\(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx\) [191]

Optimal result

Integrand size = 45, antiderivative size = 402 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {j x^4}{4 b}+\frac {(b d-a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {\left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {(b f-a j) \log \left (a+b x^4\right )}{4 b^2} \]

[Out]

g*x/b+1/2*h*x^2/b+1/3*i*x^3/b+1/4*j*x^4/b+1/4*(-a*j+b*f)*ln(b*x^4+a)/b^2+1/2*(-a*h+b*d)*arctan(x^2*b^(1/2)/a^(
1/2))/b^(3/2)/a^(1/2)-1/8*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-(-a*i+b*e)*a^(1/2)+(-a*g+b*c)*b
^(1/2))/a^(3/4)/b^(7/4)*2^(1/2)+1/8*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-(-a*i+b*e)*a^(1/2)+(-a
*g+b*c)*b^(1/2))/a^(3/4)/b^(7/4)*2^(1/2)+1/4*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*((-a*i+b*e)*a^(1/2)+(-a*g+b*
c)*b^(1/2))/a^(3/4)/b^(7/4)*2^(1/2)+1/4*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*((-a*i+b*e)*a^(1/2)+(-a*g+b*c)*b^(
1/2))/a^(3/4)/b^(7/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {1899, 1901, 1182, 1176, 631, 210, 1179, 642, 1833, 1824, 649, 211, 266} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (b d-a h)}{2 \sqrt {a} b^{3/2}}+\frac {(b f-a j) \log \left (a+b x^4\right )}{4 b^2}+\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {j x^4}{4 b} \]

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4),x]

[Out]

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

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

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 1824

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1833

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1)
, Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] &&
IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]

Rule 1899

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b
, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rule 1901

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c+e x^2+g x^4+i x^6}{a+b x^4}+\frac {x \left (d+f x^2+h x^4+j x^6\right )}{a+b x^4}\right ) \, dx \\ & = \int \frac {c+e x^2+g x^4+i x^6}{a+b x^4} \, dx+\int \frac {x \left (d+f x^2+h x^4+j x^6\right )}{a+b x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {d+f x+h x^2+j x^3}{a+b x^2} \, dx,x,x^2\right )+\int \left (\frac {g}{b}+\frac {i x^2}{b}+\frac {b c-a g+(b e-a i) x^2}{b \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {g x}{b}+\frac {i x^3}{3 b}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {h}{b}+\frac {j x}{b}+\frac {b d-a h+(b f-a j) x}{b \left (a+b x^2\right )}\right ) \, dx,x,x^2\right )+\frac {\int \frac {b c-a g+(b e-a i) x^2}{a+b x^4} \, dx}{b} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {j x^4}{4 b}+\frac {\text {Subst}\left (\int \frac {b d-a h+(b f-a j) x}{a+b x^2} \, dx,x,x^2\right )}{2 b}-\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{2 b^2}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 b^2} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {j x^4}{4 b}+\frac {(b d-a h) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 b}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\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}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\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}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}+\frac {(b f-a j) \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )}{2 b} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {j x^4}{4 b}+\frac {(b d-a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {(b f-a j) \log \left (a+b x^4\right )}{4 b^2}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4}} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {j x^4}{4 b}+\frac {(b d-a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {(b f-a j) \log \left (a+b x^4\right )}{4 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\frac {24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3+6 b^{3/4} j x^4+\frac {6 \left (-\sqrt {2} b^{3/2} c-2 \sqrt [4]{a} b^{5/4} d-\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+2 a^{5/4} \sqrt [4]{b} h+\sqrt {2} a^{3/2} i\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {6 \left (\sqrt {2} b^{3/2} c-2 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e-\sqrt {2} a \sqrt {b} g+2 a^{5/4} \sqrt [4]{b} h-\sqrt {2} a^{3/2} i\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {3 \sqrt {2} \left (b^{3/2} c-\sqrt {a} b e-a \sqrt {b} g+a^{3/2} i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+\frac {3 \sqrt {2} \left (b^{3/2} c-\sqrt {a} b e-a \sqrt {b} g+a^{3/2} i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+\frac {6 (b f-a j) \log \left (a+b x^4\right )}{\sqrt [4]{b}}}{24 b^{7/4}} \]

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4),x]

[Out]

