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

Optimal. Leaf size=359 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}+\sqrt {a}}} \]

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Rubi [A]  time = 0.26, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1094, 634, 618, 204, 628} \begin {gather*} -\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}+\sqrt {a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b + 2*a*x^2 + a*x^4)^(-1),x]

[Out]

-ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]]/(2*Sqrt[2]*a^(1/4)*Sqr
t[a + b]*Sqrt[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^(1/4)*Sqrt[a + b]*Sqrt[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^(1/4)*Sqrt[a + b]*Sqrt[-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^(1/4)*Sqrt[a +
b]*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 1094

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

Rubi steps

\begin {align*} \int \frac {1}{a+b+2 a x^2+a x^4} \, dx &=\frac {\int \frac {\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}-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} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}+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} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}\\ &=\frac {\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 \sqrt {a} \sqrt {a+b}}+\frac {\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 \sqrt {a} \sqrt {a+b}}-\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}+\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}\\ &=-\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}+\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\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 \sqrt {a} \sqrt {a+b}}-\frac {\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 \sqrt {a} \sqrt {a+b}}\\ &=-\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}+\frac {\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} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 119, normalized size = 0.33 \begin {gather*} \frac {i \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}}}-\frac {i \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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.12, size = 567, normalized size = 1.58 \begin {gather*} \frac {1}{4} \, \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} \log \left ({\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + b\right )} \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} \log \left (-{\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + b\right )} \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} \log \left ({\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - b\right )} \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} \log \left (-{\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - b\right )} \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.25, size = 307, normalized size = 0.86 \begin {gather*} \frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} b + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b^{2} + 3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a b\right )} {\left | a \right |} \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 )}{2 \, {\left (3 \, a^{5} b + 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} b + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b^{2} - 3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{2} - 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a b\right )} {\left | a \right |} \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 )}{2 \, {\left (3 \, a^{5} b + 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*sqrt(a^2 + sqrt(-a*b)*a)*a^2*b + 4*sqrt(a^2 + sqrt(-a*b)*a)*a*b^2 + 3*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*
b)*a^2 + 4*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a*b)*abs(a)*arctan(2*sqrt(1/2)*x/sqrt((2*a + sqrt(-4*(a + b)*a
+ 4*a^2))/a))/(3*a^5*b + 7*a^4*b^2 + 4*a^3*b^3) + 1/2*(3*sqrt(a^2 - sqrt(-a*b)*a)*a^2*b + 4*sqrt(a^2 - sqrt(-a
*b)*a)*a*b^2 - 3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^2 - 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b)*abs(a)*a
rctan(2*sqrt(1/2)*x/sqrt((2*a - sqrt(-4*(a + b)*a + 4*a^2))/a))/(3*a^5*b + 7*a^4*b^2 + 4*a^3*b^3)

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maple [B]  time = 0.08, size = 913, normalized size = 2.54 \begin {gather*} -\frac {\arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{\sqrt {a +b}\, \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}+\frac {\arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{\sqrt {a +b}\, \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}+\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {a +b}\, \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, b}-\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {a +b}\, \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, b}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \ln \left (-\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x -\sqrt {a +b}\right )}{8 \sqrt {a +b}\, b}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \ln \left (\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x +\sqrt {a +b}\right )}{8 \sqrt {a +b}\, b}+\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \sqrt {a^{2}+a b}\, \arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {a +b}\, \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, a b}-\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \sqrt {a^{2}+a b}\, \arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {a +b}\, \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, a b}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \sqrt {a^{2}+a b}\, \ln \left (-\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x -\sqrt {a +b}\right )}{8 \sqrt {a +b}\, a b}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \sqrt {a^{2}+a b}\, \ln \left (\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x +\sqrt {a +b}\right )}{8 \sqrt {a +b}\, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/(a+b)^(1/2)/a/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+b)^(1/2)/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/(a+b)^(1/2)/(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)^(1/2)/a/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+b)^(1/2)/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)+1/8/(a+b)^(1/2)/a/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+b)^(1/2)/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/(a+b)^(1/2)/(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)^(1/2)/a/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+b)^(1/2)/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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a x^{4} + 2 \, a x^{2} + a + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.16, size = 986, normalized size = 2.75 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {8\,a^3\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}-\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^4\,b^2}{a^2\,b^2+a\,b^3}-\frac {2\,a^3\,b\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}-\frac {8\,a^5\,b^2\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}-\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}-\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}-\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}+\frac {8\,a^4\,b\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}-\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}\,\sqrt {-a\,b^3}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}-\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}-\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^5\,b^2\,x\,\sqrt {\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}+\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}+\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}-\frac {8\,a^3\,x\,\sqrt {\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^4\,b^2}{a^2\,b^2+a\,b^3}+\frac {2\,a^3\,b\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}+\frac {8\,a^4\,b\,x\,\sqrt {\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}}\,\sqrt {-a\,b^3}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}+\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}+\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b + 2*a*x^2 + a*x^4),x)

[Out]

2*atanh((8*a^3*x*((a*b)/(16*(a*b^3 + a^2*b^2)) - (-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^4*b^2)/(a
*b^3 + a^2*b^2) - (2*a^3*b*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2)) - (8*a^5*b^2*x*((a*b)/(16*(a*b^3 + a^2*b^2)) - (
-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a^6*b^4)/(a*b^3 + a^2*b^2) -
(2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) - (2*a^5*b^3*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2)) + (8*a^4*b*x*((a*
b)/(16*(a*b^3 + a^2*b^2)) - (-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)))^(1/2)*(-a*b^3)^(1/2))/((2*a^5*b^5)/(a*b^3 +
a^2*b^2) + (2*a^6*b^4)/(a*b^3 + a^2*b^2) - (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) - (2*a^5*b^3*(-a*b^3)
^(1/2))/(a*b^3 + a^2*b^2)))*((a*b - (-a*b^3)^(1/2))/(16*(a*b^3 + a^2*b^2)))^(1/2) - 2*atanh((8*a^5*b^2*x*((-a*
b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)) + (a*b)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a
^6*b^4)/(a*b^3 + a^2*b^2) + (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) + (2*a^5*b^3*(-a*b^3)^(1/2))/(a*b^3 +
a^2*b^2)) - (8*a^3*x*((-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)) + (a*b)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^4*b^
2)/(a*b^3 + a^2*b^2) + (2*a^3*b*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2)) + (8*a^4*b*x*((-a*b^3)^(1/2)/(16*(a*b^3 + a
^2*b^2)) + (a*b)/(16*(a*b^3 + a^2*b^2)))^(1/2)*(-a*b^3)^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a^6*b^4)/(a
*b^3 + a^2*b^2) + (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) + (2*a^5*b^3*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2))
)*((a*b + (-a*b^3)^(1/2))/(16*(a*b^3 + a^2*b^2)))^(1/2)

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sympy [A]  time = 1.24, size = 63, normalized size = 0.18 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \left (256 a^{2} b^{2} + 256 a b^{3}\right ) - 32 t^{2} a b + 1, \left (t \mapsto t \log {\left (64 t^{3} a^{2} b + 64 t^{3} a b^{2} - 4 t a + 4 t b + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**4*(256*a**2*b**2 + 256*a*b**3) - 32*_t**2*a*b + 1, Lambda(_t, _t*log(64*_t**3*a**2*b + 64*_t**3*a*
b**2 - 4*_t*a + 4*_t*b + x)))

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