3.10.13 \(\int \frac {1}{a+b+2 a x^2+a x^4} \, dx\) [913]

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

[Out]

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

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Rubi [A]
time = 0.18, 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 = {1108, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\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 {\text {ArcTan}\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 {\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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b + 2*a*x^2 + a*x^4)^(-1),x]

[Out]

-1/2*ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]]/(Sqrt[2]*a^(1/4)*S
qrt[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 210

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

Rule 632

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 642

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 648

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 1108

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 {\text {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 {\text {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] Result contains complex when optimal does not.
time = 0.04, 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 {a-i \sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {i \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a+i \sqrt {a} \sqrt {b}} \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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(243)=486\).
time = 0.06, size = 499, normalized size = 1.39

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a}\)	\(38\)
default	\(\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \ln \left (-x^{2} \sqrt {a}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}-\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (-2 \sqrt {a}\, b +\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {-2 x \sqrt {a}+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 \sqrt {a +b}\, \sqrt {a}\, b}+\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \ln \left (x^{2} \sqrt {a}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (2 \sqrt {a}\, b -\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {2 x \sqrt {a}+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 \sqrt {a +b}\, \sqrt {a}\, b}\)	\(499\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (245) = 490\).
time = 0.34, 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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Sympy [A]
time = 0.45, size = 63, normalized size = 0.18 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \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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Giac [A]
time = 3.86, 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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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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