3.7.35 \(\int \frac {1}{(d+e x) (a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [635]

Optimal. Leaf size=255 \[ \frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2} e}+\frac {\log (d+e x)}{a^3 e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e} \]

[Out]

1/4*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2+1/4*(2*b^4-15*a*b^2*c+16*a^2*c^2+
2*b*c*(-7*a*c+b^2)*(e*x+d)^2)/a^2/(-4*a*c+b^2)^2/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/2*b*(30*a^2*c^2-10*a*b^2*c+b^
4)*arctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(5/2)/e+ln(e*x+d)/a^3/e-1/4*ln(a+b*(e*x+d)^2
+c*(e*x+d)^4)/a^3/e

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Rubi [A]
time = 0.34, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1156, 1128, 754, 836, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e}+\frac {\log (d+e x)}{a^3 e}+\frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 e \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*b^4 - 15*a*b^
2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^
4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a*c
)^(5/2)*e) + Log[d + e*x]/(a^3*e) - Log[a + b*(d + e*x)^2 + c*(d + e*x)^4]/(4*a^3*e)

Rule 212

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

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

Rule 754

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 836

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

Rule 1128

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

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]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^3} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \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 {-2 \left (b^2-4 a c\right )-3 b c x}{x \left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{4 a \left (b^2-4 a c\right ) e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \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 {2 \left (b^2-4 a c\right )^2+2 b c \left (b^2-7 a c\right ) x}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right )^2 e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \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 \left (\frac {2 \left (-b^2+4 a c\right )^2}{a x}+\frac {2 \left (-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x\right )}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right )^2 e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^3 e}+\frac {\text {Subst}\left (\int \frac {-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^3 e}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^3 e}-\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^3 \left (b^2-4 a c\right )^2 e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^3 e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2} e}+\frac {\log (d+e x)}{a^3 e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e}\\ \end {align*}

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Mathematica [A]
time = 2.64, size = 391, normalized size = 1.53 \begin {gather*} \frac {\frac {a^2 \left (-b^2+2 a c-b c (d+e x)^2\right )}{\left (-b^2+4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {a \left (2 b^4-15 a b^2 c+16 a^2 c^2+2 b^3 c (d+e x)^2-14 a b c^2 (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+4 \log (d+e x)-\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2+b^4 \sqrt {b^2-4 a c}-8 a b^2 c \sqrt {b^2-4 a c}+16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2-b^4 \sqrt {b^2-4 a c}+8 a b^2 c \sqrt {b^2-4 a c}-16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{5/2}}}{4 a^3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

((a^2*(-b^2 + 2*a*c - b*c*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (a*(2*b^4 - 1
5*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*(d + e*x)^2 - 14*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + (d + e*x)^2*(b +
c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqr
t[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/
2) + ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt
[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2))/(4*a^3*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.26, size = 966, normalized size = 3.79

