3.2.67 \(\int \frac {(c+d x^4)^4}{(a+b x^4)^2} \, dx\) [167]
Optimal. Leaf size=357 \[ \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}-\frac {(b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{17/4}}+\frac {(b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{17/4}}-\frac {(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}+\frac {(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}} \]
[Out]
d^2*(3*a^2*d^2-8*a*b*c*d+6*b^2*c^2)*x/b^4+2/5*d^3*(-a*d+2*b*c)*x^5/b^3+1/9*d^4*x^9/b^2+1/4*(-a*d+b*c)^4*x/a/b^
4/(b*x^4+a)+1/16*(-a*d+b*c)^3*(13*a*d+3*b*c)*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(7/4)/b^(17/4)*2^(1/2)+1/1
6*(-a*d+b*c)^3*(13*a*d+3*b*c)*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(7/4)/b^(17/4)*2^(1/2)-1/32*(-a*d+b*c)^3*(
13*a*d+3*b*c)*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(7/4)/b^(17/4)*2^(1/2)+1/32*(-a*d+b*c)^3*(1
3*a*d+3*b*c)*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(7/4)/b^(17/4)*2^(1/2)
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Rubi [A]
time = 0.24, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {398, 393, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^3 (13 a d+3 b c)}{8 \sqrt {2} a^{7/4} b^{17/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^3 (13 a d+3 b c)}{8 \sqrt {2} a^{7/4} b^{17/4}}-\frac {(b c-a d)^3 (13 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}+\frac {(b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}+\frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {x (b c-a d)^4}{4 a b^4 \left (a+b x^4\right )}+\frac {2 d^3 x^5 (2 b c-a d)}{5 b^3}+\frac {d^4 x^9}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x^4)^4/(a + b*x^4)^2,x]
[Out]
(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^5)/(5*b^3) + (d^4*x^9)/(9*b^2) + ((b*
c - a*d)^4*x)/(4*a*b^4*(a + b*x^4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(8*Sqrt[2]*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sq
rt[2]*a^(7/4)*b^(17/4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^
2])/(16*Sqrt[2]*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*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 393
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
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*} \int \frac {\left (c+d x^4\right )^4}{\left (a+b x^4\right )^2} \, dx &=\int \left (\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac {2 d^3 (2 b c-a d) x^4}{b^3}+\frac {d^4 x^8}{b^2}+\frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^4}{b^4 \left (a+b x^4\right )^2}\right ) \, dx\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {\int \frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^4}{\left (a+b x^4\right )^2} \, dx}{b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}+\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac {1}{a+b x^4} \, dx}{4 a b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}+\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^4}+\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}+\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{9/2}}+\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{9/2}}-\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \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}{16 \sqrt {2} a^{7/4} b^{17/4}}-\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \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}{16 \sqrt {2} a^{7/4} b^{17/4}}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}-\frac {(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}+\frac {(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}+\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{17/4}}-\frac {\left ((b c-a d)^3 (3 b c+13 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{17/4}}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^5}{5 b^3}+\frac {d^4 x^9}{9 b^2}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^4\right )}-\frac {(b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{17/4}}+\frac {(b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{17/4}}-\frac {(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}+\frac {(b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{17/4}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 341, normalized size = 0.96 \begin {gather*} \frac {1440 \sqrt [4]{b} d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x+576 b^{5/4} d^3 (2 b c-a d) x^5+160 b^{9/4} d^4 x^9+\frac {360 \sqrt [4]{b} (b c-a d)^4 x}{a \left (a+b x^4\right )}+\frac {90 \sqrt {2} (-b c+a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {90 \sqrt {2} (b c-a d)^3 (3 b c+13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {45 \sqrt {2} (-b c+a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {45 \sqrt {2} (b c-a d)^3 (3 b c+13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}}{1440 b^{17/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x^4)^4/(a + b*x^4)^2,x]
