3.914 \(\int \frac{1}{x^2 (a+b+2 a x^2+a x^4)} \, dx\)

Optimal. Leaf size=433 \[ \frac{\sqrt [4]{a} \left (2 \sqrt{a}-\sqrt{a+b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\sqrt [4]{a} \left (2 \sqrt{a}-\sqrt{a+b}\right ) \log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{1}{x (a+b)}+\frac{\sqrt [4]{a} \left (\sqrt{a+b}+2 \sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\sqrt [4]{a} \left (\sqrt{a+b}+2 \sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

[Out]

-(1/((a + b)*x)) + (a^(1/4)*(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 + b)^(3/2)*Sqrt[Sqrt[a] + Sqrt[a + b]]) - (a^(1/4)*(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 + b)^(3/2)*Sqrt[Sqrt[a] + Sqrt[a + b]]) + (a^(1/4)*(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 + b)^(3/2)*Sqrt[-Sqrt[a] + Sqrt[a + b]]) -
(a^(1/4)*(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 + b)^(3/2)*Sqrt[-Sqrt[a] + Sqrt[a + b]])

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Rubi [A]  time = 0.518721, antiderivative size = 433, 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 = {1123, 1169, 634, 618, 204, 628} \[ \frac{\sqrt [4]{a} \left (2 \sqrt{a}-\sqrt{a+b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\sqrt [4]{a} \left (2 \sqrt{a}-\sqrt{a+b}\right ) \log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{1}{x (a+b)}+\frac{\sqrt [4]{a} \left (\sqrt{a+b}+2 \sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\sqrt [4]{a} \left (\sqrt{a+b}+2 \sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

-(1/((a + b)*x)) + (a^(1/4)*(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 + b)^(3/2)*Sqrt[Sqrt[a] + Sqrt[a + b]]) - (a^(1/4)*(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 + b)^(3/2)*Sqrt[Sqrt[a] + Sqrt[a + b]]) + (a^(1/4)*(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 + b)^(3/2)*Sqrt[-Sqrt[a] + Sqrt[a + b]]) -
(a^(1/4)*(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 + b)^(3/2)*Sqrt[-Sqrt[a] + Sqrt[a + b]])

Rule 1123

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2 +
c*x^4)^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[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{1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx &=-\frac{1}{(a+b) x}+\frac{\int \frac{-2 a-a x^2}{a+b+2 a x^2+a x^4} \, dx}{a+b}\\ &=-\frac{1}{(a+b) x}+\frac{\int \frac{-2 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}-\left (-2 a+\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} \sqrt [4]{a} (a+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}+\frac{\int \frac{-2 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}+\left (-2 a+\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} \sqrt [4]{a} (a+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ &=-\frac{1}{(a+b) x}+\frac{\left (\sqrt [4]{a} \left (2 \sqrt{a}-\sqrt{a+b}\right )\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+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\left (\sqrt [4]{a} \left (2 \sqrt{a}-\sqrt{a+b}\right )\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+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\left (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+b)^{3/2}}-\frac{\left (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+b)^{3/2}}\\ &=-\frac{1}{(a+b) x}+\frac{\sqrt [4]{a} \left (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+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\sqrt [4]{a} \left (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+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}+\frac{\left (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+b)^{3/2}}+\frac{\left (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+b)^{3/2}}\\ &=-\frac{1}{(a+b) x}+\frac{\sqrt [4]{a} \left (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+b)^{3/2} \sqrt{\sqrt{a}+\sqrt{a+b}}}-\frac{\sqrt [4]{a} \left (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+b)^{3/2} \sqrt{\sqrt{a}+\sqrt{a+b}}}+\frac{\sqrt [4]{a} \left (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+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\sqrt [4]{a} \left (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+b)^{3/2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ \end{align*}

Mathematica [C]  time = 0.141398, size = 174, normalized size = 0.4 \[ \frac{1}{x (-a-b)}+\frac{\left (-\sqrt{a} \sqrt{b}+i a\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{b} \sqrt{a-i \sqrt{a} \sqrt{b}} (a+b)}+\frac{\left (-\sqrt{a} \sqrt{b}-i a\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{b} \sqrt{a+i \sqrt{a} \sqrt{b}} (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

1/((-a - b)*x) + ((I*a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/(2*Sqrt[a - I*Sqrt[
a]*Sqrt[b]]*Sqrt[b]*(a + b)) + (((-I)*a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(2
*Sqrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b))

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Maple [B]  time = 0.173, size = 3318, normalized size = 7.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a*x^4+2*a*x^2+a+b),x)

[Out]

