3.7.29 \(\int \frac {1}{(d+e x)^4 (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [629]

Optimal. Leaf size=408 \[ -\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

1/6*(14*a*c-5*b^2)/a^2/(-4*a*c+b^2)/e/(e*x+d)^3+1/2*b*(-19*a*c+5*b^2)/a^3/(-4*a*c+b^2)/e/(e*x+d)+1/2*(b^2-2*a*
c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-
(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^4-29*a*b^2*c+28*a^2*c^2+b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))/a^3/(-4*
a*c+b^2)^(3/2)/e*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2)
)^(1/2))*c^(1/2)*(5*b^4-29*a*b^2*c+28*a^2*c^2-b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(3/2)/e*2
^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 2.46, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1156, 1135, 1295, 1180, 211} \begin {gather*} \frac {b \left (5 b^2-19 a c\right )}{2 a^3 e \left (b^2-4 a c\right ) (d+e x)}-\frac {5 b^2-14 a c}{6 a^2 e \left (b^2-4 a c\right ) (d+e x)^3}+\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 e \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^3 e \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

-1/6*(5*b^2 - 14*a*c)/(a^2*(b^2 - 4*a*c)*e*(d + e*x)^3) + (b*(5*b^2 - 19*a*c))/(2*a^3*(b^2 - 4*a*c)*e*(d + e*x
)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (
Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d +
e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (Sqrt[
c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))
/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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 1135

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*
a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*
a*c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7
)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (Integ
erQ[p] || IntegerQ[m])

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1180

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

Rule 1295

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(f
*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^4 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-5 b^2+14 a c-5 b c x^2}{x^4 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{2 a \left (b^2-4 a c\right ) e}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e (d+e x)^3}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\text {Subst}\left (\int \frac {-3 b \left (5 b^2-19 a c\right )-3 c \left (5 b^2-14 a c\right ) x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{6 a^2 \left (b^2-4 a c\right ) e}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (5 b^4-24 a b^2 c+14 a^2 c^2\right )-3 b c \left (5 b^2-19 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{6 a^3 \left (b^2-4 a c\right ) e}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{4 a^3 \left (b^2-4 a c\right )^{3/2} e}+\frac {\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{4 a^3 \left (b^2-4 a c\right )^{3/2} e}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}

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Mathematica [A]
time = 1.93, size = 384, normalized size = 0.94 \begin {gather*} \frac {-\frac {4 a}{(d+e x)^3}+\frac {24 b}{d+e x}+\frac {6 (d+e x) \left (b^4-4 a b^2 c+2 a^2 c^2+b^3 c (d+e x)^2-3 a b c^2 (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-5 b^4+29 a b^2 c-28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{12 a^3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

((-4*a)/(d + e*x)^3 + (24*b)/(d + e*x) + (6*(d + e*x)*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*(d + e*x)^2 - 3*a*b
*c^2*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + (3*Sqrt[2]*Sqrt[c]*(5*b^4 - 29*a*b^
2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sq
rt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 2
9*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*
x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.21, size = 489, normalized size = 1.20

