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

Optimal. Leaf size=417 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {(a h+b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {j \log \left (a+b x^4\right )}{4 b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{4 a b \left (a+b x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.54, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1858, 1876, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 205, 260} \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {(a h+b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {j \log \left (a+b x^4\right )}{4 b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{4 a b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

[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)^2,x]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + (b*f - a*j)*x^3))/(4*a*b*(a + b*x^4)) + ((b*d + a*h)*ArcTan[
(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqr
t[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTa
n[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a
*i))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c +
a*g) - Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(
7/4)) + (j*Log[a + b*x^4])/(4*b^2)

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1858

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]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1876

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*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+197 x^6+j x^7}{\left (a+b x^4\right )^2} \, dx &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \frac {-b (3 b c+a g)-2 b (b d+a h) x-b (591 a+b e) x^2-4 a b j x^3}{a+b x^4} \, dx}{4 a b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \left (\frac {-b (3 b c+a g)-b (591 a+b e) x^2}{a+b x^4}+\frac {x \left (-2 b (b d+a h)-4 a b j x^2\right )}{a+b x^4}\right ) \, dx}{4 a b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \frac {-b (3 b c+a g)-b (591 a+b e) x^2}{a+b x^4} \, dx}{4 a b^2}-\frac {\int \frac {x \left (-2 b (b d+a h)-4 a b j x^2\right )}{a+b x^4} \, dx}{4 a b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2 b (b d+a h)-4 a b j x}{a+b x^2} \, dx,x,x^2\right )}{8 a b^2}-\frac {\left (591 a+b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{8 a b^2}+\frac {\left (591 a+b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{8 a b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {\left (591 a+b e-\frac {\sqrt {b} (3 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}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (591 a+b e-\frac {\sqrt {b} (3 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}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (591 a+b e+\frac {\sqrt {b} (3 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}{16 a b^2}+\frac {\left (591 a+b e+\frac {\sqrt {b} (3 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}{16 a b^2}+\frac {(b d+a h) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{4 a b}+\frac {j \operatorname {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )}{2 b}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\left (591 a+b e-\frac {\sqrt {b} (3 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 )}{16 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\left (591 a+b e-\frac {\sqrt {b} (3 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 )}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {j \log \left (a+b x^4\right )}{4 b^2}+\frac {\left (591 a+b e+\frac {\sqrt {b} (3 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 )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\left (591 a+b e+\frac {\sqrt {b} (3 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 )}{8 \sqrt {2} a^{5/4} b^{7/4}}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(197 a-b e) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}-\frac {\left (591 a+b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (591 a+b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (591 a+b e-\frac {\sqrt {b} (3 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 )}{16 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\left (591 a+b e-\frac {\sqrt {b} (3 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 )}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {j \log \left (a+b x^4\right )}{4 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 460, normalized size = 1.10 \[ \frac {-\frac {2 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i+4 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+3 \sqrt {2} b^{3/2} c\right )}{a^{7/4}}+\frac {2 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i-4 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+3 \sqrt {2} b^{3/2} c\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 a^{3/2} i+\sqrt {a} b e-a \sqrt {b} g-3 b^{3/2} c\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-3 a^{3/2} i-\sqrt {a} b e+a \sqrt {b} g+3 b^{3/2} c\right )}{a^{7/4}}+\frac {8 \left (a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{a \left (a+b x^4\right )}+8 j \log \left (a+b x^4\right )}{32 b^2} \]

Antiderivative was successfully verified.

[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)^2,x]

[Out]

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

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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((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)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.22, size = 617, normalized size = 1.48 \[ \frac {3}{32} \, 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 )}{a 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 )}{a b^{4}}\right )} + \frac {3}{32} \, 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 )}{a 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 )}{a b^{4}}\right )} + \frac {j \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b^{2}} - \frac {{\left (a i - b e\right )} x^{3} - {\left (b d - a h\right )} x^{2} - {\left (b c - a g\right )} x + \frac {a b f - a^{2} j}{b}}{4 \, {\left (b x^{4} + a\right )} a b} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 2 \, \sqrt {2} \sqrt {a b} a b h + 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 + 2 \, \sqrt {2} \sqrt {a b} a b h + 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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.06, size = 675, normalized size = 1.62 \[ \frac {d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{4 \sqrt {a b}\, a}+\frac {h \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{4 \sqrt {a b}\, b}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a 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 )}{32 \left (\frac {a}{b}\right )^{\frac {1}{4}} a 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 )}{16 a 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 )}{16 a 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 )}{32 a b}+\frac {3 \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 )}{16 a^{2}}+\frac {3 \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 )}{16 a^{2}}+\frac {3 \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 )}{32 a^{2}}+\frac {3 \sqrt {2}\, i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {3 \sqrt {2}\, i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {3 \sqrt {2}\, 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 )}{32 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {j \ln \left (b \,x^{4}+a \right )}{4 b^{2}}+\frac {-\frac {\left (a i -b e \right ) x^{3}}{4 a b}-\frac {\left (a h -b d \right ) x^{2}}{4 a b}-\frac {\left (a g -b c \right ) x}{4 a b}+\frac {a j -b f}{4 b^{2}}}{b \,x^{4}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^2,x)

