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

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

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Rubi [A]  time = 0.52, antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1123, 1169, 634, 618, 204, 628} \begin {gather*} \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}}} \end {gather*}

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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.15, size = 174, normalized size = 0.40 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.76, size = 1582, normalized size = 3.65

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))*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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giac [B]  time = 0.38, size = 742, normalized size = 1.71 \begin {gather*} \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 \right |} {\left | -a - b \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 {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a^{2} + 2 \, a b + \sqrt {-4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a^{2} + a b\right )} + 4 \, {\left (a^{2} + a b\right )}^{2}}}{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 \right |} {\left | -a - b \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 {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a^{2} + 2 \, a b - \sqrt {-4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a^{2} + a b\right )} + 4 \, {\left (a^{2} + a b\right )}^{2}}}{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} \end {gather*}

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)*abs(-a - b) - (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))*arct
an(2*sqrt(1/2)*x/sqrt((2*a^2 + 2*a*b + sqrt(-4*(a^2 + 2*a*b + b^2)*(a^2 + a*b) + 4*(a^2 + a*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)*abs(-a - b) - (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(2*sqrt(1/2)*x/sqrt((2*a^2 +
2*a*b - sqrt(-4*(a^2 + 2*a*b + b^2)*(a^2 + a*b) + 4*(a^2 + a*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 = 0.07, size = 3318, normalized size = 7.66 \begin {gather*} \text {output too large to display} \end {gather*}

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^(3/2)/(a+b)^2/b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)*
a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(
a^2+a*b)^(1/2))^(1/2)-1/2*a^2/(a+b)^(5/2)/b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(
1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^
(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)-1/4/a^(1/2)/(a+b)^2/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^
(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))
*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)+1/2*a^2/(a+b)^(5/2)/b/(2*a+4*(a
+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1
/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)-1/4*a^(3/2
)/(a+b)^2/b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(
1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1
/2))^(1/2)+1/4/a^(1/2)/(a+b)^2/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+
2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*
(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/4/(a+b)^(5/2)*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x
+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/8*a^(1/2)/(a+b)^2*ln(-a^(1/2)*x^2+(-2*a+2*((a+b
)*a)^(1/2))^(1/2)*x-(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)+1/8*a^(1/2)/(a+b)^2*ln(a^(1/2)*x^2+(-2*a+2*((a
+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)-1/4*a/(a+b)^(5/2)*ln(a^(1/2)*x^2+(-2*a+2*((a
+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)-2*a^2/(a+b)^(5/2)/(2*a+4*(a+b)^(1/2)*a^(1/2)
-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a
+b)*a)^(1/2))^(1/2))+1/4*a/(a+b)^(5/2)*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^(1/2))*(-2*a+2*(
a^2+a*b)^(1/2))^(1/2)+2*a^2/(a+b)^(5/2)/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)
*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))+1/4/(a+b)^(5/2)*ln(-a^
(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/(a+b)
/x-1/2*a/(a+b)^(5/2)/b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)
*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*
(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)+1/2*a/(a+b)^(5/2)/b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)
*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*
a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/4*a^(1/2)/(a+b)^2/b/(2*a+4*(a+b)^(
1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a
^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)
+1/4*a^(1/2)/(a+b)^2/b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)
*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*
(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-2*a/(a+b)^(5/2)*b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*a
rctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))+1/8/a^
(1/2)/(a+b)^2*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2
+a*b)^(1/2)-1/8/a^(1/2)/(a+b)^2*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^(1/2))*(-2*a+2*(a^2+a*b
)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/8*a^(3/2)/(a+b)^2/b*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^(1
/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)+1/4*a^2/(a+b)^(5/2)/b*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+
b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)+2*a/(a+b)^(5/2)*b/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)
*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))+1/2
*a/(a+b)^(5/2)/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2)
)^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)
^(1/2))^(1/2)-1/4*a^(1/2)/(a+b)^2/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*
a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2
)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)-1/8*a^(1/2)/(a+b)^2/b*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^
(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/2/(a+b)^(5/2)/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(
1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))
^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)+1/2/(a+b)^(5/2)/(2*a+4*(
a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(
1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)
^(1/2)-1/2*a/(a+b)^(5/2)/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+2*((a+
b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+
2*(a^2+a*b)^(1/2))^(1/2)+1/4*a/(a+b)^(5/2)/b*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^(1/2))*(-2
*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)-1/4*a/(a+b)^(5/2)/b*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*
x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)+1/8*a^(1/2)/(a+b)^2/b*ln(a^(1/2)*x^2+(-2*a+2*((a
+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)*(a^2+a*b)^(1/2)+1/4*a^(1/2)/(a+b)^2/(2*a+4*(
a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^
(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)+1/8*a^(3
/2)/(a+b)^2/b*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)-1/4*
a^2/(a+b)^(5/2)/b*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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} \end {gather*}

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 = 5.27, size = 2848, normalized size = 6.58

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]

- 1/(x*(a + b)) - atan((((-(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)) - 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
)*1i - ((-(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)) + 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)*1i)/(6*a^6*b +
2*a^7 + ((-(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)) - 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) + ((-(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)) + 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) + 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 - ata
n((((-(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)) - 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)*1i - ((-(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 + 3
2*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 + 6
40*a^7*b^3 + 320*a^8*b^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)*1i)/(6*a^6*b + 2*a^7 + ((-(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)) - 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) + ((-(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)) + 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) + 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 = 4.54, size = 134, normalized size = 0.31 \begin {gather*} \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 {gather*}

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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