3.7.32 \(\int \frac {(d+e x)^2}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [632]

Optimal. Leaf size=363 \[ -\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) (d+e x)^2\right )}{8 a \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (b^2+20 a c+\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}} e}+\frac {\sqrt {c} \left (b^2+20 a c-\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

-1/4*(e*x+d)*(b+2*c*(e*x+d)^2)/(-4*a*c+b^2)/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2+1/8*(e*x+d)*(b*(8*a*c+b^2)+c*(20*a
*c+b^2)*(e*x+d)^2)/a/(-4*a*c+b^2)^2/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/16*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a
*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^2+20*a*c+b*(-52*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^2/e*2^(1/2)/(b-(-
4*a*c+b^2)^(1/2))^(1/2)+1/16*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^2+20*a*c-
b*(-52*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^2/e*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.71, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1156, 1133, 1192, 1180, 211} \begin {gather*} \frac {\sqrt {c} \left (\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a e \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{8 \sqrt {2} a e \left (b^2-4 a c\right )^2 \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (c \left (20 a c+b^2\right ) (d+e x)^2+b \left (8 a c+b^2\right )\right )}{8 a e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((d + e*x)*(b + 2*c*(d + e*x)^2))/((b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + ((d + e*x)*(b
*(b^2 + 8*a*c) + c*(b^2 + 20*a*c)*(d + e*x)^2))/(8*a*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) +
(Sqrt[c]*(b^2 + 20*a*c + (b*(b^2 - 52*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqr
t[b^2 - 4*a*c]]])/(8*Sqrt[2]*a*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) + (Sqrt[c]*(b^2 + 20*a*c - (b*(b
^2 - 52*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a
*(b^2 - 4*a*c)^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 1133

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(d*x)^(m - 1)*(b + 2*c*x
^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Dist[d^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(d*x
)^(m - 2)*(b*(m - 1) + 2*c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntegerQ[2*p] && (IntegerQ[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 1192

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e}\\ &=-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {\text {Subst}\left (\int \frac {b-10 c x^2}{\left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{4 \left (b^2-4 a c\right ) e}\\ &=-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) (d+e x)^2\right )}{8 a \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-b \left (b^2-16 a c\right )-c \left (b^2+20 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{8 a \left (b^2-4 a c\right )^2 e}\\ &=-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) (d+e