method	result	size
default	\(-\frac {1}{3 a^{2} e \left (e x +d \right )^{3}}+\frac {2 b}{a^{3} e \left (e x +d \right )}-\frac {\frac {-\frac {b c \,e^{2} \left (3 a c -b^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right )}-\frac {3 d b c e \left (3 a c -b^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (-9 a b \,c^{2} d^{2}+3 b^{3} c \,d^{2}+2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{8 a c -2 b^{2}}+\frac {d \left (-3 a b \,c^{2} d^{2}+b^{3} c \,d^{2}+2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (b c \,e^{2} \left (-19 a c +5 b^{2}\right ) \textit {\_R}^{2}+2 b c d e \left (-19 a c +5 b^{2}\right ) \textit {\_R} -19 a b \,c^{2} d^{2}+5 b^{3} c \,d^{2}+14 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{4 \left (4 a c -b^{2}\right ) e}}{a^{3}}\)	\(489\)
risch	\(\frac {\frac {\left (19 a c -5 b^{2}\right ) b c \,e^{5} x^{6}}{2 \left (4 a c -b^{2}\right ) a^{3}}+\frac {3 d \,e^{4} \left (19 a c -5 b^{2}\right ) b c \,x^{5}}{\left (4 a c -b^{2}\right ) a^{3}}-\frac {\left (-855 a b \,c^{2} d^{2}+225 b^{3} c \,d^{2}+14 a^{2} c^{2}-62 a \,b^{2} c +15 b^{4}\right ) e^{3} x^{4}}{6 \left (4 a c -b^{2}\right ) a^{3}}-\frac {2 \left (-285 a b \,c^{2} d^{2}+75 b^{3} c \,d^{2}+14 a^{2} c^{2}-62 a \,b^{2} c +15 b^{4}\right ) d \,e^{2} x^{3}}{3 \left (4 a c -b^{2}\right ) a^{3}}+\frac {e \left (855 a b \,c^{2} d^{4}-225 b^{3} c \,d^{4}-84 a^{2} c^{2} d^{2}+372 a \,b^{2} c \,d^{2}-90 b^{4} d^{2}+40 a^{2} b c -10 a \,b^{3}\right ) x^{2}}{6 \left (4 a c -b^{2}\right ) a^{3}}+\frac {d \left (171 a b \,c^{2} d^{4}-45 b^{3} c \,d^{4}-28 a^{2} c^{2} d^{2}+124 a \,b^{2} c \,d^{2}-30 b^{4} d^{2}+40 a^{2} b c -10 a \,b^{3}\right ) x}{3 \left (4 a c -b^{2}\right ) a^{3}}-\frac {-57 a b \,c^{2} d^{6}+15 b^{3} c \,d^{6}+14 a^{2} c^{2} d^{4}-62 a \,b^{2} c \,d^{4}+15 b^{4} d^{4}-40 a^{2} b c \,d^{2}+10 a \,b^{3} d^{2}+8 a^{3} c -2 a^{2} b^{2}}{6 e \left (4 a c -b^{2}\right ) a^{3}}}{\left (e x +d \right )^{3} \left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4096 a^{13} c^{6} e^{4}-6144 a^{12} b^{2} c^{5} e^{4}+3840 a^{11} b^{4} c^{4} e^{4}-1280 