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

Optimal. Leaf size=516 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{(a h+5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}-\frac{8 a f-x \left (2 x (a h+5 b d)+3 x^2 (a i+3 b e)+a g+11 b c\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(12*a*b*(a + b*x^4)^3) + (x*(7*(11*b*c + a*g) + 12
*(5*b*d + a*h)*x + 15*(3*b*e + a*i)*x^2))/(384*a^3*b*(a + b*x^4)) - (8*a*f - x*(11*b*c + a*g + 2*(5*b*d + a*h)
*x + 3*(3*b*e + a*i)*x^2))/(96*a^2*b*(a + b*x^4)^2) + ((5*b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2
)*b^(3/2)) - ((7*Sqrt[b]*(11*b*c + a*g) + 5*Sqrt[a]*(3*b*e + a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2
56*Sqrt[2]*a^(15/4)*b^(7/4)) + ((7*Sqrt[b]*(11*b*c + a*g) + 5*Sqrt[a]*(3*b*e + a*i))*ArcTan[1 + (Sqrt[2]*b^(1/
4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(7/4)) - ((7*Sqrt[b]*(11*b*c + a*g) - 5*Sqrt[a]*(3*b*e + a*i))*Log[Sqr
t[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(7/4)) + ((7*Sqrt[b]*(11*b*c + a*g) -
5*Sqrt[a]*(3*b*e + a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(7/4
))

________________________________________________________________________________________

Rubi [A]  time = 0.850028, antiderivative size = 516, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1858, 1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{(a h+5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}-\frac{8 a f-x \left (2 x (a h+5 b d)+3 x^2 (a i+3 b e)+a g+11 b c\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]

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)^4,x]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(12*a*b*(a + b*x^4)^3) + (x*(7*(11*b*c + a*g) + 12
*(5*b*d + a*h)*x + 15*(3*b*e + a*i)*x^2))/(384*a^3*b*(a + b*x^4)) - (8*a*f - x*(11*b*c + a*g + 2*(5*b*d + a*h)
*x + 3*(3*b*e + a*i)*x^2))/(96*a^2*b*(a + b*x^4)^2) + ((5*b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2
)*b^(3/2)) - ((7*Sqrt[b]*(11*b*c + a*g) + 5*Sqrt[a]*(3*b*e + a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2
56*Sqrt[2]*a^(15/4)*b^(7/4)) + ((7*Sqrt[b]*(11*b*c + a*g) + 5*Sqrt[a]*(3*b*e + a*i))*ArcTan[1 + (Sqrt[2]*b^(1/
4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(7/4)) - ((7*Sqrt[b]*(11*b*c + a*g) - 5*Sqrt[a]*(3*b*e + a*i))*Log[Sqr
t[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(7/4)) + ((7*Sqrt[b]*(11*b*c + a*g) -
5*Sqrt[a]*(3*b*e + a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(7/4
))

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 1854

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

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 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

Rule 275

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

Rule 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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+208 x^6}{\left (a+b x^4\right )^4} \, dx &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac{\int \frac{-b (11 b c+a g)-2 b (5 b d+a h) x-3 b (208 a+3 b e) x^2-8 b^2 f x^3}{\left (a+b x^4\right )^3} \, dx}{12 a b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{\int \frac{7 b (11 b c+a g)+12 b (5 b d+a h) x+15 b (208 a+3 b e) x^2}{\left (a+b x^4\right )^2} \, dx}{96 a^2 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c+a g)-24 b (5 b d+a h) x-15 b (208 a+3 b e) x^2}{a+b x^4} \, dx}{384 a^3 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\int \left (-\frac{24 b (5 b d+a h) x}{a+b x^4}+\frac{-21 b (11 b c+a g)-15 b (208 a+3 b e) x^2}{a+b x^4}\right ) \, dx}{384 a^3 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c+a g)-15 b (208 a+3 b e) x^2}{a+b x^4} \, dx}{384 a^3 b^2}+\frac{(5 b d+a h) \int \frac{x}{a+b x^4} \, dx}{16 a^3 b}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 b c+a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{256 a^3 b^2}+\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 b c+a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{256 a^3 b^2}+\frac{(5 b d+a h) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3 b}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{(5 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 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}{512 \sqrt{2} a^{13/4} b^{7/4}}+\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 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}{512 \sqrt{2} a^{13/4} b^{7/4}}+\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 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}{512 a^3 b^2}+\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 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}{512 a^3 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{(5 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 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 )}{512 \sqrt{2} a^{13/4} b^{7/4}}-\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 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 )}{512 \sqrt{2} a^{13/4} b^{7/4}}+\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 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 )}{256 \sqrt{2} a^{13/4} b^{7/4}}-\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 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 )}{256 \sqrt{2} a^{13/4} b^{7/4}}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{(5 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}-\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{13/4} b^{7/4}}+\frac{\left (5 (208 a+3 b e)+\frac{7 \sqrt{b} (11 b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{13/4} b^{7/4}}+\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 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 )}{512 \sqrt{2} a^{13/4} b^{7/4}}-\frac{\left (5 (208 a+3 b e)-\frac{7 \sqrt{b} (11 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 )}{512 \sqrt{2} a^{13/4} b^{7/4}}\\ \end{align*}