method	result	size
default	\(\frac {\ln \left (e x +d \right )}{a^{3} e}-\frac {\frac {\frac {c^{2} e^{5} \left (7 a c -b^{2}\right ) a b \,x^{6}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {3 \left (7 a c -b^{2}\right ) a b \,c^{2} d \,e^{4} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {e^{3} a c \left (-210 a b \,c^{2} d^{2}+30 b^{3} c \,d^{2}+16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {c d \,e^{2} a \left (-70 a b \,c^{2} d^{2}+10 b^{3} c \,d^{2}+16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {e a \left (105 a b \,c^{3} d^{4}-15 b^{3} c^{2} d^{4}-48 a^{2} c^{3} d^{2}+87 a \,b^{2} c^{2} d^{2}-12 b^{4} c \,d^{2}+a^{2} b \,c^{2}+6 a \,b^{3} c -b^{5}\right ) x^{2}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {d a \left (21 a b \,c^{3} d^{4}-3 b^{3} c^{2} d^{4}-16 a^{2} c^{3} d^{2}+29 a \,b^{2} c^{2} d^{2}-4 b^{4} c \,d^{2}+a^{2} b \,c^{2}+6 a \,b^{3} c -b^{5}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {a \left (-14 a b \,c^{3} d^{6}+2 b^{3} c^{2} d^{6}+16 a^{2} c^{3} d^{4}-29 a \,b^{2} c^{2} d^{4}+4 b^{4} c \,d^{4}-2 a^{2} b \,c^{2} d^{2}-12 a \,b^{3} c \,d^{2}+2 b^{5} d^{2}+24 a^{3} c^{2}-21 a^{2} b^{2} c +3 b^{4} a \right )}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\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 (e^{3} c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \textit {\_R}^{3}+3 d \,e^{2} c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \textit {\_R}^{2}+e \left (48 a^{2} c^{3} d^{2}-24 a \,b^{2} c^{2} d^{2}+3 b^{4} c \,d^{2}+23 a^{2} b \,c^{2}-9 a \,b^{3} c +b^{5}\right ) \textit {\_R} +16 a^{2} c^{3} d^{3}-8 a \,b^{2} c^{2} d^{3}+b^{4} c \,d^{3}+23 a^{2} b \,c^{2} d -9 a \,b^{3} c d +b^{5} d \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}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}}{a^{3}}\)	\(966\)
risch	\(\text {Expression too large to display}\)	\(1591\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x,method=_RETURNVERBOSE)

[Out]

ln(e*x+d)/a^3/e-1/a^3*((1/2*c^2*e^5*(7*a*c-b^2)*a*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+3*(7*a*c-b^2)*a*b*c^2*d*e^4
/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5-1/4*e^3*a*c*(-210*a*b*c^2*d^2+30*b^3*c*d^2+16*a^2*c^2-29*a*b^2*c+4*b^4)/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^4-c*d*e^2*a*(-70*a*b*c^2*d^2+10*b^3*c*d^2+16*a^2*c^2-29*a*b^2*c+4*b^4)/(16*a^2*c^2-8*a*
b^2*c+b^4)*x^3+1/2*e*a*(105*a*b*c^3*d^4-15*b^3*c^2*d^4-48*a^2*c^3*d^2+87*a*b^2*c^2*d^2-12*b^4*c*d^2+a^2*b*c^2+
6*a*b^3*c-b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+d*a*(21*a*b*c^3*d^4-3*b^3*c^2*d^4-16*a^2*c^3*d^2+29*a*b^2*c^2*d^
2-4*b^4*c*d^2+a^2*b*c^2+6*a*b^3*c-b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/4/e*a*(-14*a*b*c^3*d^6+2*b^3*c^2*d^6+16*
a^2*c^3*d^4-29*a*b^2*c^2*d^4+4*b^4*c*d^4-2*a^2*b*c^2*d^2-12*a*b^3*c*d^2+2*b^5*d^2+24*a^3*c^2-21*a^2*b^2*c+3*a*
b^4)/(16*a^2*c^2-8*a*b^2*c+b^4))/(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+1/2/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((e^3*c*(16*a^2*c^2-8*a*b^2*c+b^4)*_R^3+3*d*e^2*c*(16*a^2*c^2
-8*a*b^2*c+b^4)*_R^2+e*(48*a^2*c^3*d^2-24*a*b^2*c^2*d^2+3*b^4*c*d^2+23*a^2*b*c^2-9*a*b^3*c+b^5)*_R+16*a^2*c^3*
d^3-8*a*b^2*c^2*d^3+b^4*c*d^3+23*a^2*b*c^2*d-9*a*b^3*c*d+b^5*d)/(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)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")

[Out]