a^{10} b^{6} c^{3} e^{4}+240 a^{9} b^{8} c^{2} e^{4}-24 a^{8} b^{10} c \,e^{4}+a^{7} b^{12} e^{4}\right ) \textit {\_Z}^{4}+\left (-80640 a^{7} b \,c^{7} e^{2}+215040 a^{6} b^{3} c^{6} e^{2}-219744 a^{5} b^{5} c^{5} e^{2}+116928 a^{4} b^{7} c^{4} e^{2}-35767 a^{3} b^{9} c^{3} e^{2}+6366 a^{2} b^{11} c^{2} e^{2}-615 a \,b^{13} c \,e^{2}+25 b^{15} e^{2}\right ) \textit {\_Z}^{2}+38416 a^{2} c^{9}-17640 a \,b^{2} c^{8}+2025 b^{4} c^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10240 a^{13} c^{6} e^{5}-15872 a^{12} b^{2} c^{5} e^{5}+10240 a^{11} b^{4} c^{4} e^{5}-3520 a^{10} b^{6} c^{3} e^{5}+680 a^{9} b^{8} c^{2} e^{5}-70 a^{8} b^{10} c \,e^{5}+3 a^{7} b^{12} e^{5}\right ) \textit {\_R}^{4}+\left (-175168 a^{7} b \,c^{7} e^{3}+451120 a^{6} b^{3} c^{6} e^{3}-451644 a^{5} b^{5} c^{5} e^{3}+237257 a^{4} b^{7} c^{4} e^{3}-71999 a^{3} b^{9} c^{3} e^{3}+12757 a^{2} b^{11} c^{2} e^{3}-1230 a \,b^{13} c \,e^{3}+50 b^{15} e^{3}\right ) \textit {\_R}^{2}+76832 e \,a^{2} c^{9}-35280 a \,b^{2} e \,c^{8}+4050 b^{4} c^{7} e \right ) x +\left (10240 a^{13} c^{6} d \,e^{4}-15872 a^{12} b^{2} c^{5} d \,e^{4}+10240 a^{11} b^{4} c^{4} d \,e^{4}-3520 a^{10} b^{6} c^{3} d \,e^{4}+680 a^{9} b^{8} c^{2} d \,e^{4}-70 a^{8} b^{10} c d \,e^{4}+3 a^{7} b^{12} d \,e^{4}\right ) \textit {\_R}^{4}+\left (-8448 a^{10} b \,c^{6} e^{3}+15872 a^{9} b^{3} c^{5} e^{3}-11872 a^{8} b^{5} c^{4} e^{3}+4592 a^{7} b^{7} c^{3} e^{3}-977 a^{6} b^{9} c^{2} e^{3}+109 a^{5} b^{11} c \,e^{3}-5 a^{4} b^{13} e^{3}\right ) \textit {\_R}^{3}+\left (-175168 a^{7} b \,c^{7} d \,e^{2}+451120 a^{6} b^{3} c^{6} d \,e^{2}-451644 a^{5} b^{5} c^{5} d \,e^{2}+237257 a^{4} b^{7} c^{4} d \,e^{2}-71999 a^{3} b^{9} c^{3} d \,e^{2}+12757 a^{2} b^{11} c^{2} d \,e^{2}-1230 a \,b^{13} c d \,e^{2}+50 b^{15} d \,e^{2}\right ) \textit {\_R}^{2}+\left (10976 a^{5} c^{8} e -9184 a^{4} b^{2} c^{7} e +2510 a^{3} b^{4} c^{6} e -225 a^{2} b^{6} c^{5} e \right ) \textit {\_R} +76832 a^{2} c^{9} d -35280 a \,b^{2} c^{8} d +4050 b^{4} c^{7} d \right )\right )}{4}\)	\(1394\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x,method=_RETURNVERBOSE)

