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

Optimal. Leaf size=384 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {(b d-a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}+\frac {f \log \left (a+b x^4\right )}{4 b}+\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b} \]

[Out]

g*x/b+1/2*h*x^2/b+1/3*i*x^3/b+1/4*f*ln(b*x^4+a)/b+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]  time = 0.57, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {1885, 1819, 1810, 635, 205, 260, 1887, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {(b d-a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}+\frac {f \log \left (a+b x^4\right )}{4 b}+\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*x)/b + (h*x^2)/(2*b) + (i*x^3)/(3*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*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)) + ((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)) + (f*Log[a + b*x^4])/(4*b)

Rule 204

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

Rule 205

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

Rule 260

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 617

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 628

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 635

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 1162

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 1165

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 1168

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 1810

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 1819

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 1885

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 1887

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*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+190 x^6}{a+b x^4} \, dx &=\int \left (\frac {x \left (d+f x^2+h x^4\right )}{a+b x^4}+\frac {c+e x^2+g x^4+190 x^6}{a+b x^4}\right ) \, dx\\ &=\int \frac {x \left (d+f x^2+h x^4\right )}{a+b x^4} \, dx+\int \frac {c+e x^2+g x^4+190 x^6}{a+b x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+f x+h x^2}{a+b x^2} \, dx,x,x^2\right )+\int \left (\frac {g}{b}+\frac {190 x^2}{b}+\frac {b c-a g-(190 a-b e) x^2}{b \left (a+b x^4\right )}\right ) \, dx\\ &=\frac {g x}{b}+\frac {190 x^3}{3 b}+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {h}{b}+\frac {b d-a h+b f x}{b \left (a+b x^2\right )}\right ) \, dx,x,x^2\right )+\frac {\int \frac {b c-a g+(-190 a+b e) x^2}{a+b x^4} \, dx}{b}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {190 x^3}{3 b}+\frac {\operatorname {Subst}\left (\int \frac {b d-a h+b f x}{a+b x^2} \, dx,x,x^2\right )}{2 b}-\frac {\left (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 b^2}+\frac {\left (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 {190 x^3}{3 b}+\frac {1}{2} f \operatorname {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )-\frac {\left (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 {(b d-a h) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 b}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {190 x^3}{3 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 (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 {f \log \left (a+b x^4\right )}{4 b}-\frac {\left (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\right ) \operatorname {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 (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\right ) \operatorname {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 {190 x^3}{3 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 (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 (190 a-b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}\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 {f \log \left (a+b x^4\right )}{4 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.37, size = 427, normalized size = 1.11 \[ \frac {\frac {6 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (2 a^{5/4} \sqrt [4]{b} h+\sqrt {2} a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g-\sqrt {2} b^{3/2} c\right )}{a^{3/4}}+\frac {6 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (2 a^{5/4} \sqrt [4]{b} h-\sqrt {2} a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e-\sqrt {2} a \sqrt {b} g+\sqrt {2} b^{3/2} c\right )}{a^{3/4}}-\frac {3 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (a^{3/2} i-\sqrt {a} b e-a \sqrt {b} g+b^{3/2} c\right )}{a^{3/4}}+\frac {3 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (a^{3/2} i-\sqrt {a} b e-a \sqrt {b} g+b^{3/2} c\right )}{a^{3/4}}+6 b^{3/4} f \log \left (a+b x^4\right )+24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3}{24 b^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(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*(-(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) + (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^(3/4)*f*Log[a + b*x^4])/(24*b^(7/4))

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

________________________________________________________________________________________

giac [A]  time = 0.21, size = 562, normalized size = 1.46 \[ -\frac {1}{8} \, i {\left (\frac {2 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \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 )}{b^{4}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{b^{4}}\right )} - \frac {1}{8} \, i {\left (\frac {2 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \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 )}{b^{4}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{b^{4}}\right )} + \frac {f \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} d + \sqrt {2} \sqrt {a b} a b h + \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 )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} d + \sqrt {2} \sqrt {a b} a b h + \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 )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\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 )}{8 \, a b^{3}} - \frac {\sqrt {2} {\left (\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 )}{8 \, a b^{3}} + \frac {2 \, b^{2} i x^{3} + 3 \, b^{2} h x^{2} + 6 \, b^{2} g x}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

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

________________________________________________________________________________________

maple [B]  time = 0.06, size = 603, normalized size = 1.57 \[ \frac {i \,x^{3}}{3 b}-\frac {a h \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{2 \sqrt {a b}\, b}+\frac {h \,x^{2}}{2 b}+\frac {d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{2 \sqrt {a b}}-\frac {\sqrt {2}\, a i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {\sqrt {2}\, a i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {\sqrt {2}\, a i \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 a}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {f \ln \left (b \,x^{4}+a \right )}{4 b}+\frac {g x}{b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.08, size = 399, normalized size = 1.04 \[ \frac {2 \, i x^{3} + 3 \, h x^{2} + 6 \, g x}{6 \, b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f + b^{2} c - \sqrt {a} b^{\frac {3}{2}} e - a b g + a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f - b^{2} c + \sqrt {a} b^{\frac {3}{2}} e + a b g - a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i - 2 \, \sqrt {a} b^{2} d + 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i + 2 \, \sqrt {a} b^{2} d - 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

1/6*(2*i*x^3 + 3*h*x^2 + 6*g*x)/b + 1/8*(sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*f + 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)) + sqrt(2)*(sqr
t(2)*a^(3/4)*b^(5/4)*f - 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

________________________________________________________________________________________

mupad [B]  time = 5.05, size = 3798, normalized size = 9.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

________________________________________________________________________________________