(24*b^(3/4)*g*x + 12*b^(3/4)*h*x^2 + 8*b^(3/4)*i*x^3 + 6*b^(3/4)*j*x^4 + (6*(-(Sqrt[2]*b^(3/2)*c) - 2*a^(1/4)*
b^(5/4)*d - Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (S
qrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (6*(Sqrt[2]*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e - Sqrt
[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) -
(3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[
b]*x^2])/a^(3/4) + (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (6*(b*f - a*j)*Log[a + b*x^4])/b^(1/4))/(24*b^(7/4))

Maple [C] (verified)

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

Time = 1.52 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.26

method	result	size
risch	\(\frac {j \,x^{4}}{4 b}+\frac {i \,x^{3}}{3 b}+\frac {h \,x^{2}}{2 b}+\frac {g x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c -a g +\left (-a h +b d \right ) \textit {\_R} +\left (-a i +b e \right ) \textit {\_R}^{2}+\left (-a j +b f \right ) \textit {\_R}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\)	\(103\)
default	\(\frac {\frac {1}{4} j \,x^{4}+\frac {1}{3} i \,x^{3}+\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\frac {\left (-a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-a h +b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {\left (-a i +b e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\left (-a j +b f \right ) \ln \left (b \,x^{4}+a \right )}{4 b}}{b}\)	\(299\)

[In]

int((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\text {Timed out} \]

[In]

integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\text {Timed out} \]

[In]

integrate((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\frac {3 \, j x^{4} + 4 \, i x^{3} + 6 \, h x^{2} + 12 \, g x}{12 \, b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f - \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} j + b^{2} c - \sqrt {a} b^{\frac {3}{2}} e - a b g + a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f - \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} j - b^{2} c + \sqrt {a} b^{\frac {3}{2}} e + a b g - a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i - 2 \, \sqrt {a} b^{2} d + 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i + 2 \, \sqrt {a} b^{2} d - 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \]

[In]

integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

1/12*(3*j*x^4 + 4*i*x^3 + 6*h*x^2 + 12*g*x)/b + 1/8*(sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*f - sqrt(2)*a^(7/4)*b^(1
/4)*j + b^2*c - sqrt(a)*b^(3/2)*e - a*b*g + a^(3/2)*sqrt(b)*i)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + s
qrt(a))/(a^(3/4)*b^(5/4)) + sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*f - sqrt(2)*a^(7/4)*b^(1/4)*j - b^2*c + sqrt(a)*b
^(3/2)*e + a*b*g - a^(3/2)*sqrt(b)*i)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4))
+ 2*(sqrt(2)*a^(1/4)*b^(9/4)*c + sqrt(2)*a^(3/4)*b^(7/4)*e - sqrt(2)*a^(5/4)*b^(5/4)*g - sqrt(2)*a^(7/4)*b^(3
/4)*i - 2*sqrt(a)*b^2*d + 2*a^(3/2)*b*h)*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^(5/4)) + 2*(sqrt(2)*a^(1/4)*b^(9/4)*c + sqrt(2)*a^(3/4)*b^(7/4)*
e - sqrt(2)*a^(5/4)*b^(5/4)*g - sqrt(2)*a^(7/4)*b^(3/4)*i + 2*sqrt(a)*b^2*d - 2*a^(3/2)*b*h)*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^(5/4)))/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\frac {{\left (b f - a j\right )} \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b^{2}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{3} d + \sqrt {2} \sqrt {a b} a b^{2} h - \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{3} d + \sqrt {2} \sqrt {a b} a b^{2} h - \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} + \frac {3 \, b^{3} j x^{4} + 4 \, b^{3} i x^{3} + 6 \, b^{3} h x^{2} + 12 \, b^{3} g x}{12 \, b^{4}} \]

[In]

integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a),x, algorithm="giac")

[Out]

1/4*(b*f - a*j)*log(abs(b*x^4 + a))/b^2 - 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^3*d + sqrt(2)*sqrt(a*b)*a*b^2*h - (
a*b^3)^(1/4)*b^3*c + (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e + (a*b^3)^(3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x +
sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) - 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^3*d + sqrt(2)*sqrt(a*b)*a*b^2*h -
(a*b^3)^(1/4)*b^3*c + (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e + (a*b^3)^(3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x
- sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c - (a*b^3)^(1/4)*a*b^2*g - (a*b^
3)^(3/4)*b*e + (a*b^3)^(3/4)*a*i)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^4) - 1/8*sqrt(2)*((a*b^3)^
(1/4)*b^3*c - (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e + (a*b^3)^(3/4)*a*i)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) +
sqrt(a/b))/(a*b^4) + 1/12*(3*b^3*j*x^4 + 4*b^3*i*x^3 + 6*b^3*h*x^2 + 12*b^3*g*x)/b^4

