\(\int \frac {c+d x+e x^2}{(a+b x^4)^2} \, dx\) [128]

Optimal result

Integrand size = 20, antiderivative size = 308 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\frac {x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+3 \sqrt {b} c\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+3 \sqrt {b} c\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}}+\frac {x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )} \]

[In]

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

[Out]

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

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 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b
}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 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 (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int \frac {-3 c-2 d x-e x^2}{a+b x^4} \, dx}{4 a} \\ & = \frac {x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int \left (-\frac {2 d x}{a+b x^4}+\frac {-3 c-e x^2}{a+b x^4}\right ) \, dx}{4 a} \\ & = \frac {x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int \frac {-3 c-e x^2}{a+b x^4} \, dx}{4 a}+\frac {d \int \frac {x}{a+b x^4} \, dx}{2 a} \\ & = \frac {x \left (c+d x+e x^2\right )}{4 a \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 (\frac {3 \sqrt {b} c}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{8 a b}+\frac {\left (\frac {3 \sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{8 a b} \\ & = \frac {x \left (c+d x+e x^2\right )}{4 a \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 (\frac {3 \sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b}+\frac {\left (\frac {3 \sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\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^{3/4}}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\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^{3/4}} \\ & = \frac {x \left (c+d x+e x^2\right )}{4 a \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 \sqrt {b} c-\sqrt {a} e\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^{3/4}}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\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^{3/4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\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^{3/4}}-\frac {\left (3 \sqrt {b} c+\sqrt {a} e\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^{3/4}} \\ & = \frac {x \left (c+d x+e x^2\right )}{4 a \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 \sqrt {b} c+\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\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^{3/4}}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\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^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\frac {\frac {8 a x (c+x (d+e x))}{a+b x^4}-\frac {2 \sqrt [4]{a} \left (3 \sqrt {2} \sqrt {b} c+4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt {2} \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {2 \sqrt [4]{a} \left (3 \sqrt {2} \sqrt {b} c-4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt {2} \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {\sqrt {2} \left (-3 \sqrt [4]{a} \sqrt {b} c+a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {\sqrt {2} \left (3 \sqrt [4]{a} \sqrt {b} c-a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{32 a^2} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 1.66 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.26

method	result	size
risch	\(\frac {\frac {e \,x^{3}}{4 a}+\frac {d \,x^{2}}{4 a}+\frac {c x}{4 a}}{b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +2 \textit {\_R} d +3 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b a}\)	\(80\)
default	\(c \left (\frac {x}{4 a \left (b \,x^{4}+a \right )}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )+d \left (\frac {x^{2}}{4 a \left (b \,x^{4}+a \right )}+\frac {\arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{4 a \sqrt {a b}}\right )+e \left (\frac {x^{3}}{4 a \left (b \,x^{4}+a \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\)	\(287\)

[In]

int((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*c/a*x)/(b*x^4+a)+1/16/b/a*sum((_R^2*e+2*_R*d+3*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 = 3.38 (sec) , antiderivative size = 124258, normalized size of antiderivative = 403.44 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [A] (verification not implemented)

Time = 40.16 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.64 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{3} + t^{2} \cdot \left (3072 a^{4} b^{2} c e + 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} - 1152 a^{2} b^{2} c^{2} d\right ) + a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} + 81 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{7} b^{2} e^{3} - 36864 t^{3} a^{6} b^{3} c^{2} e + 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e + 13824 t^{2} a^{4} b^{3} c^{3} d + 144 t a^{4} b c e^{4} + 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} + 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} + 120 a^{2} b c d^{3} e^{2} - 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} - 9 a^{2} b c^{2} e^{4} + 96 a^{2} b c d^{2} e^{3} - 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} + 729 b^{3} c^{6}} \right )} \right )\right )} + \frac {c x + d x^{2} + e x^{3}}{4 a^{2} + 4 a b x^{4}} \]

[In]

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

[Out]