x)^2\right )}{8 a \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\left (c \left (b^2+20 a c-\frac {b \left (b^2-52 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 )}{16 a \left (b^2-4 a c\right )^2 e}+\frac {\left (c \left (b^2+20 a c+\frac {b \left (b^2-52 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 )}{16 a \left (b^2-4 a c\right )^2 e}\\ &=-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) (d+e x)^2\right )}{8 a \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (b^2+20 a c+\frac {b \left (b^2-52 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 )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}} e}+\frac {\sqrt {c} \left (b^2+20 a c-\frac {b \left (b^2-52 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 )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}

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Mathematica [A]
time = 3.21, size = 382, normalized size = 1.05 \begin {gather*} \frac {-\frac {4 \left (b (d+e x)+2 c (d+e x)^3\right )}{\left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 (d+e x) \left (b^3+8 a b c+b^2 c (d+e x)^2+20 a c^2 (d+e x)^2\right )}{a \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a 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 )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^3+52 a b c+b^2 \sqrt {b^2-4 a c}+20 a 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 )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(b*(d + e*x) + 2*c*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*(d + e*x)*(b^
3 + 8*a*b*c + b^2*c*(d + e*x)^2 + 20*a*c^2*(d + e*x)^2))/(a*(b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4
)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sq
rt[c]*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/
Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.25, size = 885, normalized size = 2.44

method	result	size
default	\(\frac {\frac {c^{2} e^{6} \left (20 a c +b^{2}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {7 c^{2} d \,e^{5} \left (20 a c +b^{2}\right ) x^{6}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (420 a \,c^{2} d^{2}+21 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) c \,e^{4} x^{5}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {5 c d \,e^{3} \left (140 a \,c^{2} d^{2}+7 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {e^{2} \left (700 a \,c^{3} d^{4}+35 b^{2} c^{2} d^{4}+280 a b \,c^{2} d^{2}+20 b^{3} c \,d^{2}+36 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d e \left (420 a \,c^{3} d^{4}+21 b^{2} c^{2} d^{4}+280 a b \,c^{2} d^{2}+20 b^{3} c \,d^{2}+108 a^{2} c^{2}+15 a \,b^{2} c +3 b^{4}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (140 a \,c^{3} d^{6}+7 b^{2} c^{2} d^{6}+140 a b \,c^{2} d^{4}+10 b^{3} c \,d^{4}+108 a^{2} c^{2} d^{2}+15 a \,b^{2} c \,d^{2}+3 b^{4} d^{2}+16 a^{2} b c -a \,b^{3}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d \left (20 a \,c^{3} d^{6}+b^{2} c^{2} d^{6}+28 a b \,c^{2} d^{4}+2 b^{3} c \,d^{4}+36 a^{2} c^{2} d^{2}+5 a \,b^{2} c \,d^{2}+b^{4} d^{2}+16 a^{2} b c -a \,b^{3}\right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\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 )^{2}}+\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 (c \,e^{2} \left (20 a c +b^{2}\right ) \textit {\_R}^{2}+2 c d e \left (20 a c +b^{2}\right ) \textit {\_R} +20 a \,c^{2} d^{2}+b^{2} c \,d^{2}-16 a b c +b^{3}\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}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a e}\)	\(885\)
risch	\(\frac {\frac {c^{2} e^{6} \left (20 a c +b^{2}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {7 c^{2} d \,e^{5} \left (20 a c +b^{2}\right ) x^{6}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (420 a \,c^{2} d^{2}+21 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) c \,e^{4} x^{5}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {5 c d \,e^{3} \left (140 a \,c^{2} d^{2}+7 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {e^{2} \left (700 a \,c^{3} d^{4}+35 b^{2} c^{2} d^{4}+280 a b \,c^{2} d^{2}+20 b^{3} c \,d^{2}+36 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d e \left (420 a \,c^{3} d^{4}+21 b^{2} c^{2} d^{4}+280 a b \,c^{2} d^{2}+20 b^{3} c \,d^{2}+108 a^{2} c^{2}+15 a \,b^{2} c +3 b^{4}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (140 a \,c^{3} d^{6}+7 b^{2} c^{2} d^{6}+140 a b \,c^{2} d^{4}+10 b^{3} c \,d^{4}+108 a^{2} c^{2} d^{2}+15 a \,b^{2} c \,d^{2}+3 b^{4} d^{2}+16 a^{2} b c -a \,b^{3}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d \left (20 a \,c^{3} d^{6}+b^{2} c^{2} d^{6}+28 a b \,c^{2} d^{4}+2 b^{3} c \,d^{4}+36 a^{2} c^{2} d^{2}+5 a \,b^{2} c \,d^{2}+b^{4} d^{2}+16 a^{2} b c -a \,b^{3}\right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\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 )^{2}}+\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 (\frac {c \,e^{2} \left (20 a c +b^{2}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 c d e \left (20 a c +b^{2}\right ) \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {-20 a \,c^{2} d^{2}-b^{2} c \,d^{2}+16 a b c -b^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +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}}{16 a e}\)	\(933\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/8*c^2*e^6*(20*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^7+7/8*c^2*d*e^5*(20*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^
4)/a*x^6+1/8*(420*a*c^2*d^2+21*b^2*c*d^2+28*a*b*c+2*b^3)*c*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^5+5/8*c*d*e^3*(1
40*a*c^2*d^2+7*b^2*c*d^2+28*a*b*c+2*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^4+1/8*e^2*(700*a*c^3*d^4+35*b^2*c^2*d^
4+280*a*b*c^2*d^2+20*b^3*c*d^2+36*a^2*c^2+5*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^3+1/8*d*e*(420*a*c^3*d
^4+21*b^2*c^2*d^4+280*a*b*c^2*d^2+20*b^3*c*d^2+108*a^2*c^2+15*a*b^2*c+3*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^2+
1/8*(140*a*c^3*d^6+7*b^2*c^2*d^6+140*a*b*c^2*d^4+10*b^3*c*d^4+108*a^2*c^2*d^2+15*a*b^2*c*d^2+3*b^4*d^2+16*a^2*