1/4*(2*(b^3*c^2 - 7*a*b*c^3)*d^6 + 12*(b^3*c^2*e^5 - 7*a*b*c^3*e^5)*d*x^5 + 2*(b^3*c^2*e^6 - 7*a*b*c^3*e^6)*x^
6 + 3*a*b^4 - 21*a^2*b^2*c + 24*a^3*c^2 + (4*b^4*c - 29*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (4*b^4*c*e^4 - 29*a*b^2*
c^2*e^4 + 16*a^2*c^3*e^4 + 30*(b^3*c^2*e^4 - 7*a*b*c^3*e^4)*d^2)*x^4 + 4*(10*(b^3*c^2*e^3 - 7*a*b*c^3*e^3)*d^3
+ (4*b^4*c*e^3 - 29*a*b^2*c^2*e^3 + 16*a^2*c^3*e^3)*d)*x^3 + 2*(b^5 - 6*a*b^3*c - a^2*b*c^2)*d^2 + 2*(b^5*e^2
- 6*a*b^3*c*e^2 - a^2*b*c^2*e^2 + 15*(b^3*c^2*e^2 - 7*a*b*c^3*e^2)*d^4 + 3*(4*b^4*c*e^2 - 29*a*b^2*c^2*e^2 +
16*a^2*c^3*e^2)*d^2)*x^2 + 4*(3*(b^3*c^2*e - 7*a*b*c^3*e)*d^5 + (4*b^4*c*e - 29*a*b^2*c^2*e + 16*a^2*c^3*e)*d^
3 + (b^5*e - 6*a*b^3*c*e - a^2*b*c^2*e)*d)*x)/((a^2*b^4*c^2*e - 8*a^3*b^2*c^3*e + 16*a^4*c^4*e)*d^8 + 8*(a^2*b
^4*c^2*e^8 - 8*a^3*b^2*c^3*e^8 + 16*a^4*c^4*e^8)*d*x^7 + (a^2*b^4*c^2*e^9 - 8*a^3*b^2*c^3*e^9 + 16*a^4*c^4*e^9
)*x^8 + a^4*b^4*e - 8*a^5*b^2*c*e + 16*a^6*c^2*e + 2*(a^2*b^5*c*e - 8*a^3*b^3*c^2*e + 16*a^4*b*c^3*e)*d^6 + 2*
(a^2*b^5*c*e^7 - 8*a^3*b^3*c^2*e^7 + 16*a^4*b*c^3*e^7 + 14*(a^2*b^4*c^2*e^7 - 8*a^3*b^2*c^3*e^7 + 16*a^4*c^4*e
^7)*d^2)*x^6 + 4*(14*(a^2*b^4*c^2*e^6 - 8*a^3*b^2*c^3*e^6 + 16*a^4*c^4*e^6)*d^3 + 3*(a^2*b^5*c*e^6 - 8*a^3*b^3
*c^2*e^6 + 16*a^4*b*c^3*e^6)*d)*x^5 + (a^2*b^6*e - 6*a^3*b^4*c*e + 32*a^5*c^3*e)*d^4 + (a^2*b^6*e^5 - 6*a^3*b^
4*c*e^5 + 32*a^5*c^3*e^5 + 70*(a^2*b^4*c^2*e^5 - 8*a^3*b^2*c^3*e^5 + 16*a^4*c^4*e^5)*d^4 + 30*(a^2*b^5*c*e^5 -
8*a^3*b^3*c^2*e^5 + 16*a^4*b*c^3*e^5)*d^2)*x^4 + 4*(14*(a^2*b^4*c^2*e^4 - 8*a^3*b^2*c^3*e^4 + 16*a^4*c^4*e^4)
*d^5 + 10*(a^2*b^5*c*e^4 - 8*a^3*b^3*c^2*e^4 + 16*a^4*b*c^3*e^4)*d^3 + (a^2*b^6*e^4 - 6*a^3*b^4*c*e^4 + 32*a^5
*c^3*e^4)*d)*x^3 + 2*(a^3*b^5*e - 8*a^4*b^3*c*e + 16*a^5*b*c^2*e)*d^2 + 2*(a^3*b^5*e^3 - 8*a^4*b^3*c*e^3 + 16*
a^5*b*c^2*e^3 + 14*(a^2*b^4*c^2*e^3 - 8*a^3*b^2*c^3*e^3 + 16*a^4*c^4*e^3)*d^6 + 15*(a^2*b^5*c*e^3 - 8*a^3*b^3*
c^2*e^3 + 16*a^4*b*c^3*e^3)*d^4 + 3*(a^2*b^6*e^3 - 6*a^3*b^4*c*e^3 + 32*a^5*c^3*e^3)*d^2)*x^2 + 4*(2*(a^2*b^4*
c^2*e^2 - 8*a^3*b^2*c^3*e^2 + 16*a^4*c^4*e^2)*d^7 + 3*(a^2*b^5*c*e^2 - 8*a^3*b^3*c^2*e^2 + 16*a^4*b*c^3*e^2)*d
^5 + (a^2*b^6*e^2 - 6*a^3*b^4*c*e^2 + 32*a^5*c^3*e^2)*d^3 + (a^3*b^5*e^2 - 8*a^4*b^3*c*e^2 + 16*a^5*b*c^2*e^2)
*d)*x) - integrate(((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + 3*(b^4*c*e^2 - 8*a*b^2*c^2*e^2 + 16*a^2*c^3*e^2)*
d*x^2 + (b^4*c*e^3 - 8*a*b^2*c^2*e^3 + 16*a^2*c^3*e^3)*x^3 + (b^5 - 9*a*b^3*c + 23*a^2*b*c^2)*d + (b^5*e - 9*a
*b^3*c*e + 23*a^2*b*c^2*e + 3*(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e)*d^2)*x)/(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^4 - 8*a^4*b^2*c + 16*a^5*c^2
) + e^(-1)*log(x*e + d)/a^3