Mupad [B] (verification not implemented)

Time = 9.53 (sec) , antiderivative size = 5664, normalized size of antiderivative = 14.09 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx=\text {Too large to display} \]

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4),x)

[Out]

symsum(log((a^4*i^3 - a*b^3*e^3 + b^4*c*d^2 - b^4*c^2*e + a^4*g*j^2 + a^2*b^2*c*h^2 - a^2*b^2*e*g^2 + a^2*b^2*
f^2*g + 3*a^2*b^2*e^2*i - 2*a^4*h*i*j - a*b^3*c*f^2 - a*b^3*d^2*g + a*b^3*c^2*i - a^3*b*c*j^2 - 3*a^3*b*e*i^2
- a^3*b*g*h^2 + a^3*b*g^2*i + 2*a^2*b^2*c*f*j - 2*a^2*b^2*c*g*i - 2*a^2*b^2*d*e*j - 2*a^2*b^2*d*f*i + 2*a^2*b^
2*d*g*h - 2*a^2*b^2*e*f*h - 2*a*b^3*c*d*h + 2*a*b^3*c*e*g + 2*a*b^3*d*e*f + 2*a^3*b*d*i*j + 2*a^3*b*e*h*j - 2*
a^3*b*f*g*j + 2*a^3*b*f*h*i)/b^2 + root(256*a^3*b^8*z^4 + 256*a^4*b^6*j*z^3 - 256*a^3*b^7*f*z^3 - 192*a^4*b^5*
f*j*z^2 + 64*a^4*b^5*g*i*z^2 - 64*a^3*b^6*e*g*z^2 - 64*a^3*b^6*d*h*z^2 - 64*a^3*b^6*c*i*z^2 + 64*a^2*b^7*c*e*z
^2 + 96*a^5*b^4*j^2*z^2 + 32*a^4*b^5*h^2*z^2 + 96*a^3*b^6*f^2*z^2 + 32*a^2*b^7*d^2*z^2 + 32*a^5*b^3*g*i*j*z -
32*a^4*b^4*f*g*i*z + 32*a^4*b^4*e*h*i*z - 32*a^4*b^4*e*g*j*z - 32*a^4*b^4*d*h*j*z - 32*a^4*b^4*c*i*j*z + 32*a^
3*b^5*e*f*g*z + 32*a^3*b^5*d*f*h*z - 32*a^3*b^5*d*e*i*z - 32*a^3*b^5*c*g*h*z + 32*a^3*b^5*c*f*i*z + 32*a^3*b^5
*c*e*j*z - 32*a^2*b^6*c*e*f*z + 32*a^2*b^6*c*d*g*z + 16*a^5*b^3*h^2*j*z - 16*a^5*b^3*h*i^2*z - 48*a^5*b^3*f*j^
2*z + 48*a^4*b^4*f^2*j*z + 16*a^4*b^4*g^2*h*z - 16*a^4*b^4*f*h^2*z + 16*a^3*b^5*d^2*j*z + 16*a^4*b^4*d*i^2*z -
16*a^3*b^5*e^2*h*z - 16*a^3*b^5*d*g^2*z + 16*a^2*b^6*c^2*h*z - 16*a^2*b^6*d^2*f*z + 16*a^2*b^6*d*e^2*z - 16*a
*b^7*c^2*d*z + 16*a^6*b^2*j^3*z - 16*a^3*b^5*f^3*z - 8*a^5*b^2*f*g*i*j + 8*a^5*b^2*e*h*i*j - 8*a^4*b^3*e*f*h*i
+ 8*a^4*b^3*e*f*g*j + 8*a^4*b^3*d*g*h*i + 8*a^4*b^3*d*f*h*j - 8*a^4*b^3*d*e*i*j - 8*a^4*b^3*c*g*h*j + 8*a^4*b
