\(\int \frac {(c+d x^4)^4}{a+b x^4} \, dx\) [160]

Optimal result

Integrand size = 19, antiderivative size = 332 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^3 (4 b c-a d) x^9}{9 b^2}+\frac {d^4 x^{13}}{13 b}-\frac {(b c-a d)^4 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-a d)^4 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}} \]

[Out]

d*(-a*d+2*b*c)*(a^2*d^2-2*a*b*c*d+2*b^2*c^2)*x/b^4+1/5*d^2*(a^2*d^2-4*a*b*c*d+6*b^2*c^2)*x^5/b^3+1/9*d^3*(-a*d
+4*b*c)*x^9/b^2+1/13*d^4*x^13/b+1/4*(-a*d+b*c)^4*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(3/4)/b^(17/4)*2^(1/2)
+1/4*(-a*d+b*c)^4*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(3/4)/b^(17/4)*2^(1/2)-1/8*(-a*d+b*c)^4*ln(-a^(1/4)*b^
(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(3/4)/b^(17/4)*2^(1/2)+1/8*(-a*d+b*c)^4*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^
(1/2)+x^2*b^(1/2))/a^(3/4)/b^(17/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {398, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^4}{2 \sqrt {2} a^{3/4} b^{17/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^4}{2 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-a d)^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{5 b^3}+\frac {d^3 x^9 (4 b c-a d)}{9 b^2}+\frac {d^4 x^{13}}{13 b} \]

[In]

Int[(c + d*x^4)^4/(a + b*x^4),x]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^5)/(5*b^3
) + (d^3*(4*b*c - a*d)*x^9)/(9*b^2) + (d^4*x^13)/(13*b) - ((b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4
)])/(2*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*
b^(17/4)) - ((b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4)
) + ((b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

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

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{b^3}+\frac {d^3 (4 b c-a d) x^8}{b^2}+\frac {d^4 x^{12}}{b}+\frac {b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+a^4 d^4}{b^4 \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^3 (4 b c-a d) x^9}{9 b^2}+\frac {d^4 x^{13}}{13 b}+\frac {(b c-a d)^4 \int \frac {1}{a+b x^4} \, dx}{b^4} \\ & = \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^3 (4 b c-a d) x^9}{9 b^2}+\frac {d^4 x^{13}}{13 b}+\frac {(b c-a d)^4 \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{2 \sqrt {a} b^4}+\frac {(b c-a d)^4 \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{2 \sqrt {a} b^4} \\ & = \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^3 (4 b c-a d) x^9}{9 b^2}+\frac {d^4 x^{13}}{13 b}+\frac {(b c-a d)^4 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt {a} b^{9/2}}+\frac {(b c-a d)^4 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt {a} b^{9/2}}-\frac {(b c-a d)^4 \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-a d)^4 \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} b^{17/4}} \\ & = \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^3 (4 b c-a d) x^9}{9 b^2}+\frac {d^4 x^{13}}{13 b}-\frac {(b c-a d)^4 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-a d)^4 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{17/4}} \\ & = \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^3 (4 b c-a d) x^9}{9 b^2}+\frac {d^4 x^{13}}{13 b}-\frac {(b c-a d)^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-a d)^4 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-a d)^4 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{17/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=\frac {4680 \sqrt [4]{b} d \left (4 b^3 c^3-6 a b^2 c^2 d+4 a^2 b c d^2-a^3 d^3\right ) x+936 b^{5/4} d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^5+520 b^{9/4} d^3 (4 b c-a d) x^9+360 b^{13/4} d^4 x^{13}-\frac {1170 \sqrt {2} (b c-a d)^4 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {1170 \sqrt {2} (b c-a d)^4 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {585 \sqrt {2} (b c-a d)^4 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+\frac {585 \sqrt {2} (b c-a d)^4 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}}{4680 b^{17/4}} \]

[In]

Integrate[(c + d*x^4)^4/(a + b*x^4),x]

[Out]

(4680*b^(1/4)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 936*b^(5/4)*d^2*(6*b^2*c^2 - 4*a*b*c
*d + a^2*d^2)*x^5 + 520*b^(9/4)*d^3*(4*b*c - a*d)*x^9 + 360*b^(13/4)*d^4*x^13 - (1170*Sqrt[2]*(b*c - a*d)^4*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (1170*Sqrt[2]*(b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^
(1/4)])/a^(3/4) - (585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) +
(585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4))/(4680*b^(17/4))

Maple [C] (verified)

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

Time = 4.20 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.61

method	result	size
risch	\(\frac {d^{4} x^{13}}{13 b}-\frac {d^{4} x^{9} a}{9 b^{2}}+\frac {4 d^{3} x^{9} c}{9 b}-\frac {4 d^{3} a c \,x^{5}}{5 b^{2}}+\frac {6 d^{2} c^{2} x^{5}}{5 b}+\frac {d^{4} a^{2} x^{5}}{5 b^{3}}-\frac {d^{4} a^{3} x}{b^{4}}+\frac {4 d^{3} a^{2} c x}{b^{3}}-\frac {6 d^{2} a \,c^{2} x}{b^{2}}+\frac {4 d \,c^{3} x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{5}}\)	\(201\)
default	\(-\frac {d \left (-\frac {b^{3} d^{3} x^{13}}{13}+\frac {\left (\left (a d -2 b c \right ) b^{2} d^{2}-2 b^{3} c \,d^{2}\right ) x^{9}}{9}+\frac {\left (2 \left (a d -2 b c \right ) b^{2} c d -b d \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right )\right ) x^{5}}{5}+\left (a d -2 b c \right ) \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x \right )}{b^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b^{4} a}\)	\(282\)