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4859 vs. \(2 (248) = 496\).
time = 1.01, size = 9844, normalized size = 38.60 \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)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas")

[Out]

[1/4*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*
x^6*e^6 + 12*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d*x^5*e^5 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*
c^4)*d^6 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4 + 30*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^
3*b*c^4)*d^2)*x^4*e^4 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^4 + 4*(10*(a*b^5*c^2 - 1
1*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^3 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d)*x^3*e^3 + 2
*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3 + 15*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^4 + 3
*(4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^2)*x^2*e^2 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b
^3*c^2 + 4*a^4*b*c^3)*d^2 + 4*(3*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^5 + (4*a*b^6*c - 45*a^2*b^4*c^2
+ 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^3 + (a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3)*d)*x*e + ((b^5*c^
2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8*e^8 + 8*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d*x^7*e^7 + (b^5*c^2 - 10
*a*b^3*c^3 + 30*a^2*b*c^4)*d^8 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3 + 14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^
2*b*c^4)*d^2)*x^6*e^6 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^6
+ 4*(14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^3 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d)*x^5*e^5 + (
b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3 + 70*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^4 + 30*(b^6*c -
10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^2)*x^4*e^4 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^4 + 4*(14*(
b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^5 + 10*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^3 + (b^7 - 8*a*b^5*c
+ 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d)*x^3*e^3 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2 + 14*(b^5*c^2 - 10*a*b
^3*c^3 + 30*a^2*b*c^4)*d^6 + 15*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^4 + 3*(b^7 - 8*a*b^5*c + 10*a^2*b^3*
c^2 + 60*a^3*b*c^3)*d^2)*x^2*e^2 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*d^2 + 4*(2*(b^5*c^2 - 10*a*b^3*c^
3 + 30*a^2*b*c^4)*d^7 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^5 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60
*a^3*b*c^3)*d^3 + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*d)*x*e)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4*e^4 + 8*c^2
*d*x^3*e^3 + 2*c^2*d^4 + 2*b*c*d^2 + 2*(6*c^2*d^2 + b*c)*x^2*e^2 + 4*(2*c^2*d^3 + b*c*d)*x*e + b^2 - 2*a*c + (
2*c*x^2*e^2 + 4*c*d*x*e + 2*c*d^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c*d^2 + b)*x
^2*e^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a)) - ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8*e^8
+ 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*x^7*e^7 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c
^4 - 64*a^3*c^5)*d^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2
*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*x^6*e^6 + 2*(b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^
5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*x^5*e^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*
c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b^7*c