[Out]
(1440*b^(1/4)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 576*b^(5/4)*d^3*(2*b*c - a*d)*x^5 + 160*b^(9/4)*d^4*
x^9 + (360*b^(1/4)*(b*c - a*d)^4*x)/(a*(a + b*x^4)) + (90*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (90*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4
)*x)/a^(1/4)])/a^(7/4) + (45*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/a^(7/4) + (45*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/a^(7/4))/(1440*b^(17/4))
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Maple [A]
time = 0.27, size = 288, normalized size = 0.81
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {d^{4} x^{9}}{9 b^{2}}-\frac {2 d^{4} a \,x^{5}}{5 b^{3}}+\frac {4 d^{3} c \,x^{5}}{5 b^{2}}+\frac {3 d^{4} a^{2} x}{b^{4}}-\frac {8 d^{3} a c x}{b^{3}}+\frac {6 d^{2} c^{2} x}{b^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{4 a \,b^{4} \left (b \,x^{4}+a \right )}-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (13 a^{4} d^{4}-36 a^{3} b c \,d^{3}+30 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b^{5} a}\) |
\(219\) |
| default |
\(\frac {d^{2} \left (\frac {1}{9} b^{2} d^{2} x^{9}-\frac {2}{5} a b \,d^{2} x^{5}+\frac {4}{5} b^{2} c d \,x^{5}+3 a^{2} d^{2} x -8 a b c d x +6 b^{2} c^{2} x \right )}{b^{4}}-\frac {-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{4 a \left (b \,x^{4}+a \right )}+\frac {\left (13 a^{4} d^{4}-36 a^{3} b c \,d^{3}+30 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}}{b^{4}}\) |
\(288\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^4+c)^4/(b*x^4+a)^2,x,method=_RETURNVERBOSE)
[Out]
d^2/b^4*(1/9*b^2*d^2*x^9-2/5*a*b*d^2*x^5+4/5*b^2*c*d*x^5+3*a^2*d^2*x-8*a*b*c*d*x+6*b^2*c^2*x)-1/b^4*(-1/4*(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)/a*x/(b*x^4+a)+1/32*(13*a^4*d^4-36*a^3*b*c*d^3+30*a
^2*b^2*c^2*d^2-4*a*b^3*c^3*d-3*b^4*c^4)/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x
^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1)))
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Maxima [A]
time = 0.50, size = 521, normalized size = 1.46 \begin {gather*} \frac {{\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 )} x}{4 \, {\left (a b^{5} x^{4} + a^{2} b^{4}\right )}} + \frac {5 \, b^{2} d^{4} x^{9} + 18 \, {\left (2 \, b^{2} c d^{3} - a b d^{4}\right )} x^{5} + 45 \, {\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x}{45 \, b^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 30 \, a^{2} b^{2} c^{2} d^{2} + 36 \, a^{3} b c d^{3} - 13 \, 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 (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 30 \, a^{2} b^{2} c^{2} d^{2} + 36 \, a^{3} b c d^{3} - 13 \, 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 (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 30 \, a^{2} b^{2} c^{2} d^{2} + 36 \, a^{3} b c d^{3} - 13 \, 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 (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 30 \, a^{2} b^{2} c^{2} d^{2} + 36 \, a^{3} b c d^{3} - 13 \, 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}}}}{32 \, a b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^4+c)^4/(b*x^4+a)^2,x, algorithm="maxima")
[Out]
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/(a*b^5*x^4 + a^2*b^4) + 1/45*(5*
b^2*d^4*x^9 + 18*(2*b^2*c*d^3 - a*b*d^4)*x^5 + 45*(6*b^2*c^2*d^2 - 8*a*b*c*d^3 + 3*a^2*d^4)*x)/b^4 + 1/32*(2*s
qrt(2)*(3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*a^4*d^4)*arctan(1/2*sqrt(2)*(2*sq
rt(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)*(3*b^4*c
^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*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)*(3*b^4*c^4 + 4*a*b^3*c^3*
d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*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)*(3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*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)))/(a*b^4)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2580 vs.