b*c-a*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x+1/8*d/e*(20*a*c^3*d^6+b^2*c^2*d^6+28*a*b*c^2*d^4+2*b^3*c*d^4+36*a^2*
c^2*d^2+5*a*b^2*c*d^2+b^4*d^2+16*a^2*b*c-a*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/a)/(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)^2+1/16/(16*a^2*c^2-8*a*b^2*c+b^4)/a/e*sum((c*e^2*(20*a
*c+b^2)*_R^2+2*c*d*e*(20*a*c+b^2)*_R+20*a*c^2*d^2+b^2*c*d^2-16*a*b*c+b^3)/(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((e*x+d)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")

[Out]

1/8*((b^2*c^2 + 20*a*c^3)*d^7 + 7*(b^2*c^2*e^6 + 20*a*c^3*e^6)*d*x^6 + (b^2*c^2*e^7 + 20*a*c^3*e^7)*x^7 + 2*(b
^3*c + 14*a*b*c^2)*d^5 + (2*b^3*c*e^5 + 28*a*b*c^2*e^5 + 21*(b^2*c^2*e^5 + 20*a*c^3*e^5)*d^2)*x^5 + 5*(7*(b^2*
c^2*e^4 + 20*a*c^3*e^4)*d^3 + 2*(b^3*c*e^4 + 14*a*b*c^2*e^4)*d)*x^4 + (b^4 + 5*a*b^2*c + 36*a^2*c^2)*d^3 + (35
*(b^2*c^2*e^3 + 20*a*c^3*e^3)*d^4 + b^4*e^3 + 5*a*b^2*c*e^3 + 36*a^2*c^2*e^3 + 20*(b^3*c*e^3 + 14*a*b*c^2*e^3)
*d^2)*x^3 + (21*(b^2*c^2*e^2 + 20*a*c^3*e^2)*d^5 + 20*(b^3*c*e^2 + 14*a*b*c^2*e^2)*d^3 + 3*(b^4*e^2 + 5*a*b^2*
c*e^2 + 36*a^2*c^2*e^2)*d)*x^2 - (a*b^3 - 16*a^2*b*c)*d + (7*(b^2*c^2*e + 20*a*c^3*e)*d^6 + 10*(b^3*c*e + 14*a
*b*c^2*e)*d^4 - a*b^3*e + 16*a^2*b*c*e + 3*(b^4*e + 5*a*b^2*c*e + 36*a^2*c^2*e)*d^2)*x)/((a*b^4*c^2*e - 8*a^2*
b^2*c^3*e + 16*a^3*c^4*e)*d^8 + 8*(a*b^4*c^2*e^8 - 8*a^2*b^2*c^3*e^8 + 16*a^3*c^4*e^8)*d*x^7 + (a*b^4*c^2*e^9
- 8*a^2*b^2*c^3*e^9 + 16*a^3*c^4*e^9)*x^8 + a^3*b^4*e - 8*a^4*b^2*c*e + 16*a^5*c^2*e + 2*(a*b^5*c*e - 8*a^2*b^
3*c^2*e + 16*a^3*b*c^3*e)*d^6 + 2*(a*b^5*c*e^7 - 8*a^2*b^3*c^2*e^7 + 16*a^3*b*c^3*e^7 + 14*(a*b^4*c^2*e^7 - 8*
a^2*b^2*c^3*e^7 + 16*a^3*c^4*e^7)*d^2)*x^6 + 4*(14*(a*b^4*c^2*e^6 - 8*a^2*b^2*c^3*e^6 + 16*a^3*c^4*e^6)*d^3 +
3*(a*b^5*c*e^6 - 8*a^2*b^3*c^2*e^6 + 16*a^3*b*c^3*e^6)*d)*x^5 + (a*b^6*e - 6*a^2*b^4*c*e + 32*a^4*c^3*e)*d^4 +
(a*b^6*e^5 - 6*a^2*b^4*c*e^5 + 32*a^4*c^3*e^5 + 70*(a*b^4*c^2*e^5 - 8*a^2*b^2*c^3*e^5 + 16*a^3*c^4*e^5)*d^4 +
30*(a*b^5*c*e^5 - 8*a^2*b^3*c^2*e^5 + 16*a^3*b*c^3*e^5)*d^2)*x^4 + 4*(14*(a*b^4*c^2*e^4 - 8*a^2*b^2*c^3*e^4 +
16*a^3*c^4*e^4)*d^5 + 10*(a*b^5*c*e^4 - 8*a^2*b^3*c^2*e^4 + 16*a^3*b*c^3*e^4)*d^3 + (a*b^6*e^4 - 6*a^2*b^4*c*
e^4 + 32*a^4*c^3*e^4)*d)*x^3 + 2*(a^2*b^5*e - 8*a^3*b^3*c*e + 16*a^4*b*c^2*e)*d^2 + 2*(a^2*b^5*e^3 - 8*a^3*b^3
*c*e^3 + 16*a^4*b*c^2*e^3 + 14*(a*b^4*c^2*e^3 - 8*a^2*b^2*c^3*e^3 + 16*a^3*c^4*e^3)*d^6 + 15*(a*b^5*c*e^3 - 8*
a^2*b^3*c^2*e^3 + 16*a^3*b*c^3*e^3)*d^4 + 3*(a*b^6*e^3 - 6*a^2*b^4*c*e^3 + 32*a^4*c^3*e^3)*d^2)*x^2 + 4*(2*(a*
b^4*c^2*e^2 - 8*a^2*b^2*c^3*e^2 + 16*a^3*c^4*e^2)*d^7 + 3*(a*b^5*c*e^2 - 8*a^2*b^3*c^2*e^2 + 16*a^3*b*c^3*e^2)
*d^5 + (a*b^6*e^2 - 6*a^2*b^4*c*e^2 + 32*a^4*c^3*e^2)*d^3 + (a^2*b^5*e^2 - 8*a^3*b^3*c*e^2 + 16*a^4*b*c^2*e^2)
*d)*x) + 1/8*integrate((b^3 - 16*a*b*c + (b^2*c + 20*a*c^2)*d^2 + 2*(b^2*c*e + 20*a*c^2*e)*d*x + (b^2*c*e^2 +
20*a*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*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7597 vs. \(2 (325) = 650\).