- 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*x^4*e^4 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^
2*c^3 - 128*a^4*c^4)*d^4 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a
*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*
c^4)*d)*x^3*e^3 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^
2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b^8 - 10*a*b^
6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2)*x^2*e^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 -
64*a^4*b*c^3)*d^2 + 4*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2
+ 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3
+ (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d)*x*e)*log(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c
*d^2 + b)*x^2*e^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a) + 4*((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*
c^5)*x^8*e^8 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*x^7*e^7 + (b^6*c^2 - 12*a*b^4*c^3 +
48*a^2*b^2*c^4 - 64*a^3*c^5)*d^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*
c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*x^6*e^6 +
2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*x^5*e^5 + (b^8 - 10*a*b^6*c +
24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^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)/(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. 1012 vs. \(2 (248) = 496\).
time = 4.04, size = 1012, normalized size = 3.97 \begin {gather*} -\frac {{\left (a^{3} b^{7} c e^{3} - 14 \, a^{4} b^{5} c^{2} e^{3} + 70 \, a^{5} b^{3} c^{3} e^{3} - 120 \, a^{6} b c^{4} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{3} b^{7} c e^{3} - 14 \, a^{4} b^{5} c^{2} e^{3} + 70 \, a^{5} b^{3} c^{3} e^{3} - 120 \, a^{6} b c^{4} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{6} b^{8} c e^{4} - 16 \, a^{7} b^{6} c^{2} e^{4} + 96 \, a^{8} b^{4} c^{3} e^{4} - 256 \, a^{9} b^{2} c^{4} e^{4} + 256 \, a^{10} c^{5} e^{4}\right )}} - \frac {e^{\left (-1\right )} \log \left ({\left | 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 \right |}\right )}{4 \, a^{3}} + \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{3}} + \frac {{\left (2 \, a b^{3} c^{2} d^{6} - 14 \, a^{2} b c^{3} d^{6} + 4 \, a b^{4} c d^{4} - 29 \, a^{2} b^{2} c^{2} d^{4} + 16 \, a^{3} c^{3} d^{4} + 2 \, a b^{5} d^{2} - 12 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b c^{2} d^{2} + 2 \, {\left (a b^{3} c^{2} e^{6} - 7 \, a^{2} b c^{3} e^{6}\right )} x^{6} + 3 \, a^{2} b^{4} - 21 \, a^{3} b^{2} c + 24 \, a^{4} c^{2} + 12 \, {\left (a b^{3} c^{2} d e^{5} - 7 \, a^{2} b c^{3} d e^{5}\right )} x^{5} + {\left (30 \, a b^{3} c^{2} d^{2} e^{4} - 210 \, a^{2} b c^{3} d^{2} e^{4} + 4 \, a b^{4} c e^{4} - 29 \, a^{2} b^{2} c^{2} e^{4} + 16 \, a^{3} c^{3} e^{4}\right )} x^{4} + 4 \, {\left (10 \, a b^{3} c^{2} d^{3} e^{3} - 70 \, a^{2} b c^{3} d^{3} e^{3} + 4 \, a b^{4} c d e^{3} - 29 \, a^{2} b^{2} c^{2} d e^{3} + 16 \, a^{3} c^{3} d e^{3}\right )} x^{3} + 2 \, {\left (15 \, a b^{3} c^{2} d^{4} e^{2} - 105 \, a^{2} b c^{3} d^{4} e^{2} + 12 \, a b^{4} c d^{2} e^{2} - 87 \, a^{2} b^{2} c^{2} d^{2} e^{2} + 48 \, a^{3} c^{3} d^{2} e^{2} + a b^{5} e^{2} - 6 \, a^{2} b^{3} c e^{2} - a^{3} b c^{2} e^{2}\right )} x^{2} + 4 \, {\left (3 \, a b^{3} c^{2} d^{5} e - 21 \, a^{2} b c^{3} d^{5} e + 4 \, a b^{4} c d^{3} e - 29 \, a^{2} b^{2} c^{2} d^{3} e + 16 \, a^{3} c^{3} d^{3} e + a b^{5} d e - 6 \, a^{2} b^{3} c d e - a^{3} b c^{2} d e\right )} x\right )} e^{\left (-1\right )}}{4 \, {\left (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\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")

