\(\int \frac {c+d x+e x^2+f x^3}{(a-b x^4)^2} \, dx\) [149]

Optimal result

Integrand size = 26, antiderivative size = 155 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=\frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}} \]

[Out]

1/4*(a*f+b*x*(e*x^2+d*x+c))/a/b/(-b*x^4+a)+1/4*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(1/2)+1/8*arctan(b^(1/
4)*x/a^(1/4))*(-e*a^(1/2)+3*c*b^(1/2))/a^(7/4)/b^(3/4)+1/8*arctanh(b^(1/4)*x/a^(1/4))*(e*a^(1/2)+3*c*b^(1/2))/
a^(7/4)/b^(3/4)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1868, 1890, 281, 214, 1181, 211} \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {b} c-\sqrt {a} e\right )}{8 a^{7/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+3 \sqrt {b} c\right )}{8 a^{7/4} b^{3/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )} \]

[In]

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

[Out]

(a*f + b*x*(c + d*x + e*x^2))/(4*a*b*(a - b*x^4)) + ((3*Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8
*a^(7/4)*b^(3/4)) + ((3*Sqrt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(3/4)) + (d*ArcTanh[
(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

Rule 211

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

Rule 214

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

Rule 281

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 1181

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 1868

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 1890

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*} \text {integral}& = \frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \frac {-3 c-2 d x-e x^2}{a-b x^4} \, dx}{4 a} \\ & = \frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \left (-\frac {2 d x}{a-b x^4}+\frac {-3 c-e x^2}{a-b x^4}\right ) \, dx}{4 a} \\ & = \frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \frac {-3 c-e x^2}{a-b x^4} \, dx}{4 a}+\frac {d \int \frac {x}{a-b x^4} \, dx}{2 a} \\ & = \frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}+\frac {d \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{4 a}-\frac {\left (\frac {3 \sqrt {b} c}{\sqrt {a}}-e\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a^{3/2}} \\ & = \frac {a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.42 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=\frac {\frac {4 a (a f+b x (c+x (d+e x)))}{a-b x^4}-2 \sqrt [4]{a} \sqrt [4]{b} \left (-3 \sqrt {b} c+\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\sqrt [4]{b} \left (3 \sqrt [4]{a} \sqrt {b} c+2 \sqrt {a} \sqrt [4]{b} d+a^{3/4} e\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt [4]{b} \left (3 \sqrt [4]{a} \sqrt {b} c-2 \sqrt {a} \sqrt [4]{b} d+a^{3/4} e\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+2 \sqrt {a} \sqrt {b} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{16 a^2 b} \]

[In]

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

[Out]

((4*a*(a*f + b*x*(c + x*(d + e*x))))/(a - b*x^4) - 2*a^(1/4)*b^(1/4)*(-3*Sqrt[b]*c + Sqrt[a]*e)*ArcTan[(b^(1/4
)*x)/a^(1/4)] - b^(1/4)*(3*a^(1/4)*Sqrt[b]*c + 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x] + b^(
1/4)*(3*a^(1/4)*Sqrt[b]*c - 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x] + 2*Sqrt[a]*Sqrt[b]*d*Lo
g[Sqrt[a] + Sqrt[b]*x^2])/(16*a^2*b)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.50 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {\frac {e \,x^{3}}{4 a}+\frac {d \,x^{2}}{4 a}+\frac {c x}{4 a}+\frac {f}{4 b}}{-b \,x^{4}+a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +2 \textit {\_R} d +3 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b a}\)	\(89\)
default	\(c \left (\frac {x}{4 a \left (-b \,x^{4}+a \right )}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{16 a^{2}}\right )+d \left (\frac {x^{2}}{4 a \left (-b \,x^{4}+a \right )}+\frac {\ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )+e \left (\frac {x^{3}}{4 a \left (-b \,x^{4}+a \right )}-\frac {2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+\frac {f \,x^{4}}{4 a \left (-b \,x^{4}+a \right )}\)	\(222\)

[In]

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

[Out]

(1/4/a*e*x^3+1/4*d/a*x^2+1/4*c/a*x+1/4*f/b)/(-b*x^4+a)-1/16/b/a*sum((_R^2*e+2*_R*d+3*c)/_R^3*ln(x-_R),_R=RootO
f(_Z^4*b-a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.00 (sec) , antiderivative size = 117016, normalized size of antiderivative = 754.94 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (139) = 278\).