time = 0.64, size = 7597, normalized size = 20.93 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*(b^2*c^2 + 20*a*c^3)*x^7*e^7 + 14*(b^2*c^2 + 20*a*c^3)*d*x^6*e^6 + 2*(b^2*c^2 + 20*a*c^3)*d^7 + 2*(2*b
^3*c + 28*a*b*c^2 + 21*(b^2*c^2 + 20*a*c^3)*d^2)*x^5*e^5 + 4*(b^3*c + 14*a*b*c^2)*d^5 + 10*(7*(b^2*c^2 + 20*a*
c^3)*d^3 + 2*(b^3*c + 14*a*b*c^2)*d)*x^4*e^4 + 2*(35*(b^2*c^2 + 20*a*c^3)*d^4 + b^4 + 5*a*b^2*c + 36*a^2*c^2 +
20*(b^3*c + 14*a*b*c^2)*d^2)*x^3*e^3 + 2*(b^4 + 5*a*b^2*c + 36*a^2*c^2)*d^3 + 2*(21*(b^2*c^2 + 20*a*c^3)*d^5
+ 20*(b^3*c + 14*a*b*c^2)*d^3 + 3*(b^4 + 5*a*b^2*c + 36*a^2*c^2)*d)*x^2*e^2 + 2*(7*(b^2*c^2 + 20*a*c^3)*d^6 +
10*(b^3*c + 14*a*b*c^2)*d^4 - a*b^3 + 16*a^2*b*c + 3*(b^4 + 5*a*b^2*c + 36*a^2*c^2)*d^2)*x*e - sqrt(1/2)*((a*b
^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8*e^9 + 8*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d*x^7*e^8 + 2*(a*b^5
*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3 + 14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^2)*x^6*e^7 + 4*(14*(a*b^4*c^
2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^3 + 3*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d)*x^5*e^6 + (a*b^6 - 6*a^2*b
^4*c + 32*a^4*c^3 + 70*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^4 + 30*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c
^3)*d^2)*x^4*e^5 + 4*(14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^5 + 10*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b
*c^3)*d^3 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d)*x^3*e^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 14*(a*b^
4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^6 + 15*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^4 + 3*(a*b^6 - 6*a^2*b
^4*c + 32*a^4*c^3)*d^2)*x^2*e^3 + 4*(2*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^7 + 3*(a*b^5*c - 8*a^2*b^3*c
^2 + 16*a^3*b*c^3)*d^5 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d)*x*
e^2 + ((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c - 8*a^2*
b^3*c^2 + 16*a^3*b*c^3)*d^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2
)*d^2)*e)*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*
c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*
a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))*e^(-2)/(a^3*b^10 - 20*a^4
*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5))*log((35*b^6*c^2 - 1491*a*b^4*c^
3 + 15000*a^2*b^2*c^4 + 10000*a^3*c^5)*x*e + (35*b^6*c^2 - 1491*a*b^4*c^3 + 15000*a^2*b^2*c^4 + 10000*a^3*c^5)
*d + 1/2*sqrt(1/2)*((a^3*b^14 - 38*a^4*b^12*c + 480*a^5*b^10*c^2 - 2720*a^6*b^8*c^3 + 6400*a^7*b^6*c^4 + 1536*
a^8*b^4*c^5 - 32768*a^9*b^2*c^6 + 40960*a^10*c^7)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8
*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5))*e - (b^11 - 53*a*b^9*c + 940*a^2*
b^7*c^2 - 6832*a^3*b^5*c^3 + 21824*a^4*b^3*c^4 - 25600*a^5*b*c^5)*e)*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2
+ 1680*a^3*b*c^3 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8
*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 128
0*a^10*b^2*c^4 - 1024*a^11*c^5)))*e^(-2)/(a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a
^7*b^2*c^4 - 1024*a^8*c^5))) + sqrt(1/2)*((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8*e^9 + 8*(a*b^4*c^2 - 8*
a^2*b^2*c^3 + 16*a^3*c^4)*d*x^7*e^8 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3 + 14*(a*b^4*c^2 - 8*a^2*b^2*c^
3 + 16*a^3*c^4)*d^2)*x^6*e^7 + 4*(14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^3 + 3*(a*b^5*c - 8*a^2*b^3*c^2
+ 16*a^3*b*c^3)*d)*x^5*e^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3 + 70*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*
d^4 + 30*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^2)*x^4*e^5 + 4*(14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4
)*d^5 + 10*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^3 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d)*x^3*e^4 + 2*(a
^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^6 + 15*(a*b^5*c - 8*a^2*b^
3*c^2 + 16*a^3*b*c^3)*d^4 + 3*(a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^2)*x^2*e^3 + 4*(2*(a*b^4*c^2 - 8*a^2*b^2*c^
3 + 16*a^3*c^4)*d^7 + 3*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^5 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^3
+ (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d)*x*e^2 + ((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^8 + a^3*b^4 -
8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)
*d^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d^2)*e)*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b
*c^3 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b
^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^
4 - 1024*a^11*c^5)))*e^(-2)/(a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 -
1024*a^8*c^5))*log((35*b^6*c^2 - 1491*a*b^4*c^3 + 15000*a^2*b^2*c^4 + 10000*a^3*c^5)*x*e + (35*b^6*c^2 - 1491*
a*b^4*c^3 + 15000*a^2*b^2*c^4 + 10000*a^3*c^5)*...