[Out]

-1/4*((a^3*b^7*c*e^3 - 14*a^4*b^5*c^2*e^3 + 70*a^5*b^3*c^3*e^3 - 120*a^6*b*c^4*e^3)*sqrt(b^2 - 4*a*c)*log(abs(
b*x^2*e^2 + 2*b*d*x*e + sqrt(b^2 - 4*a*c)*x^2*e^2 + 2*sqrt(b^2 - 4*a*c)*d*x*e + b*d^2 + sqrt(b^2 - 4*a*c)*d^2
+ 2*a)) - (a^3*b^7*c*e^3 - 14*a^4*b^5*c^2*e^3 + 70*a^5*b^3*c^3*e^3 - 120*a^6*b*c^4*e^3)*sqrt(b^2 - 4*a*c)*log(
abs(-b*x^2*e^2 - 2*b*d*x*e + sqrt(b^2 - 4*a*c)*x^2*e^2 + 2*sqrt(b^2 - 4*a*c)*d*x*e - b*d^2 + sqrt(b^2 - 4*a*c)
*d^2 - 2*a)))/(a^6*b^8*c*e^4 - 16*a^7*b^6*c^2*e^4 + 96*a^8*b^4*c^3*e^4 - 256*a^9*b^2*c^4*e^4 + 256*a^10*c^5*e^
4) - 1/4*e^(-1)*log(abs(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 + e^(-1)*log(abs(x*e + d))/a^3 + 1/4*(2*a*b^3*c^2*d^6 - 14*a^2*b*c^3*d^6 + 4*a*b^4*c*d^4
- 29*a^2*b^2*c^2*d^4 + 16*a^3*c^3*d^4 + 2*a*b^5*d^2 - 12*a^2*b^3*c*d^2 - 2*a^3*b*c^2*d^2 + 2*(a*b^3*c^2*e^6 -
7*a^2*b*c^3*e^6)*x^6 + 3*a^2*b^4 - 21*a^3*b^2*c + 24*a^4*c^2 + 12*(a*b^3*c^2*d*e^5 - 7*a^2*b*c^3*d*e^5)*x^5 +
(30*a*b^3*c^2*d^2*e^4 - 210*a^2*b*c^3*d^2*e^4 + 4*a*b^4*c*e^4 - 29*a^2*b^2*c^2*e^4 + 16*a^3*c^3*e^4)*x^4 + 4*
(10*a*b^3*c^2*d^3*e^3 - 70*a^2*b*c^3*d^3*e^3 + 4*a*b^4*c*d*e^3 - 29*a^2*b^2*c^2*d*e^3 + 16*a^3*c^3*d*e^3)*x^3
+ 2*(15*a*b^3*c^2*d^4*e^2 - 105*a^2*b*c^3*d^4*e^2 + 12*a*b^4*c*d^2*e^2 - 87*a^2*b^2*c^2*d^2*e^2 + 48*a^3*c^3*d
^2*e^2 + a*b^5*e^2 - 6*a^2*b^3*c*e^2 - a^3*b*c^2*e^2)*x^2 + 4*(3*a*b^3*c^2*d^5*e - 21*a^2*b*c^3*d^5*e + 4*a*b^
4*c*d^3*e - 29*a^2*b^2*c^2*d^3*e + 16*a^3*c^3*d^3*e + a*b^5*d*e - 6*a^2*b^3*c*d*e - a^3*b*c^2*d*e)*x)*e^(-1)/(
(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*(b^2
- 4*a*c)^2*a^3)

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Mupad [B]
time = 17.98, size = 2500, normalized size = 9.80 \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)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x)

[Out]

