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

Optimal. Leaf size=121 \[ -\frac {1}{x (a-b)}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {b} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1123, 1166, 205} \[ -\frac {1}{x (a-b)}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {b} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}} \]

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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 1166

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

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

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Mathematica [A]  time = 0.15, size = 143, normalized size = 1.18 \[ \frac {\frac {\left (\sqrt {a} \sqrt {b}+a\right ) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{\sqrt {b} \sqrt {a-\sqrt {a} \sqrt {b}}}-\frac {\left (a-\sqrt {a} \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {b} \sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {2}{x}}{2 (b-a)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.79, size = 1612, normalized size = 13.32 \[ \text {result too large to display} \]

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))*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))) -
4)/((a - b)*x)

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giac [B]  time = 0.38, size = 698, normalized size = 5.77 \[ \frac {{\left ({\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} b^{2}\right )} {\left (a - b\right )}^{2} {\left | a \right |} - 2 \, {\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} a^{3} b - 7 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{3}\right )} {\left | a - b \right |} {\left | a \right |} + {\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{4} - 10 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{3} b + 11 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} b^{2} - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} - a b + \sqrt {{\left (a^{2} - a b\right )}^{2} - {\left (a^{2} - a b\right )} {\left (a^{2} - 2 \, a b + b^{2}\right )}}}{a^{2} - a b}}}\right )}{2 \, {\left (3 \, a^{6} b - 13 \, a^{5} b^{2} + 21 \, a^{4} b^{3} - 15 \, a^{3} b^{4} + 4 \, a^{2} b^{5}\right )} {\left | a - b \right |}} - \frac {{\left ({\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} b^{2}\right )} {\left (a - b\right )}^{2} {\left | a \right |} + 2 \, {\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} a^{3} b - 7 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{3}\right )} {\left | a - b \right |} {\left | a \right |} + {\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{4} - 10 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{3} b + 11 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} b^{2} - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} - a b - \sqrt {{\left (a^{2} - a b\right )}^{2} - {\left (a^{2} - a b\right )} {\left (a^{2} - 2 \, a b + b^{2}\right )}}}{a^{2} - a b}}}\right )}{2 \, {\left (3 \, a^{6} b - 13 \, a^{5} b^{2} + 21 \, a^{4} b^{3} - 15 \, a^{3} b^{4} + 4 \, a^{2} b^{5}\right )} {\left | a - b \right |}} - \frac {1}{{\left (a - b\right )} x} \]

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]

1/2*((3*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a*b - 4*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*b^2)*(a - b)^2*abs(a) - 2*
(3*sqrt(a^2 + sqrt(a*b)*a)*a^3*b - 7*sqrt(a^2 + sqrt(a*b)*a)*a^2*b^2 + 4*sqrt(a^2 + sqrt(a*b)*a)*a*b^3)*abs(a
- b)*abs(a) + (3*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^4 - 10*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^3*b + 11*sqrt(
a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^2*b^2 - 4*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a*b^3)*abs(a))*arctan(x/sqrt((a^2 -
a*b + sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b + b^2)))/(a^2 - a*b)))/((3*a^6*b - 13*a^5*b^2 + 21*a^4*b^
3 - 15*a^3*b^4 + 4*a^2*b^5)*abs(a - b)) - 1/2*((3*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a*b - 4*sqrt(a^2 - sqrt(a*
b)*a)*sqrt(a*b)*b^2)*(a - b)^2*abs(a) + 2*(3*sqrt(a^2 - sqrt(a*b)*a)*a^3*b - 7*sqrt(a^2 - sqrt(a*b)*a)*a^2*b^2
+ 4*sqrt(a^2 - sqrt(a*b)*a)*a*b^3)*abs(a - b)*abs(a) + (3*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^4 - 10*sqrt(a^2
- sqrt(a*b)*a)*sqrt(a*b)*a^3*b + 11*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^2*b^2 - 4*sqrt(a^2 - sqrt(a*b)*a)*sqr
t(a*b)*a*b^3)*abs(a))*arctan(x/sqrt((a^2 - a*b - sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b + b^2)))/(a^2 -
a*b)))/((3*a^6*b - 13*a^5*b^2 + 21*a^4*b^3 - 15*a^3*b^4 + 4*a^2*b^5)*abs(a - b)) - 1/((a - b)*x)

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maple [B]  time = 0.01, size = 180, normalized size = 1.49 \[ \frac {a^{2} \arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (-a +\sqrt {a b}\right ) a}}+\frac {a^{2} \arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (a +\sqrt {a b}\right ) a}}+\frac {a \arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{2 \left (a -b \right ) \sqrt {\left (-a +\sqrt {a b}\right ) a}}-\frac {a \arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{2 \left (a -b \right ) \sqrt {\left (a +\sqrt {a b}\right ) a}}-\frac {1}{\left (a -b \right ) x} \]