[Out]

(-1/4*(a*i-b*e)/a/b*x^3-1/4*(a*h-b*d)/a/b*x^2-1/4*(a*g-b*c)/a/b*x+1/4*(a*j-b*f)/b^2)/(b*x^4+a)+1/16*(a/b)^(1/4
)*2^(1/2)/a/b*g*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/16*(a/b)^(1/4)*2^(1/2)/a^2*c*arctan(2^(1/2)/(a/b)^(1/4)*x+1)
+1/16*(a/b)^(1/4)*2^(1/2)/a/b*g*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+3/16*(a/b)^(1/4)*2^(1/2)/a^2*c*arctan(2^(1/2)/
(a/b)^(1/4)*x-1)+1/32*(a/b)^(1/4)*2^(1/2)/a/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)))+3/32*(a/b)^(1/4)*2^(1/2)/a^2*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)^(1/2)/b*h*arctan((1/a*b)^(1/2)*x^2)+1/4/(a*b)^(1/2)/a*d*arctan((1/a*b)^(1
/2)*x^2)+3/16/(a/b)^(1/4)*2^(1/2)/b^2*i*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/16/(a/b)^(1/4)*2^(1/2)/a/b*e*arctan(
2^(1/2)/(a/b)^(1/4)*x+1)+3/32/(a/b)^(1/4)*2^(1/2)/b^2*i*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)))+1/32/(a/b)^(1/4)*2^(1/2)/a/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)))+3/16/(a/b)^(1/4)*2^(1/2)/b^2*i*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+1/16/(a/b)^(
1/4)*2^(1/2)/a/b*e*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+1/4*j*ln(b*x^4+a)/b^2

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maxima [A]  time = 3.21, size = 458, normalized size = 1.10 \[ \frac {{\left (b^{2} e - a b i\right )} x^{3} - a b f + a^{2} j + {\left (b^{2} d - a b h\right )} x^{2} + {\left (b^{2} c - a b g\right )} x}{4 \, {\left (a b^{3} x^{4} + a^{2} b^{2}\right )}} + \frac {\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} j + 3 \, b^{2} c - \sqrt {a} b^{\frac {3}{2}} e + a b g - 3 \, a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} j - 3 \, b^{2} c + \sqrt {a} b^{\frac {3}{2}} e - a b g + 3 \, a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i - 4 \, \sqrt {a} b^{2} d - 4 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i + 4 \, \sqrt {a} b^{2} d + 4 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{32 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.84, size = 3939, normalized size = 9.45 \[ \text {result too large to display} \]

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 + j*x^7)/(a + b*x^4)^2,x)

[Out]