((x^2*(b^5*e + 48*a^2*c^3*d^2*e + 15*b^3*c^2*d^4*e - 6*a*b^3*c*e - a^2*b*c^2*e + 12*b^4*c*d^2*e - 105*a*b*c^3*
d^4*e - 87*a*b^2*c^2*d^2*e))/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x^4*(4*b^4*c*e^3 + 16*a^2*c^3*e^3 - 2
9*a*b^2*c^2*e^3 + 30*b^3*c^2*d^2*e^3 - 210*a*b*c^3*d^2*e^3))/(4*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x^3*(
16*a^2*c^3*d*e^2 + 10*b^3*c^2*d^3*e^2 + 4*b^4*c*d*e^2 - 29*a*b^2*c^2*d*e^2 - 70*a*b*c^3*d^3*e^2))/(a^2*b^4 + 1
6*a^4*c^2 - 8*a^3*b^2*c) + (3*x^5*(b^3*c^2*d*e^4 - 7*a*b*c^3*d*e^4))/(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c) + (x
^6*(b^3*c^2*e^5 - 7*a*b*c^3*e^5))/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x*(b^5*d + 4*b^4*c*d^3 + 16*a^2*
c^3*d^3 + 3*b^3*c^2*d^5 - 29*a*b^2*c^2*d^3 - 6*a*b^3*c*d - a^2*b*c^2*d - 21*a*b*c^3*d^5))/(a^2*b^4 + 16*a^4*c^
2 - 8*a^3*b^2*c) + (3*a*b^4 + 24*a^3*c^2 + 2*b^5*d^2 - 21*a^2*b^2*c + 4*b^4*c*d^4 + 16*a^2*c^3*d^4 + 2*b^3*c^2
*d^6 - 2*a^2*b*c^2*d^2 - 29*a*b^2*c^2*d^4 - 12*a*b^3*c*d^2 - 14*a*b*c^3*d^6)/(4*e*(a^2*b^4 + 16*a^4*c^2 - 8*a^
3*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) + log(d + e*x)/(a^3*e) - (log((((a^3*e*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^
6*e^2*(4*a*c - b^2)^5))^(1/2) + 1)*(((a^3*e*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*e^2*(4*a*c - b^2)^5
))^(1/2) + 1)*((2*b*c^2*e^16*(2*b^5 + 46*a^2*b*c^2 + b^4*c*d^2 + 10*a^2*c^3*d^2 - 18*a*b^3*c - 2*a*b^2*c^2*d^2
))/(a^2*(4*a*c - b^2)^2) + (b*c^2*e^16*(a^3*e*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*e^2*(4*a*c - b^2)
^5))^(1/2) + 1)*(a*b + 3*b^2*d^2 + 3*b^2*e^2*x^2 - 10*a*c*d^2 + 6*b^2*d*e*x - 10*a*c*e^2*x^2 - 20*a*c*d*e*x))/
a^3 + (2*b*c^3*e^18*x^2*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*(4*a*c - b^2)^2) + (4*b*c^3*d*e^17*x*(b^4 + 10*a^
2*c^2 - 2*a*b^2*c))/(a^2*(4*a*c - b^2)^2)))/(4*a^3*e) + (b*c^3*e^15*(7*a*c - b^2)*(4*b^5 + 71*a^2*b*c^2 + 6*b^
4*c*d^2 + 80*a^2*c^3*d^2 - 33*a*b^3*c - 47*a*b^2*c^2*d^2))/(a^4*(4*a*c - b^2)^4) - (b*c^4*e^17*x^2*(6*b^6 - 56
0*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*(4*a*c - b^2)^4) - (2*b*c^4*d*e^16*x*(6*b^6 - 560*a^3*c^3 + 40
9*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*(4*a*c - b^2)^4)))/(4*a^3*e) - (b^3*c^5*e^16*x^2*(7*a*c - b^2)^3)/(a^6*(4*a*
c - b^2)^6) + (b^2*c^4*e^14*(7*a*c - b^2)^2*(b^4 + 16*a^2*c^2 + b^3*c*d^2 - 8*a*b^2*c - 7*a*b*c^2*d^2))/(a^6*(