RootSum(65536*_t**4*a**7*b**3 + _t**2*(3072*a**4*b**2*c*e + 2048*a**4*b**2*d**2) + _t*(128*a**3*b*d*e**2 - 115
2*a**2*b**2*c**2*d) + a**2*e**4 + 18*a*b*c**2*e**2 - 48*a*b*c*d**2*e + 16*a*b*d**4 + 81*b**2*c**4, Lambda(_t,
_t*log(x + (4096*_t**3*a**7*b**2*e**3 - 36864*_t**3*a**6*b**3*c**2*e + 98304*_t**3*a**6*b**3*c*d**2 + 4608*_t*
*2*a**5*b**2*c*d*e**2 - 4096*_t**2*a**5*b**2*d**3*e + 13824*_t**2*a**4*b**3*c**3*d + 144*_t*a**4*b*c*e**4 + 19
2*_t*a**4*b*d**2*e**3 - 1728*_t*a**3*b**2*c**3*e**2 + 5184*_t*a**3*b**2*c**2*d**2*e + 1536*_t*a**3*b**2*c*d**4
+ 3888*_t*a**2*b**3*c**5 + 6*a**3*d*e**5 + 120*a**2*b*c*d**3*e**2 - 64*a**2*b*d**5*e + 810*a*b**2*c**4*d*e -
1080*a*b**2*c**3*d**3)/(a**3*e**6 - 9*a**2*b*c**2*e**4 + 96*a**2*b*c*d**2*e**3 - 64*a**2*b*d**4*e**2 - 81*a*b*
*2*c**4*e**2 + 864*a*b**2*c**3*d**2*e - 576*a*b**2*c**2*d**4 + 729*b**3*c**6)))) + (c*x + d*x**2 + e*x**3)/(4*
a**2 + 4*a*b*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\frac {e x^{3} + d x^{2} + c x}{4 \, {\left (a b x^{4} + a^{2}\right )}} + \frac {\frac {\sqrt {2} {\left (3 \, \sqrt {b} c - \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, \sqrt {b} c - \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 4 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 4 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{32 \, a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\frac {e x^{3} + d x^{2} + c x}{4 \, {\left (b x^{4} + a\right )} a} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{3}} \]

[In]

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

[Out]

1/4*(e*x^3 + d*x^2 + c*x)/((b*x^4 + a)*a) + 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)^(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)^(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)^(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)^(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 = 0.36 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.53 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx=\frac {\frac {d\,x^2}{4\,a}+\frac {e\,x^3}{4\,a}+\frac {c\,x}{4\,a}}{b\,x^4+a}+\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (65536\,a^7\,b^3\,z^4+3072\,a^4\,b^2\,c\,e\,z^2+2048\,a^4\,b^2\,d^2\,z^2-1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4+81\,b^2\,c^4+a^2\,e^4,z,k\right )\,\left (\mathrm {root}\left (65536\,a^7\,b^3\,z^4+3072\,a^4\,b^2\,c\,e\,z^2+2048\,a^4\,b^2\,d^2\,z^2-1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4+81\,b^2\,c^4+a^2\,e^4,z,k\right )\,\left (12\,b^3\,c-8\,b^3\,d\,x\right )+\frac {x\,\left (36\,a\,b^3\,c^2-4\,a^2\,b^2\,e^2\right )}{16\,a^3}+\frac {b^2\,d\,e}{a}\right )-\frac {9\,b^2\,c^2\,e-12\,b^2\,c\,d^2+a\,b\,e^3}{64\,a^3}+\frac {x\,\left (2\,b^2\,d^3-3\,b^2\,c\,d\,e\right )}{16\,a^3}\right )\,\mathrm {root}\left (65536\,a^7\,b^3\,z^4+3072\,a^4\,b^2\,c\,e\,z^2+2048\,a^4\,b^2\,d^2\,z^2-1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4+81\,b^2\,c^4+a^2\,e^4,z,k\right )\right ) \]

[In]

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

[Out]

((d*x^2)/(4*a) + (e*x^3)/(4*a) + (c*x)/(4*a))/(a + b*x^4) + symsum(log((x*(2*b^2*d^3 - 3*b^2*c*d*e))/(16*a^3)
- (9*b^2*c^2*e - 12*b^2*c*d^2 + a*b*e^3)/(64*a^3) - root(65536*a^7*b^3*z^4 + 3072*a^4*b^2*c*e*z^2 + 2048*a^4*b
^2*d^2*z^2 - 1152*a^2*b^2*c^2*d*z + 128*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^2 + 16*a*b*d^4 + 81*b^2*
c^4 + a^2*e^4, z, k)*(root(65536*a^7*b^3*z^4 + 3072*a^4*b^2*c*e*z^2 + 2048*a^4*b^2*d^2*z^2 - 1152*a^2*b^2*c^2*
d*z + 128*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^2 + 16*a*b*d^4 + 81*b^2*c^4 + a^2*e^4, z, k)*(12*b^3*c
- 8*b^3*d*x) + (x*(36*a*b^3*c^2 - 4*a^2*b^2*e^2))/(16*a^3) + (b^2*d*e)/a))*root(65536*a^7*b^3*z^4 + 3072*a^4*
b^2*c*e*z^2 + 2048*a^4*b^2*d^2*z^2 - 1152*a^2*b^2*c^2*d*z + 128*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^
2 + 16*a*b*d^4 + 81*b^2*c^4 + a^2*e^4, z, k), k, 1, 4)