Mathematica [A]  time = 0.702842, size = 530, normalized size = 1.03 \[ \frac{-\frac{256 a^{11/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^3}+\frac{32 a^{7/4} b^{3/4} x (a g+a x (2 h+3 i x)+11 b c+b x (10 d+9 e x))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} b^{3/4} x (7 a g+3 a x (4 h+5 i x)+77 b c+15 b x (4 d+3 e x))}{a+b x^4}-6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 a^{5/4} \sqrt [4]{b} h+5 \sqrt{2} a^{3/2} i+80 \sqrt [4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right )+6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 a^{5/4} \sqrt [4]{b} h+5 \sqrt{2} a^{3/2} i-80 \sqrt [4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right )+3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (5 a^{3/2} i+15 \sqrt{a} b e-7 a \sqrt{b} g-77 b^{3/2} c\right )+3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 a^{3/2} i-15 \sqrt{a} b e+7 a \sqrt{b} g+77 b^{3/2} c\right )}{3072 a^{15/4} b^{7/4}} \]

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)^4,x]

[Out]

((32*a^(7/4)*b^(3/4)*x*(11*b*c + a*g + b*x*(10*d + 9*e*x) + a*x*(2*h + 3*i*x)))/(a + b*x^4)^2 + (8*a^(3/4)*b^(
3/4)*x*(77*b*c + 7*a*g + 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))/(a + b*x^4) - (256*a^(11/4)*b^(3/4)*(-(b
*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^3 - 6*(77*Sqrt[2]*b^(3/2)*c + 80*a^(1/4)*b^(
5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*Sqrt[2]*a*Sqrt[b]*g + 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1
- (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*(77*Sqrt[2]*b^(3/2)*c - 80*a^(1/4)*b^(5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*
Sqrt[2]*a*Sqrt[b]*g - 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 3*
Sqrt[2]*(-77*b^(3/2)*c + 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2] + 3*Sqrt[2]*(77*b^(3/2)*c - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g - 5*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b^(7/4))

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Maple [A]  time = 0.014, size = 767, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

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)^4,x)

[Out]

(5/128*(a*i+3*b*e)/a^3*b*x^11+1/32*(a*h+5*b*d)/a^3*b*x^10+7/384*(a*g+11*b*c)/a^3*b*x^9+7/64*(a*i+3*b*e)/a^2*x^
7+1/12/a^2*(a*h+5*b*d)*x^6+3/64/a^2*(a*g+11*b*c)*x^5-1/384*(5*a*i-113*b*e)/a/b*x^3-1/32*(a*h-11*b*d)/a/b*x^2-1
/128*(7*a*g-51*b*c)/a/b*x-1/12*f/b)/(b*x^4+a)^3+7/512/b/a^3*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)
*x+1)*g+77/512/a^4*c*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+7/512/b/a^3*(1/b*a)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*g+77/512/a^4*c*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)+7/10
24/b/a^3*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/
b*a)^(1/2)))*g+77/1024/a^4*c*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)
^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))+1/32/b/a^2/(a*b)^(1/2)*arctan(x^2*(b/a)^(1/2))*h+5/32/a^3*d/(a*b)^(1/2)*arcta
n(x^2*(b/a)^(1/2))+5/512/b^2/a^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*i+15/512/a^3*e/b/(1/b
*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+5/512/b^2/a^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)
^(1/4)*x-1)*i+15/512/a^3*e/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)+5/1024/b^2/a^2/(1/b*a)^(1
/4)*2^(1/2)*ln((x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*i+15/1
024/a^3*e/b/(1/b*a)^(1/4)*2^(1/2)*ln((x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2+(1/b*a)^(1/4)*x*2^(1/2)+
(1/b*a)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)^4,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.10734, size = 992, normalized size = 1.92 \begin{align*} \text{result too large to display} \end{align*}

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)^4,x, algorithm="giac")

[Out]

5/1024*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^3*b^4) - sqrt
(2)*(a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4)) + 5/1024*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^3*b^4) + sqrt(2)*(a*b^3)^(3/4)*log(x^2 - sqrt(2
)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4)) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 8*sqrt(2)*sqrt(a*b)*a*b*
h + 77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/
b)^(1/4))/(a/b)^(1/4))/(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 8*sqrt(2)*sqrt(a*b)*a*b*h + 77*
(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4
))/(a/b)^(1/4))/(a^4*b^3) + 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g - 15*(a*b^3)^(3/4)*
e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) - 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)
^(1/4)*a*b*g - 15*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) + 1/384*(15*a*b^2*i*
x^11 + 45*b^3*x^11*e + 60*b^3*d*x^10 + 12*a*b^2*h*x^10 + 77*b^3*c*x^9 + 7*a*b^2*g*x^9 + 42*a^2*b*i*x^7 + 126*a
*b^2*x^7*e + 160*a*b^2*d*x^6 + 32*a^2*b*h*x^6 + 198*a*b^2*c*x^5 + 18*a^2*b*g*x^5 - 5*a^3*i*x^3 + 113*a^2*b*x^3
*e + 132*a^2*b*d*x^2 - 12*a^3*h*x^2 + 153*a^2*b*c*x - 21*a^3*g*x - 32*a^3*f)/((b*x^4 + a)^3*a^3*b)