Time = 43.72 (sec) , antiderivative size = 520, normalized size of antiderivative = 3.35 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{3} + t^{2} \left (- 3072 a^{4} b^{2} c e - 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} + 1152 a^{2} b^{2} c^{2} d\right ) - a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} - 81 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{7} b^{2} e^{3} + 36864 t^{3} a^{6} b^{3} c^{2} e - 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e - 13824 t^{2} a^{4} b^{3} c^{3} d - 144 t a^{4} b c e^{4} - 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} - 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} - 120 a^{2} b c d^{3} e^{2} + 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} + 9 a^{2} b c^{2} e^{4} - 96 a^{2} b c d^{2} e^{3} + 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} - 729 b^{3} c^{6}} \right )} \right )\right )} + \frac {- a f - b c x - b d x^{2} - b e x^{3}}{- 4 a^{2} b + 4 a b^{2} x^{4}} \]

[In]

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

[Out]

RootSum(65536*_t**4*a**7*b**3 + _t**2*(-3072*a**4*b**2*c*e - 2048*a**4*b**2*d**2) + _t*(128*a**3*b*d*e**2 + 11
52*a**2*b**2*c**2*d) - a**2*e**4 + 18*a*b*c**2*e**2 - 48*a*b*c*d**2*e + 16*a*b*d**4 - 81*b**2*c**4, Lambda(_t,
_t*log(x + (4096*_t**3*a**7*b**2*e**3 + 36864*_t**3*a**6*b**3*c**2*e - 98304*_t**3*a**6*b**3*c*d**2 + 4608*_t
**2*a**5*b**2*c*d*e**2 - 4096*_t**2*a**5*b**2*d**3*e - 13824*_t**2*a**4*b**3*c**3*d - 144*_t*a**4*b*c*e**4 - 1
92*_t*a**4*b*d**2*e**3 - 1728*_t*a**3*b**2*c**3*e**2 + 5184*_t*a**3*b**2*c**2*d**2*e + 1536*_t*a**3*b**2*c*d**
4 - 3888*_t*a**2*b**3*c**5 + 6*a**3*d*e**5 - 120*a**2*b*c*d**3*e**2 + 64*a**2*b*d**5*e + 810*a*b**2*c**4*d*e -
1080*a*b**2*c**3*d**3)/(a**3*e**6 + 9*a**2*b*c**2*e**4 - 96*a**2*b*c*d**2*e**3 + 64*a**2*b*d**4*e**2 - 81*a*b
**2*c**4*e**2 + 864*a*b**2*c**3*d**2*e - 576*a*b**2*c**2*d**4 - 729*b**3*c**6)))) + (-a*f - b*c*x - b*d*x**2 -
b*e*x**3)/(-4*a**2*b + 4*a*b**2*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.29 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=-\frac {b e x^{3} + b d x^{2} + b c x + a f}{4 \, {\left (a b^{2} x^{4} - a^{2} b\right )}} + \frac {\frac {2 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {2 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (3 \, \sqrt {b} c - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (3 \, \sqrt {b} c + \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{16 \, a} \]

[In]

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

[Out]

-1/4*(b*e*x^3 + b*d*x^2 + b*c*x + a*f)/(a*b^2*x^4 - a^2*b) + 1/16*(2*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqr
t(b)) - 2*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) + 2*(3*sqrt(b)*c - sqrt(a)*e)*arctan(sqrt(b)*x/sqrt(s
qrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (3*sqrt(b)*c + sqrt(a)*e)*log((sqrt(b)*x - sqrt(sqr
t(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (116) = 232\).