________________________________________________________________________________________

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2295 vs. \(2 (325) = 650\).
time = 3.38, size = 2295, normalized size = 6.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(((d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^2*c*e^2 + 20*(d*e^(-1) + sq
rt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*c^2*e^2 - 2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 +
sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^2*c*d*e - 40*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e
^(-4)/c))*a*c^2*d*e + b^2*c*d^2 + 20*a*c^2*d^2 + b^3 - 16*a*b*c)*log(d*e^(-1) + x + sqrt(1/2)*sqrt(-(b*e^2 + s
qrt(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))) + ((d*e^(-1) -
sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^2*c*e^2 + 20*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^
2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*c^2*e^2 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e
^2)*e^(-4)/c))*b^2*c*d*e - 40*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*c^2*d*e
+ b^2*c*d^2 + 20*a*c^2*d^2 + b^3 - 16*a*b*c)*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))) + ((d*e^(-1) + sqrt(1/2)*sqrt(-(b
*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^2*c*e^2 + 20*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*
c)*e^2)*e^(-4)/c))^2*a*c^2*e^2 - 2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^2*
c*d*e - 40*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*c^2*d*e + b^2*c*d^2 + 20*a
*c^2*d^2 + b^3 - 16*a*b*c)*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) + sq
rt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))) + ((d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4
*a*c)*e^2)*e^(-4)/c))^2*b^2*c*e^2 + 20*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^
2*a*c^2*e^2 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^2*c*d*e - 40*(d*e^(-1
) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*c^2*d*e + b^2*c*d^2 + 20*a*c^2*d^2 + b^3 - 16
*a*b*c)*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*b^4 - 8*a^2*b^2*c + 16*a^3*c^2) + 1/8*(b^2*c^2*x^7*e^7 + 20*a*c^3*x
^7*e^7 + 7*b^2*c^2*d*x^6*e^6 + 140*a*c^3*d*x^6*e^6 + 21*b^2*c^2*d^2*x^5*e^5 + 420*a*c^3*d^2*x^5*e^5 + 35*b^2*c
^2*d^3*x^4*e^4 + 700*a*c^3*d^3*x^4*e^4 + 35*b^2*c^2*d^4*x^3*e^3 + 700*a*c^3*d^4*x^3*e^3 + 21*b^2*c^2*d^5*x^2*e
^2 + 420*a*c^3*d^5*x^2*e^2 + 7*b^2*c^2*d^6*x*e + 140*a*c^3*d^6*x*e + b^2*c^2*d^7 + 20*a*c^3*d^7 + 2*b^3*c*x^5*
e^5 + 28*a*b*c^2*x^5*e^5 + 10*b^3*c*d*x^4*e^4 + 140*a*b*c^2*d*x^4*e^4 + 20*b^3*c*d^2*x^3*e^3 + 280*a*b*c^2*d^2
*x^3*e^3 + 20*b^3*c*d^3*x^2*e^2 + 280*a*b*c^2*d^3*x^2*e^2 + 10*b^3*c*d^4*x*e + 140*a*b*c^2*d^4*x*e + 2*b^3*c*d