[Out]

-1/3/a^2/e/(e*x+d)^3+2/a^3*b/e/(e*x+d)-1/a^3*((-1/2*b*c*e^2*(3*a*c-b^2)/(4*a*c-b^2)*x^3-3/2*d*b*c*e*(3*a*c-b^2
)/(4*a*c-b^2)*x^2+1/2*(-9*a*b*c^2*d^2+3*b^3*c*d^2+2*a^2*c^2-4*a*b^2*c+b^4)/(4*a*c-b^2)*x+1/2*d/e*(-3*a*b*c^2*d
^2+b^3*c*d^2+2*a^2*c^2-4*a*b^2*c+b^4)/(4*a*c-b^2))/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*
x^2+c*d^4+2*b*d*e*x+b*d^2+a)+1/4/(4*a*c-b^2)/e*sum((b*c*e^2*(-19*a*c+5*b^2)*_R^2+2*b*c*d*e*(-19*a*c+5*b^2)*_R-
19*a*b*c^2*d^2+5*b^3*c*d^2+14*a^2*c^2-24*a*b^2*c+5*b^4)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b
*e+b*d)*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2
*b+a)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="maxima")

[Out]

1/6*(3*(5*b^3*c - 19*a*b*c^2)*d^6 + 18*(5*b^3*c*e^5 - 19*a*b*c^2*e^5)*d*x^5 + 3*(5*b^3*c*e^6 - 19*a*b*c^2*e^6)
*x^6 + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d^4 + (15*b^4*e^4 - 62*a*b^2*c*e^4 + 14*a^2*c^2*e^4 + 45*(5*b^3*c*e^
4 - 19*a*b*c^2*e^4)*d^2)*x^4 - 2*a^2*b^2 + 8*a^3*c + 4*(15*(5*b^3*c*e^3 - 19*a*b*c^2*e^3)*d^3 + (15*b^4*e^3 -
62*a*b^2*c*e^3 + 14*a^2*c^2*e^3)*d)*x^3 + 10*(a*b^3 - 4*a^2*b*c)*d^2 + (45*(5*b^3*c*e^2 - 19*a*b*c^2*e^2)*d^4
+ 10*a*b^3*e^2 - 40*a^2*b*c*e^2 + 6*(15*b^4*e^2 - 62*a*b^2*c*e^2 + 14*a^2*c^2*e^2)*d^2)*x^2 + 2*(9*(5*b^3*c*e
- 19*a*b*c^2*e)*d^5 + 2*(15*b^4*e - 62*a*b^2*c*e + 14*a^2*c^2*e)*d^3 + 10*(a*b^3*e - 4*a^2*b*c*e)*d)*x)/((a^3*
b^2*c*e - 4*a^4*c^2*e)*d^7 + 7*(a^3*b^2*c*e^7 - 4*a^4*c^2*e^7)*d*x^6 + (a^3*b^2*c*e^8 - 4*a^4*c^2*e^8)*x^7 + (
a^3*b^3*e - 4*a^4*b*c*e)*d^5 + (a^3*b^3*e^6 - 4*a^4*b*c*e^6 + 21*(a^3*b^2*c*e^6 - 4*a^4*c^2*e^6)*d^2)*x^5 + 5*
(7*(a^3*b^2*c*e^5 - 4*a^4*c^2*e^5)*d^3 + (a^3*b^3*e^5 - 4*a^4*b*c*e^5)*d)*x^4 + (a^4*b^2*e - 4*a^5*c*e)*d^3 +
(a^4*b^2*e^4 - 4*a^5*c*e^4 + 35*(a^3*b^2*c*e^4 - 4*a^4*c^2*e^4)*d^4 + 10*(a^3*b^3*e^4 - 4*a^4*b*c*e^4)*d^2)*x^
3 + (21*(a^3*b^2*c*e^3 - 4*a^4*c^2*e^3)*d^5 + 10*(a^3*b^3*e^3 - 4*a^4*b*c*e^3)*d^3 + 3*(a^4*b^2*e^3 - 4*a^5*c*
e^3)*d)*x^2 + (7*(a^3*b^2*c*e^2 - 4*a^4*c^2*e^2)*d^6 + 5*(a^3*b^3*e^2 - 4*a^4*b*c*e^2)*d^4 + 3*(a^4*b^2*e^2 -
4*a^5*c*e^2)*d^2)*x) - 1/2*integrate(-(5*b^4 - 24*a*b^2*c + 14*a^2*c^2 + (5*b^3*c - 19*a*b*c^2)*d^2 + 2*(5*b^3
*c*e - 19*a*b*c^2*e)*d*x + (5*b^3*c*e^2 - 19*a*b*c^2*e^2)*x^2)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + b*d^2 + (6
*c*d^2*e^2 + b*e^2)*x^2 + 2*(2*c*d^3*e + b*d*e)*x + a), x)/(a^3*b^2 - 4*a^4*c)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5636 vs. \(2 (361) = 722\).
time = 0.73, size = 5636, normalized size = 13.81 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="fricas")

[Out]

1/12*(6*(5*b^3*c - 19*a*b*c^2)*x^6*e^6 + 36*(5*b^3*c - 19*a*b*c^2)*d*x^5*e^5 + 6*(5*b^3*c - 19*a*b*c^2)*d^6 +
2*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2 + 45*(5*b^3*c - 19*a*b*c^2)*d^2)*x^4*e^4 + 2*(15*b^4 - 62*a*b^2*c + 14*a^2
*c^2)*d^4 + 8*(15*(5*b^3*c - 19*a*b*c^2)*d^3 + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d)*x^3*e^3 - 4*a^2*b^2 + 16*
a^3*c + 2*(45*(5*b^3*c - 19*a*b*c^2)*d^4 + 10*a*b^3 - 40*a^2*b*c + 6*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d^2)*x
^2*e^2 + 20*(a*b^3 - 4*a^2*b*c)*d^2 + 4*(9*(5*b^3*c - 19*a*b*c^2)*d^5 + 2*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d
^3 + 10*(a*b^3 - 4*a^2*b*c)*d)*x*e - 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7*e^8 + 7*(a^3*b^2*c - 4*a^4*c^2)*
d*x^6*e^7 + (a^3*b^3 - 4*a^4*b*c + 21*(a^3*b^2*c - 4*a^4*c^2)*d^2)*x^5*e^6 + 5*(7*(a^3*b^2*c - 4*a^4*c^2)*d^3
+ (a^3*b^3 - 4*a^4*b*c)*d)*x^4*e^5 + (a^4*b^2 - 4*a^5*c + 35*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 10*(a^3*b^3 - 4*a^4
*b*c)*d^2)*x^3*e^4 + (21*(a^3*b^2*c - 4*a^4*c^2)*d^5 + 10*(a^3*b^3 - 4*a^4*b*c)*d^3 + 3*(a^4*b^2 - 4*a^5*c)*d)
*x^2*e^3 + (7*(a^3*b^2*c - 4*a^4*c^2)*d^6 + 5*(a^3*b^3 - 4*a^4*b*c)*d^4 + 3*(a^4*b^2 - 4*a^5*c)*d^2)*x*e^2 + (
(a^3*b^2*c - 4*a^4*c^2)*d^7 + (a^3*b^3 - 4*a^4*b*c)*d^5 + (a^4*b^2 - 4*a^5*c)*d^3)*e)*sqrt(-(25*b^9 - 315*a*b^
7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^1
0*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*
b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*e^(-2)/(a^7*b^6 - 12*a^8*
b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log((1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2
*c^7 + 9604*a^4*c^8)*x*e + (1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*
c^8)*d + 1/2*sqrt(1/2)*((5*a^7*b^11 - 94*a^8*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 -
3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 -
24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3))*e - (125*b^14 -
2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*
b^2*c^6 - 10976*a^7*c^7)*e)*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4
+ (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2
- 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^1
6*b^2*c^2 - 64*a^17*c^3)))*e^(-2)/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))) + 3*sqrt(1/2)*((a^
3*b^2*c - 4*a^4*c^2)*x^7*e^8 + 7*(a^3*b^2*c - 4*a^4*c^2)*d*x^6*e^7 + (a^3*b^3 - 4*a^4*b*c + 21*(a^3*b^2*c - 4*
a^4*c^2)*d^2)*x^5*e^6 + 5*(7*(a^3*b^2*c - 4*a^4*c^2)*d^3 + (a^3*b^3 - 4*a^4*b*c)*d)*x^4*e^5 + (a^4*b^2 - 4*a^5
*c + 35*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 10*(a^3*b^3 - 4*a^4*b*c)*d^2)*x^3*e^4 + (21*(a^3*b^2*c - 4*a^4*c^2)*d^5
+ 10*(a^3*b^3 - 4*a^4*b*c)*d^3 + 3*(a^4*b^2 - 4*a^5*c)*d)*x^2*e^3 + (7*(a^3*b^2*c - 4*a^4*c^2)*d^6 + 5*(a^3*b^
3 - 4*a^4*b*c)*d^4 + 3*(a^4*b^2 - 4*a^5*c)*d^2)*x*e^2 + ((a^3*b^2*c - 4*a^4*c^2)*d^7 + (a^3*b^3 - 4*a^4*b*c)*d
^5 + (a^4*b^2 - 4*a^5*c)*d^3)*e)*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*
b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8
*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 4
8*a^16*b^2*c^2 - 64*a^17*c^3)))*e^(-2)/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log((1125*b^8*
c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*x*e + (1125*b^8*c^4 - 12325*a*b^
6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*d - 1/2*sqrt(1/2)*((5*a^7*b^11 - 94*a^8*b^9*c +
700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*c + 39
525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^1
5*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3))*e - (125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c
^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7)*e)*sqrt(-(25*b^9 - 315*a*b^7
*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10
*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b
^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*e^(-2)/(a^7*b^6 - 12*a^8*b
^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))) + 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7*e^8 + 7*(a^3*b^2*c - 4*a^4*c
^2)*d*x^6*e^7 + (a^3*b^3 - 4*a^4*b*c + 21*(a^3*b^2*c - 4*a^4*c^2)*d^2)*x^5*e^6 + 5*(7*(a^3*b^2*c - 4*a^4*c^2)*
d^3 + (a^3*b^3 - 4*a^4*b*c)*d)*x^4*e^5 + (a^4*b^2 - 4*a^5*c + 35*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 10*(a^3*b^3 - 4
*a^4*b*c)*d^2)*x^3*e^4 + (21*(a^3*b^2*c - 4*a^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(1/(e*x+d)**4/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1987 vs. \(2 (361) = 722\).
time = 3.37, size = 1987, normalized size = 4.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="giac")