\(2 (286) = 572\).
time = 3.33, size = 2580, normalized size = 7.23 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^4+c)^4/(b*x^4+a)^2,x, algorithm="fricas")
[Out]
1/720*(80*a*b^3*d^4*x^13 + 16*(36*a*b^3*c*d^3 - 13*a^2*b^2*d^4)*x^9 + 144*(30*a*b^3*c^2*d^2 - 36*a^2*b^2*c*d^3
+ 13*a^3*b*d^4)*x^5 - 180*(a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2
- 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 7721
12*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a
^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*
a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4)*arctan((sqrt(a^4*b^8*sqrt(-(81*b^16*c^16 + 432*a*b^15*c^15*
d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 4348
08*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^1
0*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*
a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17)) + (9*b^8*c^8 + 24*a*b^7*c^7*d - 164*a^
2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 - 936*
a^7*b*c*d^7 + 169*a^8*d^8)*x^2)*a^5*b^13*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a
^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b
^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*
c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c
*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(3/4) + (3*a^5*b^17*c^4 + 4*a^6*b^16*c^3*d - 30*a^7*b^15*c^2*d^2 + 36*a^8
*b^14*c*d^3 - 13*a^9*b^13*d^4)*x*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*
c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d
^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11
+ 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 +
28561*a^16*d^16)/(a^7*b^17))^(3/4))/(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13
*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*
d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^1
1 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 +
28561*a^16*d^16)) - 45*(a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8
304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*
a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11
*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^1
5*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4)*log(a^2*b^4*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^1
4*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10
*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 -
11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^1
4 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4) - (3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^
2 + 36*a^3*b*c*d^3 - 13*a^4*d^4)*x) + 45*(a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*
b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c
^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^1
0 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*
d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4)*log(-a^2*b^4*(-(81*b^16*c^16 + 432*a*b^15*c^1
5*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 43
4808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a
^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 157778
4*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4) - (3*b^4*c^4 + 4*a*b^3*c^3*d -
30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*a^4*d^4)*x) + 180*(b^4*c^4 - 4*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 - 36
*a^3*b*c*d^3 + 13*a^4*d^4)*x)/(a*b^5*x^4 + a^2*b^4)
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**4+c)**4/(b*x**4+a)**2,x)
[Out]
Timed out
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 642 vs.
\(2 (286) = 572\).
time = 0.65, size = 642, normalized size = 1.80 \begin {gather*} \frac {\sqrt {2} {\left (3 \, \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 - 30 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} + 36 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} - 13 \, \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 )}{16 \, a^{2} b^{5}} + \frac {\sqrt {2} {\left (3 \, \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 - 30 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} + 36 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} - 13 \, \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 )}{16 \, a^{2} b^{5}} + \frac {\sqrt {2} {\left (3 \, \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 - 30 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} + 36 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} - 13 \, \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 )}{32 \, a^{2} b^{5}} - \frac {\sqrt {2} {\left (3 \, \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 - 30 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d^{2} + 36 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} b c d^{3} - 13 \, \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 )}{32 \, a^{2} b^{5}} + \frac {b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{4 \, {\left (b x^{4} + a\right )} a b^{4}} + \frac {5 \, b^{16} d^{4} x^{9} + 36 \, b^{16} c d^{3} x^{5} - 18 \, a b^{15} d^{4} x^{5} + 270 \, b^{16} c^{2} d^{2} x - 360 \, a b^{15} c d^{3} x + 135 \, a^{2} b^{14} d^{4} x}{45 \, b^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^4+c)^4/(b*x^4+a)^2,x, algorithm="giac")
