3.121 \(\int \frac {c+d x}{(a-b x^4)^4} \, dx\)

Optimal. Leaf size=162 \[ \frac {77 c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac {77 c \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3} \]

[Out]

1/12*x*(d*x+c)/a/(-b*x^4+a)^3+1/96*x*(10*d*x+11*c)/a^2/(-b*x^4+a)^2+1/384*x*(60*d*x+77*c)/a^3/(-b*x^4+a)+77/25
6*c*arctan(b^(1/4)*x/a^(1/4))/a^(15/4)/b^(1/4)+77/256*c*arctanh(b^(1/4)*x/a^(1/4))/a^(15/4)/b^(1/4)+5/32*d*arc
tanh(x^2*b^(1/2)/a^(1/2))/a^(7/2)/b^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1855, 1876, 212, 208, 205, 275} \[ \frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {77 c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac {77 c \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a - b*x^4)^4,x]

[Out]

(x*(c + d*x))/(12*a*(a - b*x^4)^3) + (x*(11*c + 10*d*x))/(96*a^2*(a - b*x^4)^2) + (x*(77*c + 60*d*x))/(384*a^3
*(a - b*x^4)) + (77*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(1/4)) + (77*c*ArcTanh[(b^(1/4)*x)/a^(1/4)]
)/(256*a^(15/4)*b^(1/4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

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 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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 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

Rubi steps

\begin {align*} \int \frac {c+d x}{\left (a-b x^4\right )^4} \, dx &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}-\frac {\int \frac {-11 c-10 d x}{\left (a-b x^4\right )^3} \, dx}{12 a}\\ &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {\int \frac {77 c+60 d x}{\left (a-b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}-\frac {\int \frac {-231 c-120 d x}{a-b x^4} \, dx}{384 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}-\frac {\int \left (-\frac {231 c}{a-b x^4}-\frac {120 d x}{a-b x^4}\right ) \, dx}{384 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac {(77 c) \int \frac {1}{a-b x^4} \, dx}{128 a^3}+\frac {(5 d) \int \frac {x}{a-b x^4} \, dx}{16 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac {(77 c) \int \frac {1}{\sqrt {a}-\sqrt {b} x^2} \, dx}{256 a^{7/2}}+\frac {(77 c) \int \frac {1}{\sqrt {a}+\sqrt {b} x^2} \, dx}{256 a^{7/2}}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a-b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac {77 c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac {77 c \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 217, normalized size = 1.34 \[ \frac {\frac {128 a^3 x (c+d x)}{\left (a-b x^4\right )^3}+\frac {16 a^2 x (11 c+10 d x)}{\left (a-b x^4\right )^2}+\frac {4 a x (77 c+60 d x)}{a-b x^4}-\frac {3 \left (77 \sqrt [4]{a} \sqrt [4]{b} c+40 \sqrt {a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt {b}}+\frac {3 \left (77 \sqrt [4]{a} \sqrt [4]{b} c-40 \sqrt {a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt {b}}+\frac {462 \sqrt [4]{a} c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac {120 \sqrt {a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {b}}}{1536 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a - b*x^4)^4,x]

[Out]

((128*a^3*x*(c + d*x))/(a - b*x^4)^3 + (16*a^2*x*(11*c + 10*d*x))/(a - b*x^4)^2 + (4*a*x*(77*c + 60*d*x))/(a -
b*x^4) + (462*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(1/4) - (3*(77*a^(1/4)*b^(1/4)*c + 40*Sqrt[a]*d)*Log[a
^(1/4) - b^(1/4)*x])/Sqrt[b] + (3*(77*a^(1/4)*b^(1/4)*c - 40*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sqrt[b] + (1
20*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1536*a^4)

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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((d*x+c)/(-b*x^4+a)^4,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.28, size = 296, normalized size = 1.83 \[ \frac {77 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, a^{4} b} - \frac {77 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, a^{4} b} - \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {-a b} b d - 77 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {-a b} b d - 77 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{2}} - \frac {60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} - 160 \, a b d x^{6} - 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