4*a*c - b^2)^6) - (2*b^3*c^5*d*e^15*x*(7*a*c - b^2)^3)/(a^6*(4*a*c - b^2)^6))*(((a^3*e*(-(b^2*(b^4 + 30*a^2*c^
2 - 10*a*b^2*c)^2)/(a^6*e^2*(4*a*c - b^2)^5))^(1/2) - 1)*(((a^3*e*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a
^6*e^2*(4*a*c - b^2)^5))^(1/2) - 1)*((2*b*c^2*e^16*(2*b^5 + 46*a^2*b*c^2 + b^4*c*d^2 + 10*a^2*c^3*d^2 - 18*a*b
^3*c - 2*a*b^2*c^2*d^2))/(a^2*(4*a*c - b^2)^2) - (b*c^2*e^16*(a^3*e*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/
(a^6*e^2*(4*a*c - b^2)^5))^(1/2) - 1)*(a*b + 3*b^2*d^2 + 3*b^2*e^2*x^2 - 10*a*c*d^2 + 6*b^2*d*e*x - 10*a*c*e^2
*x^2 - 20*a*c*d*e*x))/a^3 + (2*b*c^3*e^18*x^2*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*(4*a*c - b^2)^2) + (4*b*c^3
*d*e^17*x*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*(4*a*c - b^2)^2)))/(4*a^3*e) - (b*c^3*e^15*(7*a*c - b^2)*(4*b^5
+ 71*a^2*b*c^2 + 6*b^4*c*d^2 + 80*a^2*c^3*d^2 - 33*a*b^3*c - 47*a*b^2*c^2*d^2))/(a^4*(4*a*c - b^2)^4) + (b*c^
4*e^17*x^2*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*(4*a*c - b^2)^4) + (2*b*c^4*d*e^16*x*(6*
b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*(4*a*c - b^2)^4)))/(4*a^3*e) - (b^3*c^5*e^16*x^2*(7*a*
c - b^2)^3)/(a^6*(4*a*c - b^2)^6) + (b^2*c^4*e^14*(7*a*c - b^2)^2*(b^4 + 16*a^2*c^2 + b^3*c*d^2 - 8*a*b^2*c -
7*a*b*c^2*d^2))/(a^6*(4*a*c - b^2)^6) - (2*b^3*c^5*d*e^15*x*(7*a*c - b^2)^3)/(a^6*(4*a*c - b^2)^6)))*(2*b^10*e
- 2048*a^5*c^5*e + 320*a^2*b^6*c^2*e - 1280*a^3*b^4*c^3*e + 2560*a^4*b^2*c^4*e - 40*a*b^8*c*e))/(2*(4*a^3*b^1
0*e^2 - 4096*a^8*c^5*e^2 - 80*a^4*b^8*c*e^2 + 640*a^5*b^6*c^2*e^2 - 2560*a^6*b^4*c^3*e^2 + 5120*a^7*b^2*c^4*e^
2)) - (b*atan((x*((((((b*((2*(5120*a^10*b*c^9*d*e^17 + 2*a^4*b^13*c^3*d*e^17 - 36*a^5*b^11*c^4*d*e^17 + 276*a^
6*b^9*c^5*d*e^17 - 1216*a^7*b^7*c^6*d*e^17 + 3456*a^8*b^5*c^7*d*e^17 - 6144*a^9*b^3*c^8*d*e^17))/(a^6*b^12 + 4
096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5) - (
(2*b^10*e - 2048*a^5*c^5*e + 320*a^2*b^6*c^2*e - 1280*a^3*b^4*c^3*e + 2560*a^4*b^2*c^4*e - 40*a*b^8*c*e)*(1638
40*a^13*b*c^9*d*e^18 - 12*a^6*b^15*c^2*d*e^18 + 328*a^7*b^13*c^3*d*e^18 - 3840*a^8*b^11*c^4*d*e^18 + 24960*a^9
*b^9*c^5*d*e^18 - 97280*a^10*b^7*c^6*d*e^18 + 227328*a^11*b^5*c^7*d*e^18 - 294912*a^12*b^3*c^8*d*e^18))/((4*a^
3*b^10*e^2 - 4096*a^8*c^5*e^2 - 80*a^4*b^8*c*e^2 + 640*a^5*b^6*c^2*e^2 - 2560*a^6*b^4*c^3*e^2 + 5120*a^7*b^2*c
^4*e^2)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10...

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