a+b}+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}+\frac{\left (a+b+2 \sqrt{a} \sqrt{a+b}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right )-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x\right )}{2 a^{3/2} \sqrt{a+b}}+\frac{\left (a+b+2 \sqrt{a} \sqrt{a+b}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right )-x^2} \, dx,x,\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x\right )}{2 a^{3/2} \sqrt{a+b}}\\ &=\frac{x}{a}+\frac{\left (a+b+2 \sqrt{a} \sqrt{a+b}\right ) \tan ^{-1}\left (\frac{\sqrt{-\sqrt{a}+\sqrt{a+b}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{a+b}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a}+\sqrt{a+b}}}-\frac{\left (a+b+2 \sqrt{a} \sqrt{a+b}\right ) \tan ^{-1}\left (\frac{\sqrt{-\sqrt{a}+\sqrt{a+b}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{a+b}}}\right )}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a}+\sqrt{a+b}}}+\frac{\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) \log \left (\sqrt{a+b}-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\left (a+b-2 \sqrt{a} \sqrt{a+b}\right ) \log \left (\sqrt{a+b}+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ \end{align*}

Mathematica [C]  time = 0.09335, 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]*Sq
rt[b]]*Sqrt[b]) + ((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])

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Maple [B]  time = 0.23, size = 1658, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

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]

Exception raised: AttributeError

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Fricas [A]  time = 1.60866, size = 1303, normalized size = 3.02 \begin{align*} \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} \end{align*}

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*sq
rt(-(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

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Sympy [A]  time = 1.06445, size = 105, normalized size = 0.24 \begin{align*} \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} \end{align*}

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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: NotImplementedError