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

Optimal. Leaf size=162 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 e \left (b^2-4 a c\right )^{3/2}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\frac{\log (d+e x)}{a^2 e}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^2 - 6*a*c)*A
rcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)*e) + Log[d + e*x]/(a^2*e) - Log[a
+ b*(d + e*x)^2 + c*(d + e*x)^4]/(4*a^2*e)

________________________________________________________________________________________

Rubi [A]  time = 0.294215, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1142, 1114, 740, 800, 634, 618, 206, 628} \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 e \left (b^2-4 a c\right )^{3/2}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\frac{\log (d+e x)}{a^2 e}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^2 - 6*a*c)*A
rcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)*e) + Log[d + e*x]/(a^2*e) - Log[a
+ b*(d + e*x)^2 + c*(d + e*x)^4]/(4*a^2*e)

Rule 1142

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 1114

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 740

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 800

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 634

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 618

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 206

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac{b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-b^2+4 a c-b c x}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 a \left (b^2-4 a c\right ) e}\\ &=\frac{b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{-b^2+4 a c}{a x}+\frac{b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,(d+e x)^2\right )}{2 a \left (b^2-4 a c\right ) e}\\ &=\frac{b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\log (d+e x)}{a^2 e}-\frac{\operatorname{Subst}\left (\int \frac{b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^2 \left (b^2-4 a c\right ) e}\\ &=\frac{b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\log (d+e x)}{a^2 e}-\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 e}-\frac{\left (b \left (b^2-6 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right ) e}\\ &=\frac{b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\log (d+e x)}{a^2 e}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\frac{\left (b \left (b^2-6 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a^2 \left (b^2-4 a c\right ) e}\\ &=\frac{b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e}+\frac{\log (d+e x)}{a^2 e}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}\\ \end{align*}

Mathematica [A]  time = 0.428707, size = 235, normalized size = 1.45 \[ \frac{\frac{2 a \left (-2 a c+b^2+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}-\frac{\left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+4 \log (d+e x)}{4 a^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*(b^2 - 2*a*c + b*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + 4*Log[d + e*x]
- ((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*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)^(3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*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)^(3/2))/(4*a^2*e)

________________________________________________________________________________________

Maple [C]  time = 0.03, size = 693, normalized size = 4.3 \begin{align*}{\frac{\ln \left ( ex+d \right ) }{{a}^{2}e}}-{\frac{bce{x}^{2}}{2\,a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{bcdx}{a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{c{d}^{2}b}{2\,a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{c}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{2}}{2\,a \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{1}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( c{e}^{3} \left ( 4\,ac-{b}^{2} \right ){{\it \_R}}^{3}+3\,cd{e}^{2} \left ( 4\,ac-{b}^{2} \right ){{\it \_R}}^{2}+e \left ( 12\,a{c}^{2}{d}^{2}-3\,{b}^{2}c{d}^{2}+5\,abc-{b}^{3} \right ){\it \_R}+4\,a{c}^{2}{d}^{3}-{b}^{2}c{d}^{3}+5\,abcd-{b}^{3}d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,c{d}^{2}e{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}} \end{align*}

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)^2,x)

[Out]

ln(e*x+d)/a^2/e-1/2/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)*
b*c*e/(4*a*c-b^2)*x^2-1/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)*b*c*d/(4*a*c-b^2)*x-1/2/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)/e/(4*a*c-b^2)*c*d^2*b+1/(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)/e/(4*a*c-b^2)*c-1/2/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)/e/(4*a*c-b^2)*b^2-1/2/a^2/(4*a*c-b^2)/e*sum((c*e^3*(4*a*c-b^2)*_R^3+3*c*d*e^2*(4*a*c-b^2)*_R^2+e*(12
*a*c^2*d^2-3*b^2*c*d^2+5*a*b*c-b^3)*_R+4*a*c^2*d^3-b^2*c*d^3+5*a*b*c*d-b^3*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(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e
+2*b*d*e)*_Z+c*d^4+b*d^2+a))

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [B]  time = 3.37138, size = 5200, normalized size = 32.1 \begin{align*} \text{result too large to display} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