[In]

int((d*x^4+c)^4/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

1/13*d^4*x^13/b-1/9*d^4/b^2*x^9*a+4/9*d^3/b*x^9*c-4/5*d^3/b^2*a*c*x^5+6/5*d^2/b*c^2*x^5+1/5*d^4/b^3*a^2*x^5-d^
4/b^4*a^3*x+4*d^3/b^3*a^2*c*x-6*d^2/b^2*a*c^2*x+4*d/b*c^3*x+1/4/b^5*sum((a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d
^2-4*a*b^3*c^3*d+b^4*c^4)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 2190, normalized size of antiderivative = 6.60 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2340*(180*b^3*d^4*x^13 + 260*(4*b^3*c*d^3 - a*b^2*d^4)*x^9 + 468*(6*b^3*c^2*d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)
*x^5 + 585*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12
*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 -
11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^
3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4)*log(a*b^4*(-(b^16*c^16 -
16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11
*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a
^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*
d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*
b*c*d^3 + a^4*d^4)*x) + 585*I*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*
d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870
*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4
*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4)*log(I*
a*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4
- 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9
*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^1
3 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2
*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x) - 585*I*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2
- 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^
7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^1
1 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^
3*b^17))^(1/4)*log(-I*a*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 +
1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b
^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12
- 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4) + (b^4*c^4 -
4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x) - 585*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 1
20*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10
*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4
368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d
^15 + a^16*d^16)/(a^3*b^17))^(1/4)*log(-a*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^
3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9
*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*
a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^
(1/4) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x) + 2340*(4*b^3*c^3*d - 6*a*b
^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*x)/b^4

Sympy [A] (verification not implemented)

Time = 22.35 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=x^{9} \left (- \frac {a d^{4}}{9 b^{2}} + \frac {4 c d^{3}}{9 b}\right ) + x^{5} \left (\frac {a^{2} d^{4}}{5 b^{3}} - \frac {4 a c d^{3}}{5 b^{2}} + \frac {6 c^{2} d^{2}}{5 b}\right ) + x \left (- \frac {a^{3} d^{4}}{b^{4}} + \frac {4 a^{2} c d^{3}}{b^{3}} - \frac {6 a c^{2} d^{2}}{b^{2}} + \frac {4 c^{3} d}{b}\right ) + \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{17} + a^{16} d^{16} - 16 a^{15} b c d^{15} + 120 a^{14} b^{2} c^{2} d^{14} - 560 a^{13} b^{3} c^{3} d^{13} + 1820 a^{12} b^{4} c^{4} d^{12} - 4368 a^{11} b^{5} c^{5} d^{11} + 8008 a^{10} b^{6} c^{6} d^{10} - 11440 a^{9} b^{7} c^{7} d^{9} + 12870 a^{8} b^{8} c^{8} d^{8} - 11440 a^{7} b^{9} c^{9} d^{7} + 8008 a^{6} b^{10} c^{10} d^{6} - 4368 a^{5} b^{11} c^{11} d^{5} + 1820 a^{4} b^{12} c^{12} d^{4} - 560 a^{3} b^{13} c^{13} d^{3} + 120 a^{2} b^{14} c^{14} d^{2} - 16 a b^{15} c^{15} d + b^{16} c^{16}, \left ( t \mapsto t \log {\left (\frac {4 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac {d^{4} x^{13}}{13 b} \]

[In]

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

[Out]

x**9*(-a*d**4/(9*b**2) + 4*c*d**3/(9*b)) + x**5*(a**2*d**4/(5*b**3) - 4*a*c*d**3/(5*b**2) + 6*c**2*d**2/(5*b))
+ x*(-a**3*d**4/b**4 + 4*a**2*c*d**3/b**3 - 6*a*c**2*d**2/b**2 + 4*c**3*d/b) + RootSum(256*_t**4*a**3*b**17 +
a**16*d**16 - 16*a**15*b*c*d**15 + 120*a**14*b**2*c**2*d**14 - 560*a**13*b**3*c**3*d**13 + 1820*a**12*b**4*c*
*4*d**12 - 4368*a**11*b**5*c**5*d**11 + 8008*a**10*b**6*c**6*d**10 - 11440*a**9*b**7*c**7*d**9 + 12870*a**8*b*
*8*c**8*d**8 - 11440*a**7*b**9*c**9*d**7 + 8008*a**6*b**10*c**10*d**6 - 4368*a**5*b**11*c**11*d**5 + 1820*a**4
*b**12*c**12*d**4 - 560*a**3*b**13*c**13*d**3 + 120*a**2*b**14*c**14*d**2 - 16*a*b**15*c**15*d + b**16*c**16,
Lambda(_t, _t*log(4*_t*a*b**4/(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c*
*4) + x))) + d**4*x**13/(13*b)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=\frac {45 \, b^{3} d^{4} x^{13} + 65 \, {\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{9} + 117 \, {\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{5} + 585 \, {\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x}{585 \, b^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{8 \, b^{4}} \]

