\(\int \frac {c+d x+e x^2+f x^3+g x^4}{(a+b x^4)^2} \, dx\) [176]

Optimal result

Integrand size = 30, antiderivative size = 341 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}} \]

[Out]

1/4*x*(b*f*x^3+b*e*x^2+b*d*x-a*g+b*c)/a/b/(b*x^4+a)+1/4*d*arctan(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(1/2)-1/32*ln(
-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(3*b*c+a*g-e*a^(1/2)*b^(1/2))/a^(7/4)/b^(5/4)*2^(1/2)+1/32*ln(
a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(3*b*c+a*g-e*a^(1/2)*b^(1/2))/a^(7/4)/b^(5/4)*2^(1/2)+1/16*arct
an(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*(3*b*c+a*g+e*a^(1/2)*b^(1/2))/a^(7/4)/b^(5/4)*2^(1/2)+1/16*arctan(1+b^(1/4)*x
*2^(1/2)/a^(1/4))*(3*b*c+a*g+e*a^(1/2)*b^(1/2))/a^(7/4)/b^(5/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1872, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} \sqrt {b} e+a g+3 b c\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} \sqrt {b} e+a g+3 b c\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-\sqrt {a} \sqrt {b} e+a g+3 b c\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-\sqrt {a} \sqrt {b} e+a g+3 b c\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )} \]

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4)^2,x]

[Out]

(x*(b*c - a*g + b*d*x + b*e*x^2 + b*f*x^3))/(4*a*b*(a + b*x^4)) + (d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)
*Sqrt[b]) - ((3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(
5/4)) + ((3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(5/4)
) - ((3*b*c - Sqrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(
7/4)*b^(5/4)) + ((3*b*c - Sqrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16
*Sqrt[2]*a^(7/4)*b^(5/4))

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 211

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1176

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

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

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

Rule 1872

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1890

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \frac {-3 b c-a g-2 b d x-b e x^2}{a+b x^4} \, dx}{4 a b} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \left (-\frac {2 b d x}{a+b x^4}+\frac {-3 b c-a g-b e x^2}{a+b x^4}\right ) \, dx}{4 a b} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \frac {-3 b c-a g-b e x^2}{a+b x^4} \, dx}{4 a b}+\frac {d \int \frac {x}{a+b x^4} \, dx}{2 a} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {d \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{4 a}+\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{8 a^{3/2} b^{3/2}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{8 a^{3/2} b^{3/2}} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} b^{5/4}}-\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{3/2}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{3/2}} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e+a g\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\frac {-\frac {8 a^{3/4} \sqrt [4]{b} (a (f+g x)-b x (c+x (d+e x)))}{a+b x^4}-2 \left (3 \sqrt {2} b c+4 \sqrt [4]{a} b^{3/4} d+\sqrt {2} \sqrt {a} \sqrt {b} e+\sqrt {2} a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (3 \sqrt {2} b c-4 \sqrt [4]{a} b^{3/4} d+\sqrt {2} \sqrt {a} \sqrt {b} e+\sqrt {2} a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-3 b c+\sqrt {a} \sqrt {b} e-a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \left (3 b c-\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{32 a^{7/4} b^{5/4}} \]

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4)^2,x]

[Out]

((-8*a^(3/4)*b^(1/4)*(a*(f + g*x) - b*x*(c + x*(d + e*x))))/(a + b*x^4) - 2*(3*Sqrt[2]*b*c + 4*a^(1/4)*b^(3/4)
*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e + Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b*c - 4*a
^(1/4)*b^(3/4)*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e + Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*
(-3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b*c - S
qrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(5/4))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.52 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.31

method	result	size
risch	\(\frac {\frac {e \,x^{3}}{4 a}+\frac {d \,x^{2}}{4 a}-\frac {\left (a g -b c \right ) x}{4 a b}-\frac {f}{4 b}}{b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +2 \textit {\_R} d +\frac {a g +3 b c}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b a}\)	\(105\)
default	\(\frac {\frac {e \,x^{3}}{4 a}+\frac {d \,x^{2}}{4 a}-\frac {\left (a g -b c \right ) x}{4 a b}-\frac {f}{4 b}}{b \,x^{4}+a}+\frac {\frac {\left (a g +3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {b d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{4 b a}\)	\(291\)