^5 + 28*a*b*c^2*d^5 + b^4*x^3*e^3 + 5*a*b^2*c*x^3*e^3 + 36*a^2*c^2*x^3*e^3 + 3*b^4*d*x^2*e^2 + 15*a*b^2*c*d*x^
2*e^2 + 108*a^2*c^2*d*x^2*e^2 + 3*b^4*d^2*x*e + 15*a*b^2*c*d^2*x*e + 108*a^2*c^2*d^2*x*e + b^4*d^3 + 5*a*b^2*c
*d^3 + 36*a^2*c^2*d^3 - a*b^3*x*e + 16*a^2*b*c*x*e - a*b^3*d + 16*a^2*b*c*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)^2*(a*b^4*e - 8*a^2*b^2*c*e + 16*a^3*c^
2*e))

________________________________________________________________________________________

Mupad [B]
time = 7.43, size = 2500, normalized size = 6.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^5*(2*b^3*c*e^4 + 420*a*c^3*d^2*e^4 + 21*b^2*c^2*d^2*e^4 + 28*a*b*c^2*e^4))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^
2*c)) + (x^2*(3*b^4*d*e + 21*b^2*c^2*d^5*e + 108*a^2*c^2*d*e + 420*a*c^3*d^5*e + 20*b^3*c*d^3*e + 280*a*b*c^2*
d^3*e + 15*a*b^2*c*d*e))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (7*x^6*(b^2*c^2*d*e^5 + 20*a*c^3*d*e^5))/(8*a*
(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^7*(20*a*c^3*e^6 + b^2*c^2*e^6))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x
*(3*b^4*d^2 - a*b^3 + 140*a*c^3*d^6 + 10*b^3*c*d^4 + 108*a^2*c^2*d^2 + 7*b^2*c^2*d^6 + 16*a^2*b*c + 15*a*b^2*c
*d^2 + 140*a*b*c^2*d^4))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^3*(b^4*e^2 + 36*a^2*c^2*e^2 + 700*a*c^3*d^4
*e^2 + 20*b^3*c*d^2*e^2 + 35*b^2*c^2*d^4*e^2 + 5*a*b^2*c*e^2 + 280*a*b*c^2*d^2*e^2))/(8*a*(b^4 + 16*a^2*c^2 -
8*a*b^2*c)) + (b^4*d^3 + 20*a*c^3*d^7 + 2*b^3*c*d^5 + 36*a^2*c^2*d^3 + b^2*c^2*d^7 - a*b^3*d + 16*a^2*b*c*d +
5*a*b^2*c*d^3 + 28*a*b*c^2*d^5)/(8*a*e*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (5*x^4*(140*a*c^3*d^3*e^3 + 7*b^2*c^2
*d^3*e^3 + 2*b^3*c*d*e^3 + 28*a*b*c^2*d*e^3))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(6*b^2*d^2*e^2 + 28*c
^2*d^6*e^2 + 2*a*b*e^2 + 12*a*c*d^2*e^2 + 30*b*c*d^4*e^2) + x^6*(28*c^2*d^2*e^6 + 2*b*c*e^6) + x*(4*b^2*d^3*e
+ 8*c^2*d^7*e + 8*a*c*d^3*e + 12*b*c*d^5*e + 4*a*b*d*e) + x^3*(4*b^2*d*e^3 + 56*c^2*d^5*e^3 + 8*a*c*d*e^3 + 40
*b*c*d^3*e^3) + x^5*(56*c^2*d^3*e^5 + 12*b*c*d*e^5) + x^4*(b^2*e^4 + 70*c^2*d^4*e^4 + 2*a*c*e^4 + 30*b*c*d^2*e
^4) + a^2 + b^2*d^4 + c^2*d^8 + c^2*e^8*x^8 + 2*a*b*d^2 + 2*a*c*d^4 + 2*b*c*d^6 + 8*c^2*d*e^7*x^7) + atan(((((
256*a*b^13*c^2*e^12 + 4194304*a^7*b*c^8*e^12 - 9216*a^2*b^11*c^3*e^12 + 122880*a^3*b^9*c^4*e^12 - 819200*a^4*b
^7*c^5*e^12 + 2949120*a^5*b^5*c^6*e^12 - 5505024*a^6*b^3*c^7*e^12)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10
*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) + ((67108864*a^9*b*c^9*d*e^13
- 4096*a^2*b^15*c^2*d*e^13 + 114688*a^3*b^13*c^3*d*e^13 - 1376256*a^4*b^11*c^4*d*e^13 + 9175040*a^5*b^9*c^5*d*