[In]

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

[Out]

1/585*(45*b^3*d^4*x^13 + 65*(4*b^3*c*d^3 - a*b^2*d^4)*x^9 + 117*(6*b^3*c^2*d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)*x^
5 + 585*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*x)/b^4 + 1/8*(2*sqrt(2)*(b^4*c^4 - 4*a*b^3*c
^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))
/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d
^2 - 4*a^3*b*c*d^3 + a^4*d^4)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b))
)/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4
*d^4)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^4*c^4 - 4*a*b^3*c^
3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(
3/4)*b^(1/4)))/b^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (261) = 522\).

Time = 0.29 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.86 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{4} c^{4} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} c^{3} d + 6 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a^{4} d^{4}\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 )}{4 \, a b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{4} c^{4} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} c^{3} d + 6 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a^{4} d^{4}\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 )}{4 \, a b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{4} c^{4} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} c^{3} d + 6 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a^{4} d^{4}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{4} c^{4} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} c^{3} d + 6 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a^{4} d^{4}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{5}} + \frac {45 \, b^{12} d^{4} x^{13} + 260 \, b^{12} c d^{3} x^{9} - 65 \, a b^{11} d^{4} x^{9} + 702 \, b^{12} c^{2} d^{2} x^{5} - 468 \, a b^{11} c d^{3} x^{5} + 117 \, a^{2} b^{10} d^{4} x^{5} + 2340 \, b^{12} c^{3} d x - 3510 \, a b^{11} c^{2} d^{2} x + 2340 \, a^{2} b^{10} c d^{3} x - 585 \, a^{3} b^{9} d^{4} x}{585 \, b^{13}} \]

[In]

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

[Out]

1/4*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)
^(1/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^5
) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*
b^3)^(1/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a
*b^5) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4
*(a*b^3)^(1/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^5) - 1/8
*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)^(1
/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^5) + 1/585*(45*b^12
*d^4*x^13 + 260*b^12*c*d^3*x^9 - 65*a*b^11*d^4*x^9 + 702*b^12*c^2*d^2*x^5 - 468*a*b^11*c*d^3*x^5 + 117*a^2*b^1
0*d^4*x^5 + 2340*b^12*c^3*d*x - 3510*a*b^11*c^2*d^2*x + 2340*a^2*b^10*c*d^3*x - 585*a^3*b^9*d^4*x)/b^13

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 1822, normalized size of antiderivative = 5.49 \[ \int \frac {\left (c+d x^4\right )^4}{a+b x^4} \, dx=\text {Too large to display} \]

[In]

int((c + d*x^4)^4/(a + b*x^4),x)

[Out]

x*((4*c^3*d)/b - (a*((a*((a*d^4)/b^2 - (4*c*d^3)/b))/b + (6*c^2*d^2)/b))/b) - x^9*((a*d^4)/(9*b^2) - (4*c*d^3)
/(9*b)) + x^5*((a*((a*d^4)/b^2 - (4*c*d^3)/b))/(5*b) + (6*c^2*d^2)/(5*b)) + (d^4*x^13)/(13*b) + (atan(((((4*x*
(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^
6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 - (4*(a*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d
+ 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^4*1i)/(4*(-a)^(3/4)*b^(17/4)) + (((4*
x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*
a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 + (4*(a*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*
d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^4*1i)/(4*(-a)^(3/4)*b^(17/4)))/((((
4*x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 2
8*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 - (4*(a*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^
3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^4)/(4*(-a)^(3/4)*b^(17/4)) - (((4
*x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28
*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 + (4*(a*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3
*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^4)/(4*(-a)^(3/4)*b^(17/4))))*(a*d
- b*c)^4*1i)/(2*(-a)^(3/4)*b^(17/4)) + (atan(((((4*x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*
d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 - ((a
*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*4i)/((-a)^(3/4)*b^(21/
4)))*(a*d - b*c)^4)/(4*(-a)^(3/4)*b^(17/4)) + (((4*x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*
d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 + ((a
*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*4i)/((-a)^(3/4)*b^(21/
4)))*(a*d - b*c)^4)/(4*(-a)^(3/4)*b^(17/4)))/((((4*x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*
d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 - ((a
*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*4i)/((-a)^(3/4)*b^(21/
4)))*(a*d - b*c)^4*1i)/(4*(-a)^(3/4)*b^(17/4)) - (((4*x*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c
^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/b^5 +
((a*d - b*c)^4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*4i)/((-a)^(3/4)*b^(
21/4)))*(a*d - b*c)^4*1i)/(4*(-a)^(3/4)*b^(17/4))))*(a*d - b*c)^4)/(2*(-a)^(3/4)*b^(17/4))