((x*(b*c - a*g))/(4*a*b) - (b*f - a*j)/(4*b^2) + (x^2*(b*d - a*h))/(4*a*b) + (x^3*(b*e - a*i))/(4*a*b))/(a + b
*x^4) + symsum(log(- root(65536*a^7*b^8*z^4 - 65536*a^7*b^6*j*z^3 + 3072*a^6*b^5*g*i*z^2 + 9216*a^5*b^6*c*i*z^
2 + 4096*a^5*b^6*d*h*z^2 + 1024*a^5*b^6*e*g*z^2 + 3072*a^4*b^7*c*e*z^2 + 24576*a^7*b^4*j^2*z^2 + 2048*a^6*b^5*
h^2*z^2 + 2048*a^4*b^7*d^2*z^2 - 1536*a^6*b^3*g*i*j*z - 4608*a^5*b^4*c*i*j*z - 2048*a^5*b^4*d*h*j*z + 768*a^5*
b^4*e*h*i*z - 512*a^5*b^4*e*g*j*z - 1536*a^4*b^5*c*e*j*z + 768*a^4*b^5*d*e*i*z - 768*a^4*b^5*c*g*h*z - 768*a^3
*b^6*c*d*g*z - 1024*a^6*b^3*h^2*j*z + 1152*a^6*b^3*h*i^2*z - 128*a^5*b^4*g^2*h*z - 1024*a^4*b^5*d^2*j*z + 1152
*a^5*b^4*d*i^2*z + 128*a^4*b^5*e^2*h*z - 1152*a^3*b^6*c^2*h*z - 128*a^4*b^5*d*g^2*z + 128*a^3*b^6*d*e^2*z - 11
52*a^2*b^7*c^2*d*z - 4096*a^7*b^2*j^3*z - 192*a^5*b^2*e*h*i*j - 192*a^4*b^3*d*e*i*j + 192*a^4*b^3*c*g*h*j - 96
*a^4*b^3*d*g*h*i - 288*a^3*b^4*c*d*h*i + 192*a^3*b^4*c*d*g*j + 72*a^3*b^4*c*e*g*i - 32*a^3*b^4*d*e*g*h - 96*a^
2*b^5*c*d*e*h + 32*a^5*b^2*g^2*h*j - 48*a^5*b^2*g*h^2*i - 288*a^5*b^2*d*i^2*j - 32*a^4*b^3*e^2*h*j + 576*a^5*b
^2*c*i*j^2 + 256*a^5*b^2*d*h*j^2 + 64*a^5*b^2*e*g*j^2 + 288*a^3*b^4*c^2*h*j + 32*a^4*b^3*d*g^2*j + 12*a^4*b^3*
e*g^2*i - 144*a^4*b^3*c*h^2*i - 48*a^3*b^4*d^2*g*i - 16*a^4*b^3*e*g*h^2 + 108*a^4*b^3*c*g*i^2 - 32*a^3*b^4*d*e
^2*j + 192*a^4*b^3*c*e*j^2 + 288*a^2*b^5*c^2*d*j + 108*a^2*b^5*c^2*e*i - 144*a^2*b^5*c*d^2*i - 48*a^3*b^4*c*e*
h^2 - 16*a^2*b^5*d^2*e*g + 12*a^2*b^5*c*e^2*g - 288*a^6*b*h*i^2*j + 192*a^6*b*g*i*j^2 - 48*a*b^6*c*d^2*e + 108
*a*b^6*c^3*g + 18*a^5*b^2*g^2*i^2 + 128*a^4*b^3*d^2*j^2 + 54*a^4*b^3*e^2*i^2 + 162*a^3*b^4*c^2*i^2 + 96*a^3*b^
4*d^2*h^2 + 2*a^3*b^4*e^2*g^2 + 54*a^2*b^5*c^2*g^2 + 128*a^6*b*h^2*j^2 + 108*a^5*b^2*e*i^3 + 12*a^3*b^4*e^3*i
+ 64*a^4*b^3*d*h^3 + 64*a^2*b^5*d^3*h + 12*a^3*b^4*c*g^3 + 18*a*b^6*c^2*e^2 + 16*a^5*b^2*h^4 + 81*a^6*b*i^4 +
16*a*b^6*d^4 + 256*a^7*j^4 + 81*b^7*c^4 + a^4*b^3*g^4 + a^2*b^5*e^4, z, m)*(root(65536*a^7*b^8*z^4 - 65536*a^7
*b^6*j*z^3 + 3072*a^6*b^5*g*i*z^2 + 9216*a^5*b^6*c*i*z^2 + 4096*a^5*b^6*d*h*z^2 + 1024*a^5*b^6*e*g*z^2 + 3072*
a^4*b^7*c*e*z^2 + 24576*a^7*b^4*j^2*z^2 + 2048*a^6*b^5*h^2*z^2 + 2048*a^4*b^7*d^2*z^2 - 1536*a^6*b^3*g*i*j*z -
4608*a^5*b^4*c*i*j*z - 2048*a^5*b^4*d*h*j*z + 768*a^5*b^4*e*h*i*z - 512*a^5*b^4*e*g*j*z - 1536*a^4*b^5*c*e*j*