[In]

int((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

(1/4/a*e*x^3+1/4*d/a*x^2-1/4*(a*g-b*c)/a/b*x-1/4*f/b)/(b*x^4+a)+1/16/b/a*sum((_R^2*e+2*_R*d+1/b*(a*g+3*b*c))/_
R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 26.61 (sec) , antiderivative size = 352423, normalized size of antiderivative = 1033.50 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\frac {b e x^{3} + b d x^{2} - a f + {\left (b c - a g\right )} x}{4 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} + \frac {\frac {\sqrt {2} {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g - 4 \, \sqrt {a} b^{\frac {3}{2}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 4 \, \sqrt {a} b^{\frac {3}{2}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{32 \, a b} \]

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x, algorithm="maxima")

[Out]

1/4*(b*e*x^3 + b*d*x^2 - a*f + (b*c - a*g)*x)/(a*b^2*x^4 + a^2*b) + 1/32*(sqrt(2)*(3*b^(3/2)*c - sqrt(a)*b*e +
a*sqrt(b)*g)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) - sqrt(2)*(3*b^(3/2)*c
- sqrt(a)*b*e + a*sqrt(b)*g)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(3*s
qrt(2)*a^(1/4)*b^(7/4)*c + sqrt(2)*a^(3/4)*b^(5/4)*e + sqrt(2)*a^(5/4)*b^(3/4)*g - 4*sqrt(a)*b^(3/2)*d)*arctan
(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(
3/4)) + 2*(3*sqrt(2)*a^(1/4)*b^(7/4)*c + sqrt(2)*a^(3/4)*b^(5/4)*e + sqrt(2)*a^(5/4)*b^(3/4)*g + 4*sqrt(a)*b^(
3/2)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a
)*sqrt(b))*b^(3/4)))/(a*b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\frac {b e x^{3} + b d x^{2} + b c x - a g x - a f}{4 \, {\left (b x^{4} + a\right )} a b} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {1}{4}} a b g + \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {1}{4}} a b g + \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {1}{4}} a b g - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {1}{4}} a b g - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{3}} \]

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x, algorithm="giac")

[Out]

1/4*(b*e*x^3 + b*d*x^2 + b*c*x - a*g*x - a*f)/((b*x^4 + a)*a*b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 3*
(a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g + (a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a
/b)^(1/4))/(a^2*b^3) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 3*(a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g +
(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^3) + 1/32*sqrt(2)*(3*(a*b
^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g - (a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^3
) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g - (a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/
4) + sqrt(a/b))/(a^2*b^3)

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 1383, normalized size of antiderivative = 4.06 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {a^2\,e\,g^2+6\,a\,b\,c\,e\,g-4\,a\,b\,d^2\,g+a\,b\,e^3+9\,b^2\,c^2\,e-12\,b^2\,c\,d^2}{64\,a^3}-\frac {\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2+3072\,a^4\,b^4\,c\,e\,z^2+2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z-128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z-1152\,a^2\,b^4\,c^2\,d\,z-16\,a^2\,b^2\,d^2\,e\,g+12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3+54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4+81\,b^4\,c^4+a^2\,b^2\,e^4+a^4\,g^4,z,k\right )\,b\,\left (9\,b^2\,c^2\,x+a^2\,g^2\,x+\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2+3072\,a^4\,b^4\,c\,e\,z^2+2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z-128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z-1152\,a^2\,b^4\,c^2\,d\,z-16\,a^2\,b^2\,d^2\,e\,g+12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3+54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4+81\,b^4\,c^4+a^2\,b^2\,e^4+a^4\,g^4,z,k\right )\,a^3\,b\,g\,16-a\,b\,e^2\,x+\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2+3072\,a^4\,b^4\,c\,e\,z^2+2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z-128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z-1152\,a^2\,b^4\,c^2\,d\,z-16\,a^2\,b^2\,d^2\,e\,g+12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3+54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4+81\,b^4\,c^4+a^2\,b^2\,e^4+a^4\,g^4,z,k\right )\,a^2\,b^2\,c\,48+4\,a\,b\,d\,e-\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2+3072\,a^4\,b^4\,c\,e\,z^2+2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z-128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z-1152\,a^2\,b^4\,c^2\,d\,z-16\,a^2\,b^2\,d^2\,e\,g+12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3+54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4+81\,b^4\,c^4+a^2\,b^2\,e^4+a^4\,g^4,z,k\right )\,a^2\,b^2\,d\,x\,32+6\,a\,b\,c\,g\,x\right )}{a^2\,4}-\frac {b\,d\,x\,\left (-2\,b\,d^2+3\,b\,c\,e+a\,e\,g\right )}{16\,a^3}\right )\,\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2+3072\,a^4\,b^4\,c\,e\,z^2+2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z-128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z-1152\,a^2\,b^4\,c^2\,d\,z-16\,a^2\,b^2\,d^2\,e\,g+12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3+54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4+81\,b^4\,c^4+a^2\,b^2\,e^4+a^4\,g^4,z,k\right )\right )+\frac {\frac {d\,x^2}{4\,a}-\frac {f}{4\,b}+\frac {e\,x^3}{4\,a}+\frac {x\,\left (b\,c-a\,g\right )}{4\,a\,b}}{b\,x^4+a} \]

