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

Optimal. Leaf size=349 \[ \frac{x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+4 a (2 b f-a j)}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + (b*f + a*j)*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)) + (4*a*(2*b*f - a*j) + x*(b*(11*b*c
- a*g) + 2*b*(5*b*d - a*h)*x + 3*b*(3*b*e - a*i)*x^2))/(96*a^2*b^2*(a - b*x^4)^2) + (((7*Sqrt[b]*(11*b*c - a*g
))/Sqrt[a] - 5*(3*b*e - a*i))*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((15*b*e + (7*Sqrt[b]*(11*
b*c - a*g))/Sqrt[a] - 5*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((5*b*d - a*h)*ArcTanh[(Sq
rt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.523832, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1858, 1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac{x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+4 a (2 b f-a j)}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(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 + a*j)*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)) + (4*a*(2*b*f - a*j) + x*(b*(11*b*c
- a*g) + 2*b*(5*b*d - a*h)*x + 3*b*(3*b*e - a*i)*x^2))/(96*a^2*b^2*(a - b*x^4)^2) + (((7*Sqrt[b]*(11*b*c - a*g
))/Sqrt[a] - 5*(3*b*e - a*i))*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((15*b*e + (7*Sqrt[b]*(11*
b*c - a*g))/Sqrt[a] - 5*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((5*b*d - a*h)*ArcTanh[(Sq
rt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2))

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 208

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

Rule 1167

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]

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]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+206 x^6+j x^7}{\left (a-b x^4\right )^4} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) 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 (206 a-3 b e) x^2-4 b (2 b f-a j) 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+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \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 (206 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+(206 a+b e) x^2+(b f+a j) 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 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \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 (206 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+(206 a+b e) x^2+(b f+a j) 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 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \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 (206 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+(206 a+b e) x^2+(b f+a j) 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 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c-a g)+15 b (206 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+(206 a+b e) x^2+(b f+a j) 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 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac{\left (5 (206 a-3 b e)-\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{256 a^3 b}-\frac{\left (5 (206 a-3 b e)+\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{256 a^3 b}+\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+(206 a+b e) x^2+(b f+a j) 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 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac{\left (5 (206 a-3 b e)+\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}-\frac{\left (5 (206 a-3 b e)-\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.36775, size = 439, normalized size = 1.26 \[ \frac{\frac{128 a^3 \left (a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a-b x^4\right )^3}-\frac{16 a^2 \left (12 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (11 c+x (10 d+9 e x))\right )}{\left (a-b x^4\right )^2}+3 \sqrt [4]{a} \sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )+3 \sqrt [4]{a} \sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h-5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d+15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right )+6 \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 a^{3/2} i-15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right )-\frac{4 a b x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt{a} \sqrt{b} (a h-5 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{1536 a^4 b^2} \]

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)^4,x]

[Out]

((-4*a*b*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) - (16*a^2*(12*a^2*j - b
^2*x*(11*c + x*(10*d + 9*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a - b*x^4)^2 + (128*a^3*(a^2*j + b^2*x*(c + x*
(d + e*x)) + a*b*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^3 + 6*a^(1/4)*b^(1/4)*(77*b^(3/2)*c - 15*Sqrt[a]*b*e
- 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + 3*a^(1/4)*b^(1/4)*(-77*b^(3/2)*c - 40*a^(1/4)*b^(
5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + 3*a^(1
/4)*b^(1/4)*(77*b^(3/2)*c - 40*a^(1/4)*b^(5/4)*d + 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h - 5*a^
(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 24*Sqrt[a]*Sqrt[b]*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^4*b^
2)

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Maple [A]  time = 0.015, size = 538, 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((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)^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^2*(a*i-3*b*e)*x^
7-1/12/a^2*(a*h-5*b*d)*x^6-3/64/a^2*(a*g-11*b*c)*x^5-1/8*j*x^4/b-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+1/24*(a*j-2*b*f)/b^2)/(b*x^4-a)^3-7/256/b/a^3*(1/b*a)^(1/4)*arctan(x/(1/
b*a)^(1/4))*g+77/256/a^4*c*(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))-7/512/b/a^3*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4)
)/(x-(1/b*a)^(1/4)))*g+77/512/a^4*c*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))+1/64/b/a^2/(a*b)^(1/
2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))*h-5/64/a^3*d/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b
)^(1/2)))+5/256/b^2/a^2/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))*i-15/256/a^3*e/b/(1/b*a)^(1/4)*arctan(x/(1/b*a)^
(1/4))-5/512/b^2/a^2/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))*i+15/512/a^3*e/b/(1/b*a)^(1/4)*ln((
x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))

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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((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)^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((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)^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((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)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.11209, size = 1087, normalized size = 3.11 \begin{align*} \text{result too large to display} \end{align*}

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)^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^3*i*x^11 - 45*b^4*x^11*e - 60*b^4*d*x^10 + 12*a*b^3*h*x^10 - 77*b^4*c*x^9 + 7
*a*b^3*g*x^9 - 42*a^2*b^2*i*x^7 + 126*a*b^3*x^7*e + 160*a*b^3*d*x^6 - 32*a^2*b^2*h*x^6 + 198*a*b^3*c*x^5 - 18*
a^2*b^2*g*x^5 - 48*a^3*b*j*x^4 - 5*a^3*b*i*x^3 - 113*a^2*b^2*x^3*e - 132*a^2*b^2*d*x^2 - 12*a^3*b*h*x^2 - 153*
a^2*b^2*c*x - 21*a^3*b*g*x - 32*a^3*b*f + 16*a^4*j)/((b*x^4 - a)^3*a^3*b^2)