-1/4*a^(1/2)/(a+b)^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)*(a^2+a*b)^(1/2)+1/2*a/(a+b)^(5/2)/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)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/2*a/(a+b)^(5/2)/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)*(a^2+a*b)^(1/2)+1/4*a^(1/2
)/(a+b)^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)*(a^2+a*b)^(1/2)+1/8/a^(1/2)/(a+b)^2*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)*(a^2+a*b)^(1/2)+1/8*a^(3/2)/(a+b)^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)-1/4*a^2/(a+b)^(5/2)/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)+1/4*a^2/(a+b)^(5/2)/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)+2*a/(a+b)^(5/2)*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
))-1/8/a^(1/2)/(a+b)^2*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)*(a^2+a*b)^(1/2)-1/8*a^(3/2)/(a+b)^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)-2*a/(a+b)^(5/2)*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))+1/4*a/(a+b)^(5/2)*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)+2*a^2/(a+b)^(5/2)/(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))-1/8*a^(1/2)/(a+b)^2*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)+1/8*a^(1/2)/(a+b)^2*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)-1/4*a/(a+b)^(5/2)*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)-2*a^2/(a+b)^(5/2)/(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))-1/4/(a
+b)^(5/2)*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)*(a^2+a*b)^
(1/2)+1/4/(a+b)^(5/2)*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)*(a^2+a*b)^(1/2)-1/(a+b)/x-1/4/a^(1/2)/(a+b)^2/(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)*(a^2+a*b)^(1/2)-1/2*a^2/(a+b)^(5/2)/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)+1/2*a^2/(a+b)^(5/2)/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)+1/4/a^(1/2)/
(a+b)^2/(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)*(a^2+a*b)^(1/2)+1/4*a^(3/2)/(a+b)^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)-1/4*a^(3/2)/(a+b)^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)+1/2/(a+b)^(5/2)/(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)*(a^2+a*b)^(1/2)-1/2/(a+b)^(5/2)/(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)*(a^2+a*b)^(
1/2)+1/4*a^(1/2)/(a+b)^2/(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)-1/4*a/(a+b)^(5/2)/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)*(a^2+a*b)^(1/2)+1/4*a/(a+b)^(5/2)/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)*(a^2+a*b)^(1/2)-1/8*a^(1/2)/(a+b)^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)*(a^2+a*b)^(1/2)-1/4*a^(1/2)/(a+b)^2/(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^(1/2)/(a+b)^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)*(a^2+a*b)^(1/2)-1/
2*a/(a+b)^(5/2)/(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)+1/2*a/(a+b)^(5/2)/(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(1/x^2/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")

[Out]

Exception raised: AttributeError

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Fricas [B]  time = 1.61757, size = 3301, normalized size = 7.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")

[Out]

1/4*((a + b)*x*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b
+ 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*log
(-(3*a^2 - a*b)*x + (6*a^2*b - 2*a*b^2 + (a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(
a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2
*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^
5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))) - (a + b)*x*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2
+ 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6
*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*log(-(3*a^2 - a*b)*x - (6*a^2*b - 2*a*b^2 + (a^4*b + 2*a^
3*b^2 - 2*a*b^4 - b^5)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b
^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/
(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^
4))) + (a + b)*x*sqrt((a^2 - 3*a*b - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*
b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*l
og(-(3*a^2 - a*b)*x + (6*a^2*b - 2*a*b^2 - (a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)
/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b - (a^3*b + 3*a
^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*
b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))) - (a + b)*x*sqrt((a^2 - 3*a*b - (a^3*b + 3*a^2*b^
2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 +
6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*log(-(3*a^2 - a*b)*x - (6*a^2*b - 2*a*b^2 - (a^4*b + 2*
a^3*b^2 - 2*a*b^4 - b^5)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2
*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2
)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 +
b^4))) - 4)/((a + b)*x)

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Sympy [A]  time = 2.23106, size = 134, normalized size = 0.31 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} b^{2} + 768 a^{2} b^{3} + 768 a b^{4} + 256 b^{5}\right ) + t^{2} \left (- 32 a^{2} b + 96 a b^{2}\right ) + a, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} b - 128 t^{3} a^{3} b^{2} + 128 t^{3} a b^{4} + 64 t^{3} b^{5} + 4 t a^{3} - 40 t a^{2} b + 20 t a b^{2}}{3 a^{2} - a b} \right )} \right )\right )} - \frac{1}{x \left (a + b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a*x**4+2*a*x**2+a+b),x)

[Out]

RootSum(_t**4*(256*a**3*b**2 + 768*a**2*b**3 + 768*a*b**4 + 256*b**5) + _t**2*(-32*a**2*b + 96*a*b**2) + a, La
mbda(_t, _t*log(x + (-64*_t**3*a**4*b - 128*_t**3*a**3*b**2 + 128*_t**3*a*b**4 + 64*_t**3*b**5 + 4*_t*a**3 - 4
0*_t*a**2*b + 20*_t*a*b**2)/(3*a**2 - a*b)))) - 1/(x*(a + b))

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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(1/x^2/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError