3.10.11 \(\int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx\) [911]

Optimal. Leaf size=432 \[ \frac {x}{a}+\frac {\left (a+b+2 \sqrt {a} \sqrt {a+b}\right ) \tan ^{-1}\left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{5/4} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b+2 \sqrt {a} \sqrt {a+b}\right ) \tan ^{-1}\left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{5/4} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\left (a+b-2 \sqrt {a} \sqrt {a+b}\right ) \log \left (\sqrt {a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b-2 \sqrt {a} \sqrt {a+b}\right ) \log \left (\sqrt {a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}} \]

[Out]

x/a+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+b-2*a^(1/2)*(a+b)^(1/2))
/a^(5/4)*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+b-2*a^(1/2)*(a+b)^(1/2))/a^(5/4)*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+b+2*a^(1/2)*(a+
b)^(1/2))/a^(5/4)*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+b+2*a^(1/2)*(a+b)^(1/2))/a^(5/4)*2^(1/2)/(a+b)^(1/2)/(a^(1/2)+
(a+b)^(1/2))^(1/2)

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Rubi [A]
time = 0.59, antiderivative size = 432, 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 = {1136, 1183, 648, 632, 210, 642} \begin {gather*} \frac {\left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \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} a^{5/4} \sqrt {a+b} \sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {\left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \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} a^{5/4} \sqrt {a+b} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {\left (-2 \sqrt {a} \sqrt {a+b}+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^{5/4} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\left (-2 \sqrt {a} \sqrt {a+b}+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^{5/4} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b + 2*a*x^2 + a*x^4),x]

[Out]

x/a + ((a + b + 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^(5/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) - ((a + b + 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^(5/4)*S
qrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) + ((a + b - 2*Sqrt[a]*Sqrt[a + b])*Log[Sqrt[a + b] - Sqrt[2]*a^(1/4)*S
qrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) - (
(a + b - 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^(5/4)*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 1136

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*
x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1183

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 {x^4}{a+b+2 a x^2+a x^4} \, dx &=\frac {x}{a}-\frac {\int \frac {a+b+2 a x^2}{a+b+2 a x^2+a x^4} \, dx}{a}\\ &=\frac {x}{a}-\frac {\int \frac {\frac {\sqrt {2} (a+b) \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}-\left (a+b-2 \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} a^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\int \frac {\frac {\sqrt {2} (a+b) \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}+\left (a+b-2 \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} a^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}\\ &=\frac {x}{a}+\frac {\left (a+b-2 \sqrt {a} \sqrt {a+b}\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^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b-2 \sqrt {a} \sqrt {a+b}\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^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b+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^{3/2} \sqrt {a+b}}-\frac {\left (a+b+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^{3/2} \sqrt {a+b}}\\ &=\frac {x}{a}+\frac {\left (a+b-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^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b-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^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}+\frac {\left (a+b+2 \sqrt {a} \sqrt {a+b}\right ) \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 a^{3/2} \sqrt {a+b}}+\frac {\left (a+b+2 \sqrt {a} \sqrt {a+b}\right ) \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 a^{3/2} \sqrt {a+b}}\\ &=\frac {x}{a}+\frac {\left (a+b+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^{5/4} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b+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^{5/4} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\left (a+b-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^{5/4} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b-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^{5/4} \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 = 0.07, size = 164, normalized size = 0.38 \begin {gather*} \frac {x}{a}-\frac {i \left (\sqrt {a}-i \sqrt {b}\right )^2 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {a-i \sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {i \left (\sqrt {a}+i \sqrt {b}\right )^2 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {a+i \sqrt {a} \sqrt {b}} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/a - ((I/2)*(Sqrt[a] - I*Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a - I*Sqrt[a]*Sq
rt[b]]*Sqrt[b]) + ((I/2)*(Sqrt[a] + I*Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(a*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. \(776\) vs. \(2(300)=600\).
time = 0.10, size = 777, normalized size = 1.80

method	result	size
risch	\(\frac {x}{a}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (-2 \textit {\_R}^{2} a -a -b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a^{2}}\)	\(57\)
default	\(\frac {x}{a}+\frac {\frac {-\frac {\left (-a^{\frac {3}{2}} \sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a}\, \sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+2 \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\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}\, a b +\frac {\left (-a^{\frac {3}{2}} \sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a}\, \sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+2 \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\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 b a}+\frac {\frac {\left (-a^{\frac {3}{2}} \sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a}\, \sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+2 \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\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}\, a b -\frac {\left (-a^{\frac {3}{2}} \sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a}\, \sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+2 \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\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 b a}}{a}\)	\(777\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 615, normalized size = 1.42 \begin {gather*} \frac {a \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b - a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}}\right ) - a \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b - a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}}\right ) - a \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b + a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}}\right ) + a \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b + a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}}\right ) + 4 \, x}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))*log(-(3*a^2 + 2*a*b - b^2)*x + (a^
4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + 3*a^2*b - a*b^2)*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) +
a - 3*b)/(a^2*b))) - a*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))*log(-(3*a^2 + 2*a
*b - b^2)*x - (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + 3*a^2*b - a*b^2)*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b
+ b^2)/(a^5*b)) + a - 3*b)/(a^2*b))) - a*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))
*log(-(3*a^2 + 2*a*b - b^2)*x + (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - 3*a^2*b + a*b^2)*sqrt(-(a^2*b*sq
rt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))) + a*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b))
- a + 3*b)/(a^2*b))*log(-(3*a^2 + 2*a*b - b^2)*x - (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - 3*a^2*b + a*b
^2)*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))) + 4*x)/a

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Sympy [A]
time = 0.78, size = 105, normalized size = 0.24 \begin {gather*} \operatorname {RootSum} {\left (256 t^{4} a^{5} b^{2} + t^{2} \left (- 32 a^{4} b + 96 a^{3} b^{2}\right ) + a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} b + 4 t a^{3} - 24 t a^{2} b + 4 t a b^{2}}{3 a^{2} + 2 a b - b^{2}} \right )} \right )\right )} + \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**5*b**2 + _t**2*(-32*a**4*b + 96*a**3*b**2) + a**3 + 3*a**2*b + 3*a*b**2 + b**3, Lambda(_t
, _t*log(x + (-64*_t**3*a**4*b + 4*_t*a**3 - 24*_t*a**2*b + 4*_t*a*b**2)/(3*a**2 + 2*a*b - b**2)))) + x/a

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Giac [A]
time = 3.70, size = 533, normalized size = 1.23 \begin {gather*} \frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{4} + \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{3} b - 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{2} b^{2} + 2 \, {\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 )} a^{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 |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} + \sqrt {a^{4} - {\left (a^{2} + a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b + 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} - \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{4} + \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{3} b - 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{2} b^{2} + 2 \, {\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 )} a^{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 |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} - \sqrt {a^{4} - {\left (a^{2} + a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b + 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} + \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(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^4 + sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^3*b - 4*sqrt(a^2 + sqrt
(-a*b)*a)*sqrt(-a*b)*a^2*b^2 + 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^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))*arctan(x/sqrt((a^2 + sqrt(a^4 - (a^2 + a*b)*a^2))/a^2))/(3*a^6*b + 7*a^5*b^2 + 4*a^4*b^3
) - 1/2*(3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^4 + sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^3*b - 4*sqrt(a^2 -
sqrt(-a*b)*a)*sqrt(-a*b)*a^2*b^2 + 2*(3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b + 4*sqrt(a^2 - sqrt(-a*b)*a)*s
qrt(-a*b)*b^2)*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 - sqr
t(-a*b)*a)*a*b^3)*abs(a))*arctan(x/sqrt((a^2 - sqrt(a^4 - (a^2 + a*b)*a^2))/a^2))/(3*a^6*b + 7*a^5*b^2 + 4*a^4
*b^3) + x/a

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Mupad [B]
time = 4.65, size = 1147, normalized size = 2.66 \begin {gather*} \frac {x}{a}+2\,\mathrm {atanh}\left (\frac {24\,x\,\sqrt {-a^5\,b^3}\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{\frac {6\,\sqrt {-a^5\,b^3}}{a}+4\,a\,b^2+6\,a^2\,b-2\,b^3-\frac {2\,b^2\,\sqrt {-a^5\,b^3}}{a^3}+\frac {4\,b\,\sqrt {-a^5\,b^3}}{a^2}}-\frac {8\,x\,\sqrt {-a^5\,b^3}\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{\frac {4\,\sqrt {-a^5\,b^3}}{a}+\frac {6\,\sqrt {-a^5\,b^3}}{b}-2\,a\,b^2+4\,a^2\,b+6\,a^3-\frac {2\,b\,\sqrt {-a^5\,b^3}}{a^2}}-\frac {8\,a\,b^2\,x\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b+\frac {4\,\sqrt {-a^5\,b^3}}{a^2}+6\,a^2-2\,b^2+\frac {6\,\sqrt {-a^5\,b^3}}{a\,b}-\frac {2\,b\,\sqrt {-a^5\,b^3}}{a^3}}+\frac {24\,a^2\,b\,x\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b+\frac {4\,\sqrt {-a^5\,b^3}}{a^2}+6\,a^2-2\,b^2+\frac {6\,\sqrt {-a^5\,b^3}}{a\,b}-\frac {2\,b\,\sqrt {-a^5\,b^3}}{a^3}}\right )\,\sqrt {\frac {3\,a\,\sqrt {-a^5\,b^3}-b\,\sqrt {-a^5\,b^3}+a^4\,b-3\,a^3\,b^2}{16\,a^5\,b^2}}+2\,\mathrm {atanh}\left (\frac {24\,x\,\sqrt {-a^5\,b^3}\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}-\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}+\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{\frac {6\,\sqrt {-a^5\,b^3}}{a}-4\,a\,b^2-6\,a^2\,b+2\,b^3-\frac {2\,b^2\,\sqrt {-a^5\,b^3}}{a^3}+\frac {4\,b\,\sqrt {-a^5\,b^3}}{a^2}}-\frac {8\,x\,\sqrt {-a^5\,b^3}\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}-\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}+\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{\frac {4\,\sqrt {-a^5\,b^3}}{a}+\frac {6\,\sqrt {-a^5\,b^3}}{b}+2\,a\,b^2-4\,a^2\,b-6\,a^3-\frac {2\,b\,\sqrt {-a^5\,b^3}}{a^2}}-\frac {8\,a\,b^2\,x\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}-\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}+\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b-\frac {4\,\sqrt {-a^5\,b^3}}{a^2}+6\,a^2-2\,b^2-\frac {6\,\sqrt {-a^5\,b^3}}{a\,b}+\frac {2\,b\,\sqrt {-a^5\,b^3}}{a^3}}+\frac {24\,a^2\,b\,x\,\sqrt {\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}-\frac {3\,\sqrt {-a^5\,b^3}}{16\,a^4\,b^2}+\frac {\sqrt {-a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b-\frac {4\,\sqrt {-a^5\,b^3}}{a^2}+6\,a^2-2\,b^2-\frac {6\,\sqrt {-a^5\,b^3}}{a\,b}+\frac {2\,b\,\sqrt {-a^5\,b^3}}{a^3}}\right )\,\sqrt {-\frac {3\,a\,\sqrt {-a^5\,b^3}-b\,\sqrt {-a^5\,b^3}-a^4\,b+3\,a^3\,b^2}{16\,a^5\,b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b + 2*a*x^2 + a*x^4),x)

[Out]

x/a + 2*atanh((24*x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b^3)^(1/2))/(16*a^4*b^2) - (-a^5*b^3)
^(1/2)/(16*a^5*b))^(1/2))/((6*(-a^5*b^3)^(1/2))/a + 4*a*b^2 + 6*a^2*b - 2*b^3 - (2*b^2*(-a^5*b^3)^(1/2))/a^3 +
(4*b*(-a^5*b^3)^(1/2))/a^2) - (8*x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b^3)^(1/2))/(16*a^4*b
^2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((4*(-a^5*b^3)^(1/2))/a + (6*(-a^5*b^3)^(1/2))/b - 2*a*b^2 + 4*a^2*b
+ 6*a^3 - (2*b*(-a^5*b^3)^(1/2))/a^2) - (8*a*b^2*x*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b^3)^(1/2))/(16*a^4*b^
2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b + (4*(-a^5*b^3)^(1/2))/a^2 + 6*a^2 - 2*b^2 + (6*(-a^5*b^3)^(1/
2))/(a*b) - (2*b*(-a^5*b^3)^(1/2))/a^3) + (24*a^2*b*x*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b^3)^(1/2))/(16*a^4*
b^2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b + (4*(-a^5*b^3)^(1/2))/a^2 + 6*a^2 - 2*b^2 + (6*(-a^5*b^3)^(
1/2))/(a*b) - (2*b*(-a^5*b^3)^(1/2))/a^3))*((3*a*(-a^5*b^3)^(1/2) - b*(-a^5*b^3)^(1/2) + a^4*b - 3*a^3*b^2)/(1
6*a^5*b^2))^(1/2) + 2*atanh((24*x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2) - (3*(-a^5*b^3)^(1/2))/(16*a^4*b^2
) + (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((6*(-a^5*b^3)^(1/2))/a - 4*a*b^2 - 6*a^2*b + 2*b^3 - (2*b^2*(-a^5*b^3
)^(1/2))/a^3 + (4*b*(-a^5*b^3)^(1/2))/a^2) - (8*x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2) - (3*(-a^5*b^3)^(1
/2))/(16*a^4*b^2) + (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((4*(-a^5*b^3)^(1/2))/a + (6*(-a^5*b^3)^(1/2))/b + 2*a
*b^2 - 4*a^2*b - 6*a^3 - (2*b*(-a^5*b^3)^(1/2))/a^2) - (8*a*b^2*x*(1/(16*a*b) - 3/(16*a^2) - (3*(-a^5*b^3)^(1/
2))/(16*a^4*b^2) + (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b - (4*(-a^5*b^3)^(1/2))/a^2 + 6*a^2 - 2*b^2 - (6*
(-a^5*b^3)^(1/2))/(a*b) + (2*b*(-a^5*b^3)^(1/2))/a^3) + (24*a^2*b*x*(1/(16*a*b) - 3/(16*a^2) - (3*(-a^5*b^3)^(
1/2))/(16*a^4*b^2) + (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b - (4*(-a^5*b^3)^(1/2))/a^2 + 6*a^2 - 2*b^2 - (
6*(-a^5*b^3)^(1/2))/(a*b) + (2*b*(-a^5*b^3)^(1/2))/a^3))*(-(3*a*(-a^5*b^3)^(1/2) - b*(-a^5*b^3)^(1/2) - a^4*b
+ 3*a^3*b^2)/(16*a^5*b^2))^(1/2)

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