3.150 \(\int \frac{c+d x+e x^2+f x^3}{(a-b x^4)^3} \, dx\)

Optimal. Leaf size=188 \[ \frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2} \]

[Out]

(x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*(a - b*x^4)) + (a*f + b*x*(c + d*x + e*x^2))/(8*a*b*(a - b*x^4)^2) + ((21*
Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*Ar
cTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(3/4)) + (3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b])

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Rubi [A]  time = 0.154601, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4)^3,x]

[Out]

(x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*(a - b*x^4)) + (a*f + b*x*(c + d*x + e*x^2))/(8*a*b*(a - b*x^4)^2) + ((21*
Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*Ar
cTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(3/4)) + (3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b])

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}{\left (a-b x^4\right )^3} \, dx &=\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}-\frac{\int \frac{-7 c-6 d x-5 e x^2}{\left (a-b x^4\right )^2} \, dx}{8 a}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac{\int \frac{21 c+12 d x+5 e x^2}{a-b x^4} \, dx}{32 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac{\int \left (\frac{12 d x}{a-b x^4}+\frac{21 c+5 e x^2}{a-b x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac{\int \frac{21 c+5 e x^2}{a-b x^4} \, dx}{32 a^2}+\frac{(3 d) \int \frac{x}{a-b x^4} \, dx}{8 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2}-\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}+5 e\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c+5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.218945, size = 253, normalized size = 1.35 \[ \frac{-\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt{b} c+12 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt{b} c-12 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{16 a^2 (a f+b x (c+x (d+e x)))}{b \left (a-b x^4\right )^2}+\frac{2 \sqrt [4]{a} \left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac{12 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{128 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4)^3,x]

[Out]

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

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Maple [B]  time = 0.006, size = 326, normalized size = 1.7 \begin{align*}{\frac{cx}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}-a \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{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}-a \right ) }}-{\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{e{x}^{3}}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{5\,e{x}^{3}}{32\,{a}^{2} \left ( b{x}^{4}-a \right ) }}-{\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{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}}}}}}+{\frac{f{x}^{4}}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{f{x}^{4}}{8\,{a}^{2} \left ( b{x}^{4}-a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^3,x)

[Out]

1/8*c*x/a/(b*x^4-a)^2-7/32*c/a^2*x/(b*x^4-a)+21/128*c/a^3*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4))
)+21/64*c/a^3*(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))+1/8*d*x^2/a/(b*x^4-a)^2-3/16*d/a^2*x^2/(b*x^4-a)-3/32*d/a^
2/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))+1/8*e*x^3/a/(b*x^4-a)^2-5/32*e/a^2*x^3/(b*x^4-a)-5
/64*e/a^2/b/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))+5/128*e/a^2/b/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^
(1/4)))+1/8*f*x^4/a/(b*x^4-a)^2-1/8*f/a^2*x^4/(b*x^4-a)

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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((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((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 [B]  time = 66.1839, size = 583, normalized size = 3.1 \begin{align*} - \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (- 6881280 a^{6} b^{2} c e - 4718592 a^{6} b^{2} d^{2}\right ) + t \left (- 153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) - 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} - 194481 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e - 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} + 118540800 t a^{4} b^{2} c^{3} e^{2} - 365783040 t a^{4} b^{2} c^{2} d^{2} e - 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} - 4536000 a^{2} b c d^{3} e^{2} + 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} + 275625 a^{2} b c^{2} e^{4} - 3024000 a^{2} b c d^{2} e^{3} + 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} - 85766121 b^{3} c^{6}} \right )} \right )\right )} - \frac{- 4 a^{2} f - 11 a b c x - 10 a b d x^{2} - 9 a b e x^{3} + 7 b^{2} c x^{5} + 6 b^{2} d x^{6} + 5 b^{2} e x^{7}}{32 a^{4} b - 64 a^{3} b^{2} x^{4} + 32 a^{2} b^{3} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**3,x)

[Out]

-RootSum(268435456*_t**4*a**11*b**3 + _t**2*(-6881280*a**6*b**2*c*e - 4718592*a**6*b**2*d**2) + _t*(-153600*a*
*4*b*d*e**2 - 2709504*a**3*b**2*c**2*d) - 625*a**2*e**4 + 22050*a*b*c**2*e**2 - 60480*a*b*c*d**2*e + 20736*a*b
*d**4 - 194481*b**2*c**4, Lambda(_t, _t*log(x + (-262144000*_t**3*a**10*b**2*e**3 - 4624220160*_t**3*a**9*b**3
*c**2*e + 12683575296*_t**3*a**9*b**3*c*d**2 + 309657600*_t**2*a**7*b**2*c*d*e**2 - 283115520*_t**2*a**7*b**2*
d**3*e - 1820786688*_t**2*a**6*b**3*c**3*d + 5040000*_t*a**5*b*c*e**4 + 6912000*_t*a**5*b*d**2*e**3 + 11854080
0*_t*a**4*b**2*c**3*e**2 - 365783040*_t*a**4*b**2*c**2*d**2*e - 111476736*_t*a**4*b**2*c*d**4 + 522764928*_t*a
**3*b**3*c**5 + 112500*a**3*d*e**5 - 4536000*a**2*b*c*d**3*e**2 + 2488320*a**2*b*d**5*e + 58344300*a*b**2*c**4
*d*e - 80015040*a*b**2*c**3*d**3)/(15625*a**3*e**6 + 275625*a**2*b*c**2*e**4 - 3024000*a**2*b*c*d**2*e**3 + 20
73600*a**2*b*d**4*e**2 - 4862025*a*b**2*c**4*e**2 + 53343360*a*b**2*c**3*d**2*e - 36578304*a*b**2*c**2*d**4 -
85766121*b**3*c**6)))) - (-4*a**2*f - 11*a*b*c*x - 10*a*b*d*x**2 - 9*a*b*e*x**3 + 7*b**2*c*x**5 + 6*b**2*d*x**
6 + 5*b**2*e*x**7)/(32*a**4*b - 64*a**3*b**2*x**4 + 32*a**2*b**3*x**8)

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Giac [B]  time = 1.08412, size = 505, normalized size = 2.69 \begin{align*} -\frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \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 )}{128 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \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 )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \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 )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \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 )}{256 \, a^{3} b^{3}} - \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - 9 \, a b x^{3} e - 10 \, a b d x^{2} - 11 \, a b c x - 4 \, a^{2} f}{32 \,{\left (b x^{4} - a\right )}^{2} a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^3,x, algorithm="giac")

[Out]

-1/128*sqrt(2)*(12*sqrt(2)*sqrt(-a*b)*b^2*d - 21*(-a*b^3)^(1/4)*b^2*c - 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 - 21*(-a*b^
3)^(1/4)*b^2*c - 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 - 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 - 5*(-a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqr
t(-a/b))/(a^3*b^3) - 1/32*(5*b^2*x^7*e + 6*b^2*d*x^6 + 7*b^2*c*x^5 - 9*a*b*x^3*e - 10*a*b*d*x^2 - 11*a*b*c*x -
4*a^2*f)/((b*x^4 - a)^2*a^2*b)