z + 768*a^4*b^5*d*e*i*z - 768*a^4*b^5*c*g*h*z - 768*a^3*b^6*c*d*g*z - 1024*a^6*b^3*h^2*j*z + 1152*a^6*b^3*h*i^
2*z - 128*a^5*b^4*g^2*h*z - 1024*a^4*b^5*d^2*j*z + 1152*a^5*b^4*d*i^2*z + 128*a^4*b^5*e^2*h*z - 1152*a^3*b^6*c
^2*h*z - 128*a^4*b^5*d*g^2*z + 128*a^3*b^6*d*e^2*z - 1152*a^2*b^7*c^2*d*z - 4096*a^7*b^2*j^3*z - 192*a^5*b^2*e
*h*i*j - 192*a^4*b^3*d*e*i*j + 192*a^4*b^3*c*g*h*j - 96*a^4*b^3*d*g*h*i - 288*a^3*b^4*c*d*h*i + 192*a^3*b^4*c*
d*g*j + 72*a^3*b^4*c*e*g*i - 32*a^3*b^4*d*e*g*h - 96*a^2*b^5*c*d*e*h + 32*a^5*b^2*g^2*h*j - 48*a^5*b^2*g*h^2*i
- 288*a^5*b^2*d*i^2*j - 32*a^4*b^3*e^2*h*j + 576*a^5*b^2*c*i*j^2 + 256*a^5*b^2*d*h*j^2 + 64*a^5*b^2*e*g*j^2 +
288*a^3*b^4*c^2*h*j + 32*a^4*b^3*d*g^2*j + 12*a^4*b^3*e*g^2*i - 144*a^4*b^3*c*h^2*i - 48*a^3*b^4*d^2*g*i - 16
*a^4*b^3*e*g*h^2 + 108*a^4*b^3*c*g*i^2 - 32*a^3*b^4*d*e^2*j + 192*a^4*b^3*c*e*j^2 + 288*a^2*b^5*c^2*d*j + 108*
a^2*b^5*c^2*e*i - 144*a^2*b^5*c*d^2*i - 48*a^3*b^4*c*e*h^2 - 16*a^2*b^5*d^2*e*g + 12*a^2*b^5*c*e^2*g - 288*a^6
*b*h*i^2*j + 192*a^6*b*g*i*j^2 - 48*a*b^6*c*d^2*e + 108*a*b^6*c^3*g + 18*a^5*b^2*g^2*i^2 + 128*a^4*b^3*d^2*j^2
+ 54*a^4*b^3*e^2*i^2 + 162*a^3*b^4*c^2*i^2 + 96*a^3*b^4*d^2*h^2 + 2*a^3*b^4*e^2*g^2 + 54*a^2*b^5*c^2*g^2 + 12
8*a^6*b*h^2*j^2 + 108*a^5*b^2*e*i^3 + 12*a^3*b^4*e^3*i + 64*a^4*b^3*d*h^3 + 64*a^2*b^5*d^3*h + 12*a^3*b^4*c*g^
3 + 18*a*b^6*c^2*e^2 + 16*a^5*b^2*h^4 + 81*a^6*b*i^4 + 16*a*b^6*d^4 + 256*a^7*j^4 + 81*b^7*c^4 + a^4*b^3*g^4 +
a^2*b^5*e^4, z, m)*((768*a^3*b^5*c + 256*a^4*b^4*g)/(64*a^3*b^2) - (x*(128*a^3*b^5*d + 128*a^4*b^4*h))/(16*a^
3*b^2)) + (64*a^2*b^4*d*e - 384*a^3*b^3*c*j + 192*a^3*b^3*d*i + 64*a^3*b^3*e*h - 128*a^4*b^2*g*j + 192*a^4*b^2
*h*i)/(64*a^3*b^2) + (x*(36*a*b^5*c^2 - 4*a^2*b^4*e^2 + 4*a^3*b^3*g^2 - 36*a^4*b^2*i^2 + 24*a^2*b^4*c*g + 64*a
^3*b^3*d*j - 24*a^3*b^3*e*i + 64*a^4*b^2*h*j))/(16*a^3*b^2)) - (27*a^4*i^3 + a*b^3*e^3 - 12*b^4*c*d^2 + 9*b^4*
c^2*e + 16*a^4*g*j^2 - 12*a^2*b^2*c*h^2 + a^2*b^2*e*g^2 + 9*a^2*b^2*e^2*i - 48*a^4*h*i*j - 4*a*b^3*d^2*g + 27*
a*b^3*c^2*i + 48*a^3*b*c*j^2 + 27*a^3*b*e*i^2 - 4*a^3*b*g*h^2 + 3*a^3*b*g^2*i + 18*a^2*b^2*c*g*i - 16*a^2*b^2*
d*e*j - 8*a^2*b^2*d*g*h - 24*a*b^3*c*d*h + 6*a*b^3*c*e*g - 48*a^3*b*d*i*j - 16*a^3*b*e*h*j)/(64*a^3*b^2) - (x*
(9*a^4*i^2*j - 2*a^3*b*h^3 - 8*a^4*h*j^2 - 2*b^4*d^3 - 6*a^2*b^2*d*h^2 + a^2*b^2*e^2*j + 3*b^4*c*d*e - 6*a*b^3