^3*c*f*i*j - 8*a^3*b^4*d*e*g*h + 8*a^3*b^4*d*e*f*i + 8*a^3*b^4*c*f*g*h + 8*a^3*b^4*c*e*g*i - 8*a^3*b^4*c*e*f*j
- 8*a^3*b^4*c*d*h*i + 8*a^3*b^4*c*d*g*j - 8*a^2*b^5*c*d*f*g + 8*a^2*b^5*c*d*e*h + 4*a^5*b^2*g^2*h*j - 4*a^5*b
^2*g*h^2*i - 4*a^5*b^2*f*h^2*j + 4*a^5*b^2*f*h*i^2 + 4*a^5*b^2*d*i^2*j - 4*a^4*b^3*e^2*h*j - 4*a^5*b^2*e*g*j^2
- 4*a^5*b^2*d*h*j^2 - 4*a^5*b^2*c*i*j^2 + 4*a^4*b^3*f^2*g*i - 4*a^4*b^3*f*g^2*h - 4*a^4*b^3*e*g^2*i - 4*a^4*b
^3*d*g^2*j + 4*a^3*b^4*c^2*h*j + 4*a^4*b^3*e*g*h^2 + 4*a^4*b^3*c*h^2*i - 4*a^3*b^4*d^2*g*i - 4*a^3*b^4*d^2*f*j
- 4*a^4*b^3*d*f*i^2 - 4*a^4*b^3*c*g*i^2 + 4*a^3*b^4*e^2*f*h + 4*a^3*b^4*d*e^2*j + 4*a^4*b^3*c*e*j^2 - 4*a^3*b
^4*e*f^2*g - 4*a^3*b^4*d*f^2*h - 4*a^3*b^4*c*f^2*i + 4*a^3*b^4*d*f*g^2 - 4*a^2*b^5*c^2*f*h - 4*a^2*b^5*c^2*e*i
- 4*a^2*b^5*c^2*d*j - 4*a^3*b^4*c*e*h^2 + 4*a^2*b^5*d^2*e*g + 4*a^2*b^5*c*d^2*i - 4*a^2*b^5*d*e^2*f - 4*a^2*b
^5*c*e^2*g + 4*a^2*b^5*c*e*f^2 - 4*a^6*b*h*i^2*j + 4*a^6*b*g*i*j^2 + 4*a*b^6*c^2*d*f - 4*a*b^6*c*d^2*e - 4*a^6
*b*f*j^3 - 4*a*b^6*c^3*g + 6*a^5*b^2*f^2*j^2 + 2*a^5*b^2*g^2*i^2 + 6*a^4*b^3*e^2*i^2 + 2*a^4*b^3*f^2*h^2 + 2*a
^4*b^3*d^2*j^2 + 6*a^3*b^4*d^2*h^2 + 2*a^3*b^4*e^2*g^2 + 2*a^3*b^4*c^2*i^2 + 6*a^2*b^5*c^2*g^2 + 2*a^2*b^5*d^2
*f^2 + 2*a^6*b*h^2*j^2 - 4*a^4*b^3*f^3*j - 4*a^5*b^2*e*i^3 - 4*a^3*b^4*e^3*i - 4*a^4*b^3*d*h^3 - 4*a^2*b^5*d^3
*h - 4*a^3*b^4*c*g^3 + 2*a*b^6*c^2*e^2 + a^5*b^2*h^4 + a^4*b^3*g^4 + a^3*b^4*f^4 + a^2*b^5*e^4 + a^6*b*i^4 + a
*b^6*d^4 + a^7*j^4 + b^7*c^4, z, m)*((8*a*b^4*c*f - 8*a*b^4*d*e - 8*a^2*b^3*c*j + 8*a^2*b^3*d*i + 8*a^2*b^3*e*
h - 8*a^2*b^3*f*g + 8*a^3*b^2*g*j - 8*a^3*b^2*h*i)/b^2 + root(256*a^3*b^8*z^4 + 256*a^4*b^6*j*z^3 - 256*a^3*b^
7*f*z^3 - 192*a^4*b^5*f*j*z^2 + 64*a^4*b^5*g*i*z^2 - 64*a^3*b^6*e*g*z^2 - 64*a^3*b^6*d*h*z^2 - 64*a^3*b^6*c*i*
z^2 + 64*a^2*b^7*c*e*z^2 + 96*a^5*b^4*j^2*z^2 + 32*a^4*b^5*h^2*z^2 + 96*a^3*b^6*f^2*z^2 + 32*a^2*b^7*d^2*z^2 +