77/1024*sqrt(2)*(-a*b^3)^(1/4)*c*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^4*b) - 77/1024*sqrt(2)*(-a*
b^3)^(1/4)*c*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^4*b) - 1/512*sqrt(2)*(40*sqrt(2)*sqrt(-a*b)*b*d
- 77*(-a*b^3)^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^4*b^2) - 1/512*sqrt
(2)*(40*sqrt(2)*sqrt(-a*b)*b*d - 77*(-a*b^3)^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)
^(1/4))/(a^4*b^2) - 1/384*(60*b^2*d*x^10 + 77*b^2*c*x^9 - 160*a*b*d*x^6 - 198*a*b*c*x^5 + 132*a^2*d*x^2 + 153*
a^2*c*x)/((b*x^4 - a)^3*a^3)

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maple [A]  time = 0.06, size = 177, normalized size = 1.09 \[ -\frac {5 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{3}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 a^{4}}+\frac {-\frac {5 b^{2} d \,x^{10}}{32 a^{3}}-\frac {77 b^{2} c \,x^{9}}{384 a^{3}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}-\frac {11 d \,x^{2}}{32 a}-\frac {51 c x}{128 a}}{\left (b \,x^{4}-a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(-b*x^4+a)^4,x)

[Out]

(-5/32*d/a^3*b^2*x^10-77/384*c/a^3*b^2*x^9+5/12/a^2*d*b*x^6+33/64/a^2*c*b*x^5-11/32*d/a*x^2-51/128*c/a*x)/(b*x
^4-a)^3+77/512/a^4*c*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+77/256/a^4*c*(a/b)^(1/4)*arctan(1/(a/b)^(
1/4)*x)-5/64/a^3*d/(a*b)^(1/2)*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))

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maxima [A]  time = 2.97, size = 223, normalized size = 1.38 \[ -\frac {60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} - 160 \, a b d x^{6} - 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \, {\left (a^{3} b^{3} x^{12} - 3 \, a^{4} b^{2} x^{8} + 3 \, a^{5} b x^{4} - a^{6}\right )}} + \frac {\frac {154 \, c \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {40 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {40 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {77 \, c \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}}}{512 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-b*x^4+a)^4,x, algorithm="maxima")

[Out]

-1/384*(60*b^2*d*x^10 + 77*b^2*c*x^9 - 160*a*b*d*x^6 - 198*a*b*c*x^5 + 132*a^2*d*x^2 + 153*a^2*c*x)/(a^3*b^3*x
^12 - 3*a^4*b^2*x^8 + 3*a^5*b*x^4 - a^6) + 1/512*(154*c*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(
sqrt(a)*sqrt(b))) + 40*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 40*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(
a)*sqrt(b)) - 77*c*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(
sqrt(a)*sqrt(b))))/a^3

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mupad [B]  time = 4.97, size = 351, normalized size = 2.17 \[ \left (\sum _{k=1}^4\ln \left (-\frac {b^2\,\left (1925\,c\,d^2+1000\,d^3\,x+{\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4-838860800\,a^8\,b\,d^2\,z^2+485703680\,a^4\,b\,c^2\,d\,z-35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )}^2\,a^7\,b\,c\,315392+\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4-838860800\,a^8\,b\,d^2\,z^2+485703680\,a^4\,b\,c^2\,d\,z-35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )\,a^3\,b\,c^2\,x\,47432-{\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4-838860800\,a^8\,b\,d^2\,z^2+485703680\,a^4\,b\,c^2\,d\,z-35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )}^2\,a^7\,b\,d\,x\,163840\right )}{a^9\,32768}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4-838860800\,a^8\,b\,d^2\,z^2+485703680\,a^4\,b\,c^2\,d\,z-35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )\right )+\frac {\frac {11\,d\,x^2}{32\,a}+\frac {51\,c\,x}{128\,a}+\frac {77\,b^2\,c\,x^9}{384\,a^3}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}-\frac {33\,b\,c\,x^5}{64\,a^2}-\frac {5\,b\,d\,x^6}{12\,a^2}}{a^3-3\,a^2\,b\,x^4+3\,a\,b^2\,x^8-b^3\,x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a - b*x^4)^4,x)