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4)^2,x)

[Out]

symsum(log(- (9*b^2*c^2*e - 12*b^2*c*d^2 + a^2*e*g^2 + a*b*e^3 - 4*a*b*d^2*g + 6*a*b*c*e*g)/(64*a^3) - (root(6
5536*a^7*b^5*z^4 + 1024*a^5*b^3*e*g*z^2 + 3072*a^4*b^4*c*e*z^2 + 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z -
128*a^4*b^2*d*g^2*z + 128*a^3*b^3*d*e^2*z - 1152*a^2*b^4*c^2*d*z - 16*a^2*b^2*d^2*e*g + 12*a^2*b^2*c*e^2*g - 4
8*a*b^3*c*d^2*e + 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 + 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 +
16*a*b^3*d^4 + 81*b^4*c^4 + a^2*b^2*e^4 + a^4*g^4, z, k)*b*(9*b^2*c^2*x + a^2*g^2*x + 16*root(65536*a^7*b^5*z
^4 + 1024*a^5*b^3*e*g*z^2 + 3072*a^4*b^4*c*e*z^2 + 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z - 128*a^4*b^2*d*
g^2*z + 128*a^3*b^3*d*e^2*z - 1152*a^2*b^4*c^2*d*z - 16*a^2*b^2*d^2*e*g + 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*
e + 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 + 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4
+ 81*b^4*c^4 + a^2*b^2*e^4 + a^4*g^4, z, k)*a^3*b*g - a*b*e^2*x + 48*root(65536*a^7*b^5*z^4 + 1024*a^5*b^3*e*g
*z^2 + 3072*a^4*b^4*c*e*z^2 + 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z - 128*a^4*b^2*d*g^2*z + 128*a^3*b^3*d
*e^2*z - 1152*a^2*b^4*c^2*d*z - 16*a^2*b^2*d^2*e*g + 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*e + 108*a*b^3*c^3*g +
12*a^3*b*c*g^3 + 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4 + 81*b^4*c^4 + a^2*b^
2*e^4 + a^4*g^4, z, k)*a^2*b^2*c + 4*a*b*d*e - 32*root(65536*a^7*b^5*z^4 + 1024*a^5*b^3*e*g*z^2 + 3072*a^4*b^4
*c*e*z^2 + 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z - 128*a^4*b^2*d*g^2*z + 128*a^3*b^3*d*e^2*z - 1152*a^2*b
^4*c^2*d*z - 16*a^2*b^2*d^2*e*g + 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*e + 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 + 5
4*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4 + 81*b^4*c^4 + a^2*b^2*e^4 + a^4*g^4, z,
k)*a^2*b^2*d*x + 6*a*b*c*g*x))/(4*a^2) - (b*d*x*(3*b*c*e - 2*b*d^2 + a*e*g))/(16*a^3))*root(65536*a^7*b^5*z^4
+ 1024*a^5*b^3*e*g*z^2 + 3072*a^4*b^4*c*e*z^2 + 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z - 128*a^4*b^2*d*g^
2*z + 128*a^3*b^3*d*e^2*z - 1152*a^2*b^4*c^2*d*z - 16*a^2*b^2*d^2*e*g + 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*e
+ 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 + 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4 +
81*b^4*c^4 + a^2*b^2*e^4 + a^4*g^4, z, k), k, 1, 4) + ((d*x^2)/(4*a) - f/(4*b) + (e*x^3)/(4*a) + (x*(b*c - a*g
))/(4*a*b))/(a + b*x^4)