32*a^5*b^3*g*i*j*z - 32*a^4*b^4*f*g*i*z + 32*a^4*b^4*e*h*i*z - 32*a^4*b^4*e*g*j*z - 32*a^4*b^4*d*h*j*z - 32*a
^4*b^4*c*i*j*z + 32*a^3*b^5*e*f*g*z + 32*a^3*b^5*d*f*h*z - 32*a^3*b^5*d*e*i*z - 32*a^3*b^5*c*g*h*z + 32*a^3*b^
5*c*f*i*z + 32*a^3*b^5*c*e*j*z - 32*a^2*b^6*c*e*f*z + 32*a^2*b^6*c*d*g*z + 16*a^5*b^3*h^2*j*z - 16*a^5*b^3*h*i
^2*z - 48*a^5*b^3*f*j^2*z + 48*a^4*b^4*f^2*j*z + 16*a^4*b^4*g^2*h*z - 16*a^4*b^4*f*h^2*z + 16*a^3*b^5*d^2*j*z
+ 16*a^4*b^4*d*i^2*z - 16*a^3*b^5*e^2*h*z - 16*a^3*b^5*d*g^2*z + 16*a^2*b^6*c^2*h*z - 16*a^2*b^6*d^2*f*z + 16*
a^2*b^6*d*e^2*z - 16*a*b^7*c^2*d*z + 16*a^6*b^2*j^3*z - 16*a^3*b^5*f^3*z - 8*a^5*b^2*f*g*i*j + 8*a^5*b^2*e*h*i
*j - 8*a^4*b^3*e*f*h*i + 8*a^4*b^3*e*f*g*j + 8*a^4*b^3*d*g*h*i + 8*a^4*b^3*d*f*h*j - 8*a^4*b^3*d*e*i*j - 8*a^4
*b^3*c*g*h*j + 8*a^4*b^3*c*f*i*j - 8*a^3*b^4*d*e*g*h + 8*a^3*b^4*d*e*f*i + 8*a^3*b^4*c*f*g*h + 8*a^3*b^4*c*e*g
*i - 8*a^3*b^4*c*e*f*j - 8*a^3*b^4*c*d*h*i + 8*a^3*b^4*c*d*g*j - 8*a^2*b^5*c*d*f*g + 8*a^2*b^5*c*d*e*h + 4*a^5
*b^2*g^2*h*j - 4*a^5*b^2*g*h^2*i - 4*a^5*b^2*f*h^2*j + 4*a^5*b^2*f*h*i^2 + 4*a^5*b^2*d*i^2*j - 4*a^4*b^3*e^2*h
*j - 4*a^5*b^2*e*g*j^2 - 4*a^5*b^2*d*h*j^2 - 4*a^5*b^2*c*i*j^2 + 4*a^4*b^3*f^2*g*i - 4*a^4*b^3*f*g^2*h - 4*a^4
*b^3*e*g^2*i - 4*a^4*b^3*d*g^2*j + 4*a^3*b^4*c^2*h*j + 4*a^4*b^3*e*g*h^2 + 4*a^4*b^3*c*h^2*i - 4*a^3*b^4*d^2*g
*i - 4*a^3*b^4*d^2*f*j - 4*a^4*b^3*d*f*i^2 - 4*a^4*b^3*c*g*i^2 + 4*a^3*b^4*e^2*f*h + 4*a^3*b^4*d*e^2*j + 4*a^4
*b^3*c*e*j^2 - 4*a^3*b^4*e*f^2*g - 4*a^3*b^4*d*f^2*h - 4*a^3*b^4*c*f^2*i + 4*a^3*b^4*d*f*g^2 - 4*a^2*b^5*c^2*f
*h - 4*a^2*b^5*c^2*e*i - 4*a^2*b^5*c^2*d*j - 4*a^3*b^4*c*e*h^2 + 4*a^2*b^5*d^2*e*g + 4*a^2*b^5*c*d^2*i - 4*a^2
*b^5*d*e^2*f - 4*a^2*b^5*c*e^2*g + 4*a^2*b^5*c*e*f^2 - 4*a^6*b*h*i^2*j + 4*a^6*b*g*i*j^2 + 4*a*b^6*c^2*d*f - 4
*a*b^6*c*d^2*e - 4*a^6*b*f*j^3 - 4*a*b^6*c^3*g + 6*a^5*b^2*f^2*j^2 + 2*a^5*b^2*g^2*i^2 + 6*a^4*b^3*e^2*i^2 + 2