e^13 - 36700160*a^6*b^7*c^6*d*e^13 + 88080384*a^7*b^5*c^7*d*e^13 - 117440512*a^8*b^3*c^8*d*e^13)/(512*(a^2*b^1
2 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5))
+ (x*(262144*a^7*b*c^7*e^14 - 256*a^2*b^11*c^2*e^14 + 5120*a^3*b^9*c^3*e^14 - 40960*a^4*b^7*c^4*e^14 + 163840*
a^5*b^5*c^5*e^14 - 327680*a^6*b^3*c^6*e^14))/(32*(a^2*b^8 + 256*a^6*c^4 - 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256*
a^5*b^2*c^3)))*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^1
1*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*
c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20*e^2 + 1048576*a^13*c^10*e^2 - 40*a^4*b^18*c*e^2 + 720*a^5*b^16*c^2
*e^2 - 7680*a^6*b^14*c^3*e^2 + 53760*a^7*b^12*c^4*e^2 - 258048*a^8*b^10*c^5*e^2 + 860160*a^9*b^8*c^6*e^2 - 196
6080*a^10*b^6*c^7*e^2 + 2949120*a^11*b^4*c^8*e^2 - 2621440*a^12*b^2*c^9*e^2)))^(1/2))*(-(b^17 + b^2*(-(4*a*c -
b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b
^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*
b^20*e^2 + 1048576*a^13*c^10*e^2 - 40*a^4*b^18*c*e^2 + 720*a^5*b^16*c^2*e^2 - 7680*a^6*b^14*c^3*e^2 + 53760*a^
7*b^12*c^4*e^2 - 258048*a^8*b^10*c^5*e^2 + 860160*a^9*b^8*c^6*e^2 - 1966080*a^10*b^6*c^7*e^2 + 2949120*a^11*b^
4*c^8*e^2 - 2621440*a^12*b^2*c^9*e^2)))^(1/2) + (204800*a^5*c^8*d*e^11 - 16*b^10*c^3*d*e^11 + 672*a*b^8*c^4*d*
e^11 - 28160*a^2*b^6*c^5*d*e^11 + 209920*a^3*b^4*c^6*d*e^11 - 479232*a^4*b^2*c^7*d*e^11)/(512*(a^2*b^12 + 4096
*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) + (x*(80
0*a^3*c^6*e^12 - b^6*c^3*e^12 + 34*a*b^4*c^4*e^12 - 1472*a^2*b^2*c^5*e^12))/(32*(a^2*b^8 + 256*a^6*c^4 - 16*a^
3*b^6*c + 96*a^4*b^4*c^2 - 256*a^5*b^2*c^3)))*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 11
40*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^
7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20*e^2 + 1048576*a^13*c^10*e^2 - 40*a^
4*b^18*c*e^2 + 720*a^5*b^16*c^2*e^2 - 7680*a^6*b^14*c^3*e^2 + 53760*a^7*b^12*c^4*e^2 - 258048*a^8*b^10*c^5*e^2
+ 860160*a^9*b^8*c^6*e^2 - 1966080*a^10*b^6*c^7*e^2 + 2949120*a^11*b^4*c^8*e^2 - 2621440*a^12*b^2*c^9*e^2)))^
(1/2)*1i + ((204800*a^5*c^8*d*e^11 - 16*b^10*c^3*d*e^11 + 672*a*b^8*c^4*d*e^11 - 28160*a^2*b^6*c^5*d*e^11 + 20
9920*a^3*b^4*c^6*d*e^11 - 479232*a^4*b^2*c^7*d*e^11)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b
^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) - ((256*a*b^13*c^2*e^12 + 4194304*a^7*b*c^8*
e^12 - 9216*a^2*b^11*c^3*e^12 + 122880*a^3*b^9*c^4*e^12 - 819200*a^4*b^7*c^5*e^12 + 2949120*a^5*b^5*c^6*e^12 -
5505024*a^6*b^3*c^7*e^12)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3
+ 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) - ((671...

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