[Out]
1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 + 36*(a
*b^3)^(1/4)*a^3*b*c*d^3 - 13*(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^2*b^5) + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)*a^2*b^2*c
^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(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^2*b^5) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(
1/4)*a^2*b^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4
) + sqrt(a/b))/(a^2*b^5) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1
/4)*a^2*b^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*log(x^2 - sqrt(2)*x*(a/b)^(1/4)
+ sqrt(a/b))/(a^2*b^5) + 1/4*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4*d^4*x
)/((b*x^4 + a)*a*b^4) + 1/45*(5*b^16*d^4*x^9 + 36*b^16*c*d^3*x^5 - 18*a*b^15*d^4*x^5 + 270*b^16*c^2*d^2*x - 36
0*a*b^15*c*d^3*x + 135*a^2*b^14*d^4*x)/b^18
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Mupad [B]
time = 0.30, size = 2043, normalized size = 5.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^4)^4/(a + b*x^4)^2,x)
[Out]
x*((2*a*((2*a*d^4)/b^3 - (4*c*d^3)/b^2))/b - (a^2*d^4)/b^4 + (6*c^2*d^2)/b^2) - x^5*((2*a*d^4)/(5*b^3) - (4*c*
d^3)/(5*b^2)) + (d^4*x^9)/(9*b^2) + (x*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)
)/(4*a*(a*b^4 + b^5*x^4)) + (atan(((((x*(169*a^8*d^8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 +
1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2
*b^5) - ((a*d - b*c)^3*(13*a*d + 3*b*c)*(3*b^4*c^4 - 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*
b*c*d^3))/(4*(-a)^(7/4)*b^(21/4)))*(a*d - b*c)^3*(13*a*d + 3*b*c)*1i)/(16*(-a)^(7/4)*b^(17/4)) + (((x*(169*a^8
*d^8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 20
76*a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2*b^5) + ((a*d - b*c)^3*(13*a*d + 3*b*c)*(3*b^4*c
^4 - 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*b*c*d^3))/(4*(-a)^(7/4)*b^(21/4)))*(a*d - b*c)^3
*(13*a*d + 3*b*c)*1i)/(16*(-a)^(7/4)*b^(17/4)))/((((x*(169*a^8*d^8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*
b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*
c*d^7))/(4*a^2*b^5) - ((a*d - b*c)^3*(13*a*d + 3*b*c)*(3*b^4*c^4 - 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c
^3*d + 36*a^3*b*c*d^3))/(4*(-a)^(7/4)*b^(21/4)))*(a*d - b*c)^3*(13*a*d + 3*b*c))/(16*(-a)^(7/4)*b^(17/4)) - ((
(x*(169*a^8*d^8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c
^3*d^5 + 2076*a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2*b^5) + ((a*d - b*c)^3*(13*a*d + 3*b*
c)*(3*b^4*c^4 - 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*b*c*d^3))/(4*(-a)^(7/4)*b^(21/4)))*(a
*d - b*c)^3*(13*a*d + 3*b*c))/(16*(-a)^(7/4)*b^(17/4))))*(a*d - b*c)^3*(13*a*d + 3*b*c)*1i)/(8*(-a)^(7/4)*b^(1
7/4)) + (atan(((((x*(169*a^8*d^8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4
- 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2*b^5) - ((a*d - b*c)
^3*(13*a*d + 3*b*c)*(3*b^4*c^4 - 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*b*c*d^3)*1i)/(4*(-a)
^(7/4)*b^(21/4)))*(a*d - b*c)^3*(13*a*d + 3*b*c))/(16*(-a)^(7/4)*b^(17/4)) + (((x*(169*a^8*d^8 + 9*b^8*c^8 - 1
64*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 +
24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2*b^5) + ((a*d - b*c)^3*(13*a*d + 3*b*c)*(3*b^4*c^4 - 13*a^4*d^4 - 30
*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*b*c*d^3)*1i)/(4*(-a)^(7/4)*b^(21/4)))*(a*d - b*c)^3*(13*a*d + 3*b*c)
)/(16*(-a)^(7/4)*b^(17/4)))/((((x*(169*a^8*d^8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a
^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2*b^5)
- ((a*d - b*c)^3*(13*a*d + 3*b*c)*(3*b^4*c^4 - 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*b*c*d^
3)*1i)/(4*(-a)^(7/4)*b^(21/4)))*(a*d - b*c)^3*(13*a*d + 3*b*c)*1i)/(16*(-a)^(7/4)*b^(17/4)) - (((x*(169*a^8*d^
8 + 9*b^8*c^8 - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*
a^6*b^2*c^2*d^6 + 24*a*b^7*c^7*d - 936*a^7*b*c*d^7))/(4*a^2*b^5) + ((a*d - b*c)^3*(13*a*d + 3*b*c)*(3*b^4*c^4
- 13*a^4*d^4 - 30*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 36*a^3*b*c*d^3)*1i)/(4*(-a)^(7/4)*b^(21/4)))*(a*d - b*c)^3
*(13*a*d + 3*b*c)*1i)/(16*(-a)^(7/4)*b^(17/4))))*(a*d - b*c)^3*(13*a*d + 3*b*c))/(8*(-a)^(7/4)*b^(17/4))
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