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/(a-b)/x+1/2*a/(a-b)/((-a+(a*b)^(1/2))*a)^(1/2)*arctanh(1/((-a+(a*b)^(1/2))*a)^(1/2)*a*x)+1/2*a^2/(a-b)/(a*b
)^(1/2)/((-a+(a*b)^(1/2))*a)^(1/2)*arctanh(1/((-a+(a*b)^(1/2))*a)^(1/2)*a*x)-1/2*a/(a-b)/((a+(a*b)^(1/2))*a)^(
1/2)*arctan(1/((a+(a*b)^(1/2))*a)^(1/2)*a*x)+1/2*a^2/(a-b)/(a*b)^(1/2)/((a+(a*b)^(1/2))*a)^(1/2)*arctan(1/((a+
(a*b)^(1/2))*a)^(1/2)*a*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\frac {1}{2} \, {\left (\frac {{\left (6 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b - 8 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{2} - 3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} + \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b + 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} b^{2}\right )} a \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{3 \, a^{5} b - 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}} + \frac {{\left (6 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b - 8 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{2} + 3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} - \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} b^{2}\right )} a \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{3 \, a^{5} b - 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}}\right )} a}{a - b} - \frac {1}{{\left (a - b\right )} x} \]

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]

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

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mupad [B]  time = 5.12, size = 2774, normalized size = 22.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((x*(8*a^7*b - 4*a^8 + 4*a^4*b^4 - 8*a^5*b^3) + (-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))
/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^
7*b^2 - x*(-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2))
)^(1/2)*(64*a^9*b - 64*a^4*b^6 + 320*a^5*b^5 - 640*a^6*b^4 + 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b +
3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i + (x*(8*a^7*b - 4*a
^8 + 4*a^4*b^4 - 8*a^5*b^3) - (-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3
*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^7*b^2 + x*(-(3*a*b^2 +
a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*
a^4*b^6 + 320*a^5*b^5 - 640*a^6*b^4 + 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*
(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i)/(6*a^6*b - 2*a^7 + (x*(8*a^7*b - 4*a^8 +
4*a^4*b^4 - 8*a^5*b^3) + (-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*
b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^7*b^2 - x*(-(3*a*b^2 + a^2*b
+ 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*a^4*b
^6 + 320*a^5*b^5 - 640*a^6*b^4 + 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^
3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2) - (x*(8*a^7*b - 4*a^8 + 4*a^4*b^4 - 8*a^5*b^3) - (
-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32
*a^8*b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^7*b^2 + x*(-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(
a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*a^4*b^6 + 320*a^5*b^5 - 640*a^6
*b^4 + 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b
^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2) + 2*a^4*b^3 - 6*a^5*b^2))*(-(3*a*b^2 + a^2*b + 3*a*(a*b^3)^(1/2) + b*(a*b^3)
^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*2i - 1/(x*(a - b)) + atan(((x*(8*a^7*b - 4*a^8 + 4*a
^4*b^4 - 8*a^5*b^3) + (-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3
+ a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^7*b^2 - x*(-(3*a*b^2 + a^2*b -
3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*a^4*b^6
+ 320*a^5*b^5 - 640*a^6*b^4 + 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^
(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i + (x*(8*a^7*b - 4*a^8 + 4*a^4*b^4 - 8*a^5*b^3) - (
-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32
*a^8*b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^7*b^2 + x*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(
a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*a^4*b^6 + 320*a^5*b^5 - 640*a^6
*b^4 + 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b
^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i)/(6*a^6*b - 2*a^7 + (x*(8*a^7*b - 4*a^8 + 4*a^4*b^4 - 8*a^5*b^3) + (-(3*a
*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*
b + 32*a^4*b^5 - 128*a^5*b^4 + 192*a^6*b^3 - 128*a^7*b^2 - x*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3
)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*a^4*b^6 + 320*a^5*b^5 - 640*a^6*b^4
+ 640*a^7*b^3 - 320*a^8*b^2)))*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 -
3*a^2*b^3 + a^3*b^2)))^(1/2) - (x*(8*a^7*b - 4*a^8 + 4*a^4*b^4 - 8*a^5*b^3) - (-(3*a*b^2 + a^2*b - 3*a*(a*b^3)
^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 - 128*a^5*b
^4 + 192*a^6*b^3 - 128*a^7*b^2 + x*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^
5 - 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b - 64*a^4*b^6 + 320*a^5*b^5 - 640*a^6*b^4 + 640*a^7*b^3 - 320*a^8*b^
2)))*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))^(1/
2) + 2*a^4*b^3 - 6*a^5*b^2))*(-(3*a*b^2 + a^2*b - 3*a*(a*b^3)^(1/2) - b*(a*b^3)^(1/2))/(16*(3*a*b^4 - b^5 - 3*
a^2*b^3 + a^3*b^2)))^(1/2)*2i

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sympy [A]  time = 6.14, size = 134, normalized size = 1.11 \[ \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} b^{2} - 768 a^{2} b^{3} + 768 a b^{4} - 256 b^{5}\right ) + t^{2} \left (32 a^{2} b + 96 a b^{2}\right ) + a, \left (t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} b - 128 t^{3} a^{3} b^{2} + 128 t^{3} a b^{4} - 64 t^{3} b^{5} + 4 t a^{3} + 40 t a^{2} b + 20 t a b^{2}}{3 a^{2} + a b} \right )} \right )\right )} - \frac {1}{x \left (a - b\right )} \]

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, Lam
bda(_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 + 40*
_t*a**2*b + 20*_t*a*b**2)/(3*a**2 + a*b)))) - 1/(x*(a - b))

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