[Out]

-1/4*((5*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 19*(d*e^(-1) + s
qrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^
2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^
2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*log(d*e^(-1) + x +
sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c
*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)
*e^(-4)/c))) + (5*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 19*(d*e
^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(-1) - sqrt(1/2)*sq
rt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*log(d*e^(
-1) + x - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + s
qrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4
)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*
a*c)*e^2)*e^(-4)/c))) + (5*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2
- 19*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(-1) + sqr
t(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sq
rt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*
log(d*e^(-1) + x + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(
b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e
^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt
(b^2 - 4*a*c)*e^2)*e^(-4)/c))) + (5*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b
^3*c*e^2 - 19*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(
-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) - sqrt(1/2)*sqrt(-(b
*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*
a^2*c^2)*log(d*e^(-1) + x - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2
)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e
^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))))/(a^3*b^2 - 4*a^4*c) + 1/2*(b^3*c*x^3*e^3 - 3*a*b*c^2*x^3*e^3 + 3*b^3*
c*d*x^2*e^2 - 9*a*b*c^2*d*x^2*e^2 + 3*b^3*c*d^2*x*e - 9*a*b*c^2*d^2*x*e + b^3*c*d^3 - 3*a*b*c^2*d^3 + b^4*x*e
- 4*a*b^2*c*x*e + 2*a^2*c^2*x*e + b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d)/((c*x^4*e^4 + 4*c*d*x^3*e^3 + 6*c*d^2*x^2
*e^2 + 4*c*d^3*x*e + c*d^4 + b*x^2*e^2 + 2*b*d*x*e + b*d^2 + a)*(a^3*b^2*e - 4*a^4*c*e)) + 1/3*(6*b*x^2*e^2 +
12*b*d*x*e + 6*b*d^2 - a)*e^(-1)/((x*e + d)^3*a^3)

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Mupad [B]
time = 8.72, size = 2500, normalized size = 6.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x)

[Out]