*d^2*h - 9*a*b^3*c^2*j - 8*a^3*b*d*j^2 - a^3*b*g^2*j - 6*a^2*b^2*c*g*j + 9*a^2*b^2*c*h*i + 3*a^2*b^2*d*g*i + a
^2*b^2*e*g*h + 9*a*b^3*c*d*i + 3*a*b^3*c*e*h + a*b^3*d*e*g + 6*a^3*b*e*i*j + 3*a^3*b*g*h*i))/(16*a^3*b^2))*roo
t(65536*a^7*b^8*z^4 - 65536*a^7*b^6*j*z^3 + 3072*a^6*b^5*g*i*z^2 + 9216*a^5*b^6*c*i*z^2 + 4096*a^5*b^6*d*h*z^2
+ 1024*a^5*b^6*e*g*z^2 + 3072*a^4*b^7*c*e*z^2 + 24576*a^7*b^4*j^2*z^2 + 2048*a^6*b^5*h^2*z^2 + 2048*a^4*b^7*d
^2*z^2 - 1536*a^6*b^3*g*i*j*z - 4608*a^5*b^4*c*i*j*z - 2048*a^5*b^4*d*h*j*z + 768*a^5*b^4*e*h*i*z - 512*a^5*b^
4*e*g*j*z - 1536*a^4*b^5*c*e*j*z + 768*a^4*b^5*d*e*i*z - 768*a^4*b^5*c*g*h*z - 768*a^3*b^6*c*d*g*z - 1024*a^6*
b^3*h^2*j*z + 1152*a^6*b^3*h*i^2*z - 128*a^5*b^4*g^2*h*z - 1024*a^4*b^5*d^2*j*z + 1152*a^5*b^4*d*i^2*z + 128*a
^4*b^5*e^2*h*z - 1152*a^3*b^6*c^2*h*z - 128*a^4*b^5*d*g^2*z + 128*a^3*b^6*d*e^2*z - 1152*a^2*b^7*c^2*d*z - 409
6*a^7*b^2*j^3*z - 192*a^5*b^2*e*h*i*j - 192*a^4*b^3*d*e*i*j + 192*a^4*b^3*c*g*h*j - 96*a^4*b^3*d*g*h*i - 288*a
^3*b^4*c*d*h*i + 192*a^3*b^4*c*d*g*j + 72*a^3*b^4*c*e*g*i - 32*a^3*b^4*d*e*g*h - 96*a^2*b^5*c*d*e*h + 32*a^5*b
^2*g^2*h*j - 48*a^5*b^2*g*h^2*i - 288*a^5*b^2*d*i^2*j - 32*a^4*b^3*e^2*h*j + 576*a^5*b^2*c*i*j^2 + 256*a^5*b^2
*d*h*j^2 + 64*a^5*b^2*e*g*j^2 + 288*a^3*b^4*c^2*h*j + 32*a^4*b^3*d*g^2*j + 12*a^4*b^3*e*g^2*i - 144*a^4*b^3*c*
h^2*i - 48*a^3*b^4*d^2*g*i - 16*a^4*b^3*e*g*h^2 + 108*a^4*b^3*c*g*i^2 - 32*a^3*b^4*d*e^2*j + 192*a^4*b^3*c*e*j
^2 + 288*a^2*b^5*c^2*d*j + 108*a^2*b^5*c^2*e*i - 144*a^2*b^5*c*d^2*i - 48*a^3*b^4*c*e*h^2 - 16*a^2*b^5*d^2*e*g
+ 12*a^2*b^5*c*e^2*g - 288*a^6*b*h*i^2*j + 192*a^6*b*g*i*j^2 - 48*a*b^6*c*d^2*e + 108*a*b^6*c^3*g + 18*a^5*b^
2*g^2*i^2 + 128*a^4*b^3*d^2*j^2 + 54*a^4*b^3*e^2*i^2 + 162*a^3*b^4*c^2*i^2 + 96*a^3*b^4*d^2*h^2 + 2*a^3*b^4*e^
2*g^2 + 54*a^2*b^5*c^2*g^2 + 128*a^6*b*h^2*j^2 + 108*a^5*b^2*e*i^3 + 12*a^3*b^4*e^3*i + 64*a^4*b^3*d*h^3 + 64*
a^2*b^5*d^3*h + 12*a^3*b^4*c*g^3 + 18*a*b^6*c^2*e^2 + 16*a^5*b^2*h^4 + 81*a^6*b*i^4 + 16*a*b^6*d^4 + 256*a^7*j
^4 + 81*b^7*c^4 + a^4*b^3*g^4 + a^2*b^5*e^4, z, m), m, 1, 4)

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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((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)**2,x)

[Out]

Timed out

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