\(\int \frac {c+d x+e x^2+f x^3}{(a-b x^4)^3} \, dx\) [150]

Optimal result

Integrand size = 26, antiderivative size = 188 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}} \]

[Out]

1/32*x*(5*e*x^2+6*d*x+7*c)/a^2/(-b*x^4+a)+1/8*(a*f+b*x*(e*x^2+d*x+c))/a/b/(-b*x^4+a)^2+3/16*d*arctanh(x^2*b^(1
/2)/a^(1/2))/a^(5/2)/b^(1/2)+1/64*arctan(b^(1/4)*x/a^(1/4))*(-5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)+1/64*
arctanh(b^(1/4)*x/a^(1/4))*(5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1868, 1869, 1890, 281, 214, 1181, 211} \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (21 \sqrt {b} c-5 \sqrt {a} e\right )}{64 a^{11/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{64 a^{11/4} b^{3/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2} \]

[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 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 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[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 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 )}{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) \text {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] (verified)

Time = 0.21 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\frac {\frac {4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac {16 a^2 (a f+b x (c+x (d+e x)))}{b \left (a-b x^4\right )^2}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}-\frac {\left (21 \sqrt [4]{a} \sqrt {b} c+12 \sqrt {a} \sqrt [4]{b} d+5 a^{3/4} e\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{b^{3/4}}+\frac {\left (21 \sqrt [4]{a} \sqrt {b} c-12 \sqrt {a} \sqrt [4]{b} d+5 a^{3/4} e\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{b^{3/4}}+\frac {12 \sqrt {a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {b}}}{128 a^3} \]

[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)

Maple [C] (verified)

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

Time = 1.50 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.64

method	result	size
risch	\(\frac {-\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {3 b d \,x^{6}}{16 a^{2}}-\frac {7 b c \,x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {11 c x}{32 a}+\frac {f}{8 b}}{\left (-b \,x^{4}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d +21 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\)	\(120\)
default	\(c \left (\frac {x}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (-b \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{128 a^{2}}}{a}\right )+d \left (\frac {x^{2}}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (-b \,x^{4}+a \right )}+\frac {3 \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{32 a \sqrt {a b}}}{a}\right )+e \left (\frac {x^{3}}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (-b \,x^{4}+a \right )}-\frac {5 \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{128 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\right )+f \left (\frac {x^{4}}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {x^{4}}{8 a^{2} \left (-b \,x^{4}+a \right )}\right )\)	\(312\)

[In]

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

[Out]

(-5/32*b*e/a^2*x^7-3/16*b*d/a^2*x^6-7/32*b*c/a^2*x^5+9/32/a*e*x^3+5/16*d/a*x^2+11/32*c/a*x+1/8*f/b)/(-b*x^4+a)
^2-1/128/a^2/b*sum((5*_R^2*e+12*_R*d+21*c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.66 (sec) , antiderivative size = 118761, normalized size of antiderivative = 631.71 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=-\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - 9 \, a b e x^{3} - 10 \, a b d x^{2} - 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (a^{2} b^{3} x^{8} - 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {12 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {12 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, \sqrt {b} c + 5 \, \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2}} \]

[In]

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

[Out]

-1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2*c*x^5 - 9*a*b*e*x^3 - 10*a*b*d*x^2 - 11*a*b*c*x - 4*a^2*f)/(a^2*b^3*x
^8 - 2*a^3*b^2*x^4 + a^4*b) + 1/128*(12*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 12*d*log(sqrt(b)*x^2
- sqrt(a))/(sqrt(a)*sqrt(b)) + 2*(21*sqrt(b)*c - 5*sqrt(a)*e)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)
*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (21*sqrt(b)*c + 5*sqrt(a)*e)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b
)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/a^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (148) = 296\).

Time = 0.28 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.87 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=-\frac {\sqrt {2} {\left (21 \, b^{2} c - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - 9 \, a b e x^{3} - 10 \, a b d x^{2} - 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b} \]

[In]

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

[Out]

-1/128*sqrt(2)*(21*b^2*c - 12*sqrt(2)*(-a*b^3)^(1/4)*b*d + 5*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^2) - 1/128*sqrt(2)*(21*b^2*c + 12*sqrt(2)*(-a*b^3)^(1/4)*b*d -
5*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^2) - 1/256*s
qrt(2)*(21*b^2*c - 5*sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) + 1/2
56*sqrt(2)*(21*b^2*c - 5*sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) -
1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2*c*x^5 - 9*a*b*e*x^3 - 10*a*b*d*x^2 - 11*a*b*c*x - 4*a^2*f)/((b*x^4 -
a)^2*a^2*b)

Mupad [B] (verification not implemented)

Time = 9.52 (sec) , antiderivative size = 832, normalized size of antiderivative = 4.43 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (125\,a\,e^3+3024\,b\,c\,d^2-2205\,b\,c^2\,e+1728\,b\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,c\,344064+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^3\,b\,e^2\,x\,3200-2520\,b\,c\,d\,e\,x+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^2\,b^2\,c^2\,x\,56448-{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,d\,x\,196608-\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^3\,b\,d\,e\,15360\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\right )+\frac {\frac {f}{8\,b}+\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}+\frac {11\,c\,x}{32\,a}-\frac {7\,b\,c\,x^5}{32\,a^2}-\frac {3\,b\,d\,x^6}{16\,a^2}-\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2-2\,a\,b\,x^4+b^2\,x^8} \]

[In]

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

[Out]

symsum(log(-(b*(125*a*e^3 + 3024*b*c*d^2 - 2205*b*c^2*e + 1728*b*d^3*x + 344064*root(268435456*a^11*b^3*z^4 -
6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b
*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)^2*a^5*b^2*c + 3200*root(268
435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4
*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)*a^3*b
*e^2*x - 2520*b*c*d*e*x + 56448*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^
2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 6
25*a^2*e^4 - 194481*b^2*c^4, z, k)*a^2*b^2*c^2*x - 196608*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^
2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c
^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)^2*a^5*b^2*d*x - 15360*root(268435456*a^11*b^3*z^4
- 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*
a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)*a^3*b*d*e))/(32768*a^6))
*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 1
53600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z,
k), k, 1, 4) + (f/(8*b) + (5*d*x^2)/(16*a) + (9*e*x^3)/(32*a) + (11*c*x)/(32*a) - (7*b*c*x^5)/(32*a^2) - (3*b
*d*x^6)/(16*a^2) - (5*b*e*x^7)/(32*a^2))/(a^2 + b^2*x^8 - 2*a*b*x^4)