*a^4*b^3*f^2*h^2 + 2*a^4*b^3*d^2*j^2 + 6*a^3*b^4*d^2*h^2 + 2*a^3*b^4*e^2*g^2 + 2*a^3*b^4*c^2*i^2 + 6*a^2*b^5*c
^2*g^2 + 2*a^2*b^5*d^2*f^2 + 2*a^6*b*h^2*j^2 - 4*a^4*b^3*f^3*j - 4*a^5*b^2*e*i^3 - 4*a^3*b^4*e^3*i - 4*a^4*b^3
*d*h^3 - 4*a^2*b^5*d^3*h - 4*a^3*b^4*c*g^3 + 2*a*b^6*c^2*e^2 + a^5*b^2*h^4 + a^4*b^3*g^4 + a^3*b^4*f^4 + a^2*b
^5*e^4 + a^6*b*i^4 + a*b^6*d^4 + a^7*j^4 + b^7*c^4, z, m)*((16*a^2*b^4*g - 16*a*b^5*c)/b^2 - (x*(16*a^2*b^4*h
- 16*a*b^5*d))/b^2) - (x*(4*b^5*c^2 - 4*a*b^4*e^2 + 4*a^2*b^3*g^2 - 4*a^3*b^2*i^2 - 8*a*b^4*c*g + 8*a*b^4*d*f
- 8*a^2*b^3*d*j + 8*a^2*b^3*e*i - 8*a^2*b^3*f*h + 8*a^3*b^2*h*j))/b^2) + (x*(b^4*d^3 - a^3*b*h^3 + b^4*c^2*f -
a^4*h*j^2 + a^4*i^2*j + 3*a^2*b^2*d*h^2 + a^2*b^2*f*g^2 - a^2*b^2*f^2*h + a^2*b^2*e^2*j - 2*b^4*c*d*e + a*b^3
*d*f^2 - a*b^3*e^2*f - 3*a*b^3*d^2*h - a*b^3*c^2*j + a^3*b*d*j^2 - a^3*b*f*i^2 - a^3*b*g^2*j + 2*a^2*b^2*c*g*j
- 2*a^2*b^2*c*h*i - 2*a^2*b^2*d*f*j - 2*a^2*b^2*d*g*i + 2*a^2*b^2*e*f*i - 2*a^2*b^2*e*g*h + 2*a*b^3*c*d*i + 2
*a*b^3*c*e*h - 2*a*b^3*c*f*g + 2*a*b^3*d*e*g - 2*a^3*b*e*i*j + 2*a^3*b*f*h*j + 2*a^3*b*g*h*i))/b^2)*root(256*a
^3*b^8*z^4 + 256*a^4*b^6*j*z^3 - 256*a^3*b^7*f*z^3 - 192*a^4*b^5*f*j*z^2 + 64*a^4*b^5*g*i*z^2 - 64*a^3*b^6*e*g
*z^2 - 64*a^3*b^6*d*h*z^2 - 64*a^3*b^6*c*i*z^2 + 64*a^2*b^7*c*e*z^2 + 96*a^5*b^4*j^2*z^2 + 32*a^4*b^5*h^2*z^2
+ 96*a^3*b^6*f^2*z^2 + 32*a^2*b^7*d^2*z^2 + 32*a^5*b^3*g*i*j*z - 32*a^4*b^4*f*g*i*z + 32*a^4*b^4*e*h*i*z - 32*
a^4*b^4*e*g*j*z - 32*a^4*b^4*d*h*j*z - 32*a^4*b^4*c*i*j*z + 32*a^3*b^5*e*f*g*z + 32*a^3*b^5*d*f*h*z - 32*a^3*b
^5*d*e*i*z - 32*a^3*b^5*c*g*h*z + 32*a^3*b^5*c*f*i*z + 32*a^3*b^5*c*e*j*z - 32*a^2*b^6*c*e*f*z + 32*a^2*b^6*c*
d*g*z + 16*a^5*b^3*h^2*j*z - 16*a^5*b^3*h*i^2*z - 48*a^5*b^3*f*j^2*z + 48*a^4*b^4*f^2*j*z + 16*a^4*b^4*g^2*h*z
- 16*a^4*b^4*f*h^2*z + 16*a^3*b^5*d^2*j*z + 16*a^4*b^4*d*i^2*z - 16*a^3*b^5*e^2*h*z - 16*a^3*b^5*d*g^2*z + 16