[Out]

symsum(log(-(b^2*(1925*c*d^2 + 1000*d^3*x + 315392*root(68719476736*a^15*b^2*z^4 - 838860800*a^8*b*d^2*z^2 + 4
85703680*a^4*b*c^2*d*z - 35153041*b*c^4 + 2560000*a*d^4, z, k)^2*a^7*b*c + 47432*root(68719476736*a^15*b^2*z^4
- 838860800*a^8*b*d^2*z^2 + 485703680*a^4*b*c^2*d*z - 35153041*b*c^4 + 2560000*a*d^4, z, k)*a^3*b*c^2*x - 163
840*root(68719476736*a^15*b^2*z^4 - 838860800*a^8*b*d^2*z^2 + 485703680*a^4*b*c^2*d*z - 35153041*b*c^4 + 25600
00*a*d^4, z, k)^2*a^7*b*d*x))/(32768*a^9))*root(68719476736*a^15*b^2*z^4 - 838860800*a^8*b*d^2*z^2 + 485703680
*a^4*b*c^2*d*z - 35153041*b*c^4 + 2560000*a*d^4, z, k), k, 1, 4) + ((11*d*x^2)/(32*a) + (51*c*x)/(128*a) + (77
*b^2*c*x^9)/(384*a^3) + (5*b^2*d*x^10)/(32*a^3) - (33*b*c*x^5)/(64*a^2) - (5*b*d*x^6)/(12*a^2))/(a^3 - b^3*x^1
2 - 3*a^2*b*x^4 + 3*a*b^2*x^8)

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sympy [A]  time = 2.06, size = 231, normalized size = 1.43 \[ \operatorname {RootSum} {\left (68719476736 t^{4} a^{15} b^{2} - 838860800 t^{2} a^{8} b d^{2} + 485703680 t a^{4} b c^{2} d + 2560000 a d^{4} - 35153041 b c^{4}, \left (t \mapsto t \log {\left (x + \frac {429496729600 t^{3} a^{12} b d^{2} + 62170071040 t^{2} a^{8} b c^{2} d - 2621440000 t a^{5} d^{4} + 17998356992 t a^{4} b c^{4} + 1897280000 a c^{2} d^{3}}{788480000 a c d^{4} + 2706784157 b c^{5}} \right )} \right )\right )} + \frac {- 153 a^{2} c x - 132 a^{2} d x^{2} + 198 a b c x^{5} + 160 a b d x^{6} - 77 b^{2} c x^{9} - 60 b^{2} d x^{10}}{- 384 a^{6} + 1152 a^{5} b x^{4} - 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-b*x**4+a)**4,x)

[Out]

RootSum(68719476736*_t**4*a**15*b**2 - 838860800*_t**2*a**8*b*d**2 + 485703680*_t*a**4*b*c**2*d + 2560000*a*d*
*4 - 35153041*b*c**4, Lambda(_t, _t*log(x + (429496729600*_t**3*a**12*b*d**2 + 62170071040*_t**2*a**8*b*c**2*d
- 2621440000*_t*a**5*d**4 + 17998356992*_t*a**4*b*c**4 + 1897280000*a*c**2*d**3)/(788480000*a*c*d**4 + 270678
4157*b*c**5)))) + (-153*a**2*c*x - 132*a**2*d*x**2 + 198*a*b*c*x**5 + 160*a*b*d*x**6 - 77*b**2*c*x**9 - 60*b**
2*d*x**10)/(-384*a**6 + 1152*a**5*b*x**4 - 1152*a**4*b**2*x**8 + 384*a**3*b**3*x**12)

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