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

Optimal. Leaf size=285 \[ \frac{x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + (b*f + a*j)*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a*(b*f - a*j) +
x*(b*(7*b*c - a*g) + 2*b*(3*b*d - a*h)*x + b*(5*b*e - 3*a*i)*x^2))/(32*a^2*b^2*(a - b*x^4)) - ((5*b*e - (3*Sq
rt[b]*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((5*b*e + (3*Sqrt[b]
*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((3*b*d - a*h)*ArcTanh[(
Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*b^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.391448, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1858, 1854, 1876, 275, 208, 1167, 205} \[ \frac{x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]

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)^3,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + (b*f + a*j)*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a*(b*f - a*j) +
x*(b*(7*b*c - a*g) + 2*b*(3*b*d - a*h)*x + b*(5*b*e - 3*a*i)*x^2))/(32*a^2*b^2*(a - b*x^4)) - ((5*b*e - (3*Sq
rt[b]*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((5*b*e + (3*Sqrt[b]
*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((3*b*d - a*h)*ArcTanh[(
Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*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 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+200 x^6+j x^7}{\left (a-b x^4\right )^3} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+(200 a+b e) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}-\frac{\int \frac{-b (7 b c-a g)-2 b (3 b d-a h) x+5 b (120 a-b e) x^2-4 b (b f-a j) x^3}{\left (a-b x^4\right )^2} \, dx}{8 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(200 a+b e) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac{4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac{\int \frac{3 b (7 b c-a g)+4 b (3 b d-a h) x-5 b (120 a-b e) x^2}{a-b x^4} \, dx}{32 a^2 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(200 a+b e) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac{4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac{\int \left (\frac{4 b (3 b d-a h) x}{a-b x^4}+\frac{3 b (7 b c-a g)-5 b (120 a-b e) x^2}{a-b x^4}\right ) \, dx}{32 a^2 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(200 a+b e) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac{4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac{\int \frac{3 b (7 b c-a g)-5 b (120 a-b e) x^2}{a-b x^4} \, dx}{32 a^2 b^2}+\frac{(3 b d-a h) \int \frac{x}{a-b x^4} \, dx}{8 a^2 b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(200 a+b e) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac{4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}-\frac{\left (600 a-5 b e-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2 b}-\frac{\left (600 a-5 b e+\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2 b}+\frac{(3 b d-a h) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2 b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(200 a+b e) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac{4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac{\left (600 a-5 b e+\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}-\frac{\left (600 a-5 b e-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.277334, size = 380, normalized size = 1.33 \[ \frac{\frac{16 a^{7/4} \left (a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a-b x^4\right )^2}-\frac{4 a^{3/4} \left (8 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (7 c+x (6 d+5 e x))\right )}{a-b x^4}+\sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt{a} b e+3 a \sqrt{b} g-21 b^{3/2} c\right )+\sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )+2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )-4 \sqrt [4]{a} \sqrt{b} (a h-3 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{128 a^{11/4} b^2} \]

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)^3,x]

[Out]

((-4*a^(3/4)*(8*a^2*j - b^2*x*(7*c + x*(6*d + 5*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a - b*x^4) + (16*a^(7/4
)*(a^2*j + b^2*x*(c + x*(d + e*x)) + a*b*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^2 + 2*b^(1/4)*(21*b^(3/2)*c -
5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + b^(1/4)*(-21*b^(3/2)*c - 12*a^(1/4
)*b^(5/4)*d - 5*Sqrt[a]*b*e + 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + b^
(1/4)*(21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i
)*Log[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*Sqrt[b]*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^2)

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Maple [B]  time = 0.012, size = 488, normalized size = 1.7 \begin{align*} -{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{2}} \left ( -{\frac{ \left ( 3\,ai-5\,be \right ){x}^{7}}{32\,{a}^{2}}}-{\frac{ \left ( ah-3\,bd \right ){x}^{6}}{16\,{a}^{2}}}-{\frac{ \left ( ag-7\,bc \right ){x}^{5}}{32\,{a}^{2}}}-{\frac{j{x}^{4}}{4\,b}}-{\frac{ \left ( ai+9\,be \right ){x}^{3}}{32\,ab}}-{\frac{ \left ( ah+5\,bd \right ){x}^{2}}{16\,ab}}-{\frac{ \left ( 3\,ag+11\,bc \right ) x}{32\,ab}}+{\frac{aj-bf}{8\,{b}^{2}}} \right ) }-{\frac{3\,g}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{3\,g}{128\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{h}{32\,ab}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,d}{32\,{a}^{2}}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,i}{64\,{b}^{2}a}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,e}{64\,b{a}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{3\,i}{128\,{b}^{2}a}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e}{128\,b{a}^{2}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \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)^3,x)

[Out]

-(-1/32*(3*a*i-5*b*e)/a^2*x^7-1/16*(a*h-3*b*d)/a^2*x^6-1/32*(a*g-7*b*c)/a^2*x^5-1/4*j*x^4/b-1/32*(a*i+9*b*e)/a
/b*x^3-1/16*(a*h+5*b*d)/a/b*x^2-1/32*(3*a*g+11*b*c)/a/b*x+1/8*(a*j-b*f)/b^2)/(b*x^4-a)^2-3/64/b/a^2*(1/b*a)^(1
/4)*arctan(x/(1/b*a)^(1/4))*g+21/64*c/a^3*(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))-3/128/b/a^2*(1/b*a)^(1/4)*ln((
x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))*g+21/128*c/a^3*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))+1/32/
b/a/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))*h-3/32*d/a^2/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))
/(-a-x^2*(a*b)^(1/2)))+3/64/b^2/a/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))*i-5/64*e/a^2/b/(1/b*a)^(1/4)*arctan(x/
(1/b*a)^(1/4))-3/128/b^2/a/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))*i+5/128*e/a^2/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)^3,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)^3,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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.12438, size = 986, normalized size = 3.46 \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)^3,x, algorithm="giac")

[Out]

-3/256*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^2*b^4) - s
qrt(2)*(-a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^4)) - 3/256*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^2*b^4) + sqrt(2)*(-a*b^3)^(3/4)*log(x^2
- sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^4)) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(-a*b)*b^2*d - 4*sqrt(2)*sq
rt(-a*b)*a*b*h + 21*(-a*b^3)^(1/4)*b^2*c - 3*(-a*b^3)^(1/4)*a*b*g + 5*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*
x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^3*b^3) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(-a*b)*b^2*d - 4*sqrt(2)*sqr
t(-a*b)*a*b*h + 21*(-a*b^3)^(1/4)*b^2*c - 3*(-a*b^3)^(1/4)*a*b*g + 5*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x
- sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt(2)*(21*(-a*b^3)^(1/4)*b^2*c - 3*(-a*b^3)^(1/4)*a
*b*g - 5*(-a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^3*b^3) - 1/256*sqrt(2)*(21*(-a*b^
3)^(1/4)*b^2*c - 3*(-a*b^3)^(1/4)*a*b*g - 5*(-a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(
a^3*b^3) + 1/32*(3*a*b^2*i*x^7 - 5*b^3*x^7*e - 6*b^3*d*x^6 + 2*a*b^2*h*x^6 - 7*b^3*c*x^5 + a*b^2*g*x^5 + 8*a^2
*b*j*x^4 + a^2*b*i*x^3 + 9*a*b^2*x^3*e + 10*a*b^2*d*x^2 + 2*a^2*b*h*x^2 + 11*a*b^2*c*x + 3*a^2*b*g*x + 4*a^2*b
*f - 4*a^3*j)/((b*x^4 - a)^2*a^2*b^2)