*a^2*b^6*c^2*h*z - 16*a^2*b^6*d^2*f*z + 16*a^2*b^6*d*e^2*z - 16*a*b^7*c^2*d*z + 16*a^6*b^2*j^3*z - 16*a^3*b^5*
f^3*z - 8*a^5*b^2*f*g*i*j + 8*a^5*b^2*e*h*i*j - 8*a^4*b^3*e*f*h*i + 8*a^4*b^3*e*f*g*j + 8*a^4*b^3*d*g*h*i + 8*
a^4*b^3*d*f*h*j - 8*a^4*b^3*d*e*i*j - 8*a^4*b^3*c*g*h*j + 8*a^4*b^3*c*f*i*j - 8*a^3*b^4*d*e*g*h + 8*a^3*b^4*d*
e*f*i + 8*a^3*b^4*c*f*g*h + 8*a^3*b^4*c*e*g*i - 8*a^3*b^4*c*e*f*j - 8*a^3*b^4*c*d*h*i + 8*a^3*b^4*c*d*g*j - 8*
a^2*b^5*c*d*f*g + 8*a^2*b^5*c*d*e*h + 4*a^5*b^2*g^2*h*j - 4*a^5*b^2*g*h^2*i - 4*a^5*b^2*f*h^2*j + 4*a^5*b^2*f*
h*i^2 + 4*a^5*b^2*d*i^2*j - 4*a^4*b^3*e^2*h*j - 4*a^5*b^2*e*g*j^2 - 4*a^5*b^2*d*h*j^2 - 4*a^5*b^2*c*i*j^2 + 4*
a^4*b^3*f^2*g*i - 4*a^4*b^3*f*g^2*h - 4*a^4*b^3*e*g^2*i - 4*a^4*b^3*d*g^2*j + 4*a^3*b^4*c^2*h*j + 4*a^4*b^3*e*
g*h^2 + 4*a^4*b^3*c*h^2*i - 4*a^3*b^4*d^2*g*i - 4*a^3*b^4*d^2*f*j - 4*a^4*b^3*d*f*i^2 - 4*a^4*b^3*c*g*i^2 + 4*
a^3*b^4*e^2*f*h + 4*a^3*b^4*d*e^2*j + 4*a^4*b^3*c*e*j^2 - 4*a^3*b^4*e*f^2*g - 4*a^3*b^4*d*f^2*h - 4*a^3*b^4*c*
f^2*i + 4*a^3*b^4*d*f*g^2 - 4*a^2*b^5*c^2*f*h - 4*a^2*b^5*c^2*e*i - 4*a^2*b^5*c^2*d*j - 4*a^3*b^4*c*e*h^2 + 4*
a^2*b^5*d^2*e*g + 4*a^2*b^5*c*d^2*i - 4*a^2*b^5*d*e^2*f - 4*a^2*b^5*c*e^2*g + 4*a^2*b^5*c*e*f^2 - 4*a^6*b*h*i^
2*j + 4*a^6*b*g*i*j^2 + 4*a*b^6*c^2*d*f - 4*a*b^6*c*d^2*e - 4*a^6*b*f*j^3 - 4*a*b^6*c^3*g + 6*a^5*b^2*f^2*j^2
+ 2*a^5*b^2*g^2*i^2 + 6*a^4*b^3*e^2*i^2 + 2*a^4*b^3*f^2*h^2 + 2*a^4*b^3*d^2*j^2 + 6*a^3*b^4*d^2*h^2 + 2*a^3*b^
4*e^2*g^2 + 2*a^3*b^4*c^2*i^2 + 6*a^2*b^5*c^2*g^2 + 2*a^2*b^5*d^2*f^2 + 2*a^6*b*h^2*j^2 - 4*a^4*b^3*f^3*j - 4*
a^5*b^2*e*i^3 - 4*a^3*b^4*e^3*i - 4*a^4*b^3*d*h^3 - 4*a^2*b^5*d^3*h - 4*a^3*b^4*c*g^3 + 2*a*b^6*c^2*e^2 + a^5*
b^2*h^4 + a^4*b^3*g^4 + a^3*b^4*f^4 + a^2*b^5*e^4 + a^6*b*i^4 + a*b^6*d^4 + a^7*j^4 + b^7*c^4, z, m), m, 1, 4)
+ (h*x^2)/(2*b) + (i*x^3)/(3*b) + (j*x^4)/(4*b) + (g*x)/b