atan(((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 +
116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^
13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2 + 40
96*a^13*c^6*e^2 - 24*a^8*b^10*c*e^2 + 240*a^9*b^8*c^2*e^2 - 1280*a^10*b^6*c^3*e^2 + 3840*a^11*b^4*c^4*e^2 - 61
44*a^12*b^2*c^5*e^2)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c
^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c -
b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2)
)/(32*(a^7*b^12*e^2 + 4096*a^13*c^6*e^2 - 24*a^8*b^10*c*e^2 + 240*a^9*b^8*c^2*e^2 - 1280*a^10*b^6*c^3*e^2 + 38
40*a^11*b^4*c^4*e^2 - 6144*a^12*b^2*c^5*e^2)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7
*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6
+ 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*
(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2 + 4096*a^13*c^6*e^2 - 24*a^8*b^10*c*e^2 + 240*a^9*b^8*c^2*e^2 - 12
80*a^10*b^6*c^3*e^2 + 3840*a^11*b^4*c^4*e^2 - 6144*a^12*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^21*b*c^8*e^14 + 256
*a^15*b^13*c^2*e^14 - 6144*a^16*b^11*c^3*e^14 + 61440*a^17*b^9*c^4*e^14 - 327680*a^18*b^7*c^5*e^14 + 983040*a^
19*b^5*c^6*e^14 - 1572864*a^20*b^3*c^7*e^14) + 1048576*a^21*b*c^8*d*e^13 + 256*a^15*b^13*c^2*d*e^13 - 6144*a^1
6*b^11*c^3*d*e^13 + 61440*a^17*b^9*c^4*d*e^13 - 327680*a^18*b^7*c^5*d*e^13 + 983040*a^19*b^5*c^6*d*e^13 - 1572
864*a^20*b^3*c^7*d*e^13) - 917504*a^19*c^9*e^12 + 320*a^12*b^14*c^2*e^12 - 7936*a^13*b^12*c^3*e^12 + 82816*a^1
4*b^10*c^4*e^12 - 468480*a^15*b^8*c^5*e^12 + 1536000*a^16*b^6*c^6*e^12 - 2867200*a^17*b^4*c^7*e^12 + 2719744*a
^18*b^2*c^8*e^12) - x*(401408*a^16*c^10*e^12 - 400*a^9*b^14*c^3*e^12 + 9440*a^10*b^12*c^4*e^12 - 92816*a^11*b^
10*c^5*e^12 + 488096*a^12*b^8*c^6*e^12 - 1458688*a^13*b^6*c^7*e^12 + 2401280*a^14*b^4*c^8*e^12 - 1871872*a^15*
b^2*c^9*e^12) - 401408*a^16*c^10*d*e^11 + 400*a^9*b^14*c^3*d*e^11 - 9440*a^10*b^12*c^4*d*e^11 + 92816*a^11*b^1
0*c^5*d*e^11 - 488096*a^12*b^8*c^6*d*e^11 + 1458688*a^13*b^6*c^7*d*e^11 - 2401280*a^14*b^4*c^8*d*e^11 + 187187
2*a^15*b^2*c^9*d*e^11)*1i + (-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2
- 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^
2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/
(32*(a^7*b^12*e^2 + 4096*a^13*c^6*e^2 - 24*a^8*b^10*c*e^2 + 240*a^9*b^8*c^2*e^2 - 1280*a^10*b^6*c^3*e^2 + 3840
*a^11*b^4*c^4*e^2 - 6144*a^12*b^2*c^5*e^2)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b
*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 +
49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-
(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2 + 4096*a^13*c^6*e^2 - 24*a^8*b^10*c*e^2 + 240*a^9*b^8*c^2*e^2 - 1280
*a^10*b^6*c^3*e^2 + 3840*a^11*b^4*c^4*e^2 - 6144*a^12*b^2*c^5*e^2)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2
)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5
+ 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9
)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2 + 4096*a^13*c^6*e^2 - 24*a^8*b^10*c*e^2 + 24
0*a^9*b^8*c^2*e^2 - 1280*a^10*b^6*c^3*e^2 + 3840*a^11*b^4*c^4*e^2 - 6144*a^12*b^2*c^5*e^2)))^(1/2)*(x*(1048576
*a^21*b*c^8*e^14 + 256*a^15*b^13*c^2*e^14 - 6144*a^16*b^11*c^3*e^14 + 61440*a^17*b^9*c^4*e^14 - 327680*a^18*b^
7*c^5*e^14 + 983040*a^19*b^5*c^6*e^14 - 1572864*a^20*b^3*c^7*e^14) + 1048576*a^21*b*c^8*d*e^13 + 256*a^15*b^13
*c^2*d*e^13 - 6144*a^16*b^11*c^3*d*e^13 + 61440*a^17*b^9*c^4*d*e^13 - 327680*a^18*b^7*c^5*d*e^13 + 983040*a^19
*b^5*c^6*d*e^13 - 1572864*a^20*b^3*c^7*d*e^13) + 917504*a^19*c^9*e^12 - 320*a^12*b^14*c^2*e^12 + 7936*a^13*b^1
2*c^3*e^12 - 82816*a^14*b^10*c^4*e^12 + 468480*a^15*b^8*c^5*e^12 - 1536000*a^16*b^6*c^6*e^12 + 2867200*a^17*b^
4*c^7*e^12 - 2719744*a^18*b^2*c^8*e^12) - x*(401408*a^16*c^10*e^12 - 400*a^9*b^14*c^3*e^12 + 9440*a^10*b^12*c^
4*e^12 - 92816*a^11*b^10*c^5*e^12 + 488096*a^12*b^8*c^6*e^12 - 1458688*a^13*b^6*c^7*e^12 + 2401280*a^14*b^4*c^
8*e^12 - 1871872*a^15*b^2*c^9*e^12) - 401408*a^16*c^10*d*e^11 + 400*a^9*b^14*c^3*d*e^11 - 9440*a^10*b^12*c^4*d
*e^11 + 92816*a^11*b^10*c^5*d*e^11 - 488096*a^12*b^8*c^6*d*e^11 + 1458688*a^13*b^6*c^7*d*e^11 - 2401280*a^14*b
^4*c^8*d*e^11 + 1871872*a^15*b^2*c^9*d*e^11)*1i)/((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c
^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 11...

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