Time = 0.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.03 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=-\frac {\sqrt {2} {\left (3 \, b^{2} c - 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c + 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} + \frac {\sqrt {2} {\left (3 \, b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {b e x^{3} + b d x^{2} + b c x + a f}{4 \, {\left (b x^{4} - a\right )} a b} \]

[In]

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

[Out]

-1/16*sqrt(2)*(3*b^2*c - 2*sqrt(2)*(-a*b^3)^(1/4)*b*d + sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/
b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/4)*a) - 1/16*sqrt(2)*(3*b^2*c + 2*sqrt(2)*(-a*b^3)^(1/4)*b*d - sqrt(-a*b)
*b*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/4)*a) - 1/32*sqrt(2)*(3*b^2*c
- sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a) + 1/32*sqrt(2)*(3*b^2*c -
sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a) - 1/4*(b*e*x^3 + b*d*x^2 +
b*c*x + a*f)/((b*x^4 - a)*a*b)

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.12 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (65536\,a^7\,b^3\,z^4-3072\,a^4\,b^2\,c\,e\,z^2-2048\,a^4\,b^2\,d^2\,z^2+1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4-81\,b^2\,c^4-a^2\,e^4,z,k\right )\,\left (\mathrm {root}\left (65536\,a^7\,b^3\,z^4-3072\,a^4\,b^2\,c\,e\,z^2-2048\,a^4\,b^2\,d^2\,z^2+1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4-81\,b^2\,c^4-a^2\,e^4,z,k\right )\,\left (12\,b^3\,c-8\,b^3\,d\,x\right )+\frac {x\,\left (4\,a^2\,b^2\,e^2+36\,a\,b^3\,c^2\right )}{16\,a^3}-\frac {b^2\,d\,e}{a}\right )-\frac {-9\,b^2\,c^2\,e+12\,b^2\,c\,d^2+a\,b\,e^3}{64\,a^3}-\frac {x\,\left (2\,b^2\,d^3-3\,b^2\,c\,d\,e\right )}{16\,a^3}\right )\,\mathrm {root}\left (65536\,a^7\,b^3\,z^4-3072\,a^4\,b^2\,c\,e\,z^2-2048\,a^4\,b^2\,d^2\,z^2+1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4-81\,b^2\,c^4-a^2\,e^4,z,k\right )\right )+\frac {\frac {f}{4\,b}+\frac {d\,x^2}{4\,a}+\frac {e\,x^3}{4\,a}+\frac {c\,x}{4\,a}}{a-b\,x^4} \]

[In]

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

[Out]

symsum(log(- root(65536*a^7*b^3*z^4 - 3072*a^4*b^2*c*e*z^2 - 2048*a^4*b^2*d^2*z^2 + 1152*a^2*b^2*c^2*d*z + 128
*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^2 + 16*a*b*d^4 - 81*b^2*c^4 - a^2*e^4, z, k)*(root(65536*a^7*b^
3*z^4 - 3072*a^4*b^2*c*e*z^2 - 2048*a^4*b^2*d^2*z^2 + 1152*a^2*b^2*c^2*d*z + 128*a^3*b*d*e^2*z - 48*a*b*c*d^2*
e + 18*a*b*c^2*e^2 + 16*a*b*d^4 - 81*b^2*c^4 - a^2*e^4, z, k)*(12*b^3*c - 8*b^3*d*x) + (x*(36*a*b^3*c^2 + 4*a^
2*b^2*e^2))/(16*a^3) - (b^2*d*e)/a) - (12*b^2*c*d^2 - 9*b^2*c^2*e + a*b*e^3)/(64*a^3) - (x*(2*b^2*d^3 - 3*b^2*
c*d*e))/(16*a^3))*root(65536*a^7*b^3*z^4 - 3072*a^4*b^2*c*e*z^2 - 2048*a^4*b^2*d^2*z^2 + 1152*a^2*b^2*c^2*d*z
+ 128*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^2 + 16*a*b*d^4 - 81*b^2*c^4 - a^2*e^4, z, k), k, 1, 4) + (
f/(4*b) + (d*x^2)/(4*a) + (e*x^3)/(4*a) + (c*x)/(4*a))/(a - b*x^4)