[1/4*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*e^2*x^2 + 4*(a*b^3*c - 4*a^2*b*c^2)*d*e*
x + 2*(a*b^3*c - 4*a^2*b*c^2)*d^2 + ((b^3*c - 6*a*b*c^2)*e^4*x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c -
6*a*b*c^2)*d^4 + (b^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*b*c^2)*d^2)*e^2*x^2 + a*b^3 - 6*a^2*b*c + (b^4 - 6*a*b^2*c)
*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b^4 - 6*a*b^2*c)*d)*e*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*e^4*x^4 + 8*c^2*d
*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*a*c + (2*
c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2
*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*
c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5
- 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*
c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(c*e^4*x^
4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^4*c - 8*a*b^2*c
^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 +
(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^
3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 -
8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(e*x + d))/((a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*e^5*x^4 + 4*(a^2*b^4
*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4*x^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 6*(a^2*b^4*c - 8*a^3*b^2*
c^2 + 16*a^4*c^3)*d^2)*e^3*x^2 + 2*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^3 + (a^2*b^5 - 8*a^3*b^3*c +
16*a^4*b*c^2)*d)*e^2*x + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^4 +
(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d^2)*e), 1/4*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + 2*(a*b^3*c - 4*a^2*
b*c^2)*e^2*x^2 + 4*(a*b^3*c - 4*a^2*b*c^2)*d*e*x + 2*(a*b^3*c - 4*a^2*b*c^2)*d^2 + 2*((b^3*c - 6*a*b*c^2)*e^4*
x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c - 6*a*b*c^2)*d^4 + (b^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*b*c^2)*d^2
)*e^2*x^2 + a*b^3 - 6*a^2*b*c + (b^4 - 6*a*b^2*c)*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b^4 - 6*a*b^2*c)*d)*e*
x)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - ((b^
4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c
+ 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*
c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3
)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2
+ b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^2
+ 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 -
8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2
)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(e*x + d))/(
(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*e^5*x^4 + 4*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4*x^3 + (a^2
*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 6*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2)*e^3*x^2 + 2*(2*(a^2*b^4*c
- 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d)*e^2*x + (a^3*b^4 - 8*a^4*b^2*c +
16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d^2)*e)]

________________________________________________________________________________________

Sympy [B]  time = 167.586, size = 1091, normalized size = 6.73 \begin{align*} \text{result too large to display} \end{align*}

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)**2,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))
- 1/(4*a**2*e))*log(2*d*x/e + x**2 + (-32*a**4*c**2*e*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(
64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2*e)) + 16*a**3*b**2*c*e*(-b*sqrt(-(4*a*c -
b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2*e)) -
2*a**2*b**4*e*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b
**4*c - b**6)) - 1/(4*a**2*e)) - 8*a**2*c**2 + 7*a*b**2*c + 6*a*b*c**2*d**2 - b**4 - b**3*c*d**2)/(6*a*b*c**2*
e**2 - b**3*c*e**2)) + (b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) - 1/(4*a**2*e))*log(2*d*x/e + x**2 + (-32*a**4*c**2*e*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*
c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2*e)) + 16*a**3*b**2*c*
e*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
) - 1/(4*a**2*e)) - 2*a**2*b**4*e*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*e*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2*e)) - 8*a**2*c**2 + 7*a*b**2*c + 6*a*b*c**2*d**2 - b**4 - b**3*c
*d**2)/(6*a*b*c**2*e**2 - b**3*c*e**2)) - (-2*a*c + b**2 + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x**2)/(8*a**3*c*e
- 2*a**2*b**2*e + 8*a**2*b*c*d**2*e + 8*a**2*c**2*d**4*e - 2*a*b**3*d**2*e - 2*a*b**2*c*d**4*e + x**4*(8*a**2
*c**2*e**5 - 2*a*b**2*c*e**5) + x**3*(32*a**2*c**2*d*e**4 - 8*a*b**2*c*d*e**4) + x**2*(8*a**2*b*c*e**3 + 48*a*
*2*c**2*d**2*e**3 - 2*a*b**3*e**3 - 12*a*b**2*c*d**2*e**3) + x*(16*a**2*b*c*d*e**2 + 32*a**2*c**2*d**3*e**2 -
4*a*b**3*d*e**2 - 8*a*b**2*c*d**3*e**2)) + log(d/e + x)/(a**2*e)

________________________________________________________________________________________

Giac [B]  time = 3.25002, size = 784, normalized size = 4.84 \begin{align*} -\frac{{\left (a^{2} b^{5} e - 10 \, a^{3} b^{3} c e + 24 \, a^{4} b c^{2} e\right )} \sqrt{b^{2} - 4 \, a c} \log \left ({\left | 4 \, a^{3} c e^{4} + 2 \,{\left (a^{2} b c + \sqrt{b^{2} - 4 \, a c} a^{2} c\right )} x^{2} e^{6} + 4 \,{\left (a^{2} b c + \sqrt{b^{2} - 4 \, a c} a^{2} c\right )} d x e^{5} + 2 \,{\left (a^{2} b c + \sqrt{b^{2} - 4 \, a c} a^{2} c\right )} d^{2} e^{4} \right |}\right )}{4 \,{\left (a^{4} b^{6} e^{2} - 12 \, a^{5} b^{4} c e^{2} + 48 \, a^{6} b^{2} c^{2} e^{2} - 64 \, a^{7} c^{3} e^{2}\right )}} + \frac{{\left (a^{2} b^{5} e - 10 \, a^{3} b^{3} c e + 24 \, a^{4} b c^{2} e\right )} \sqrt{b^{2} - 4 \, a c} \log \left ({\left | -4 \, a^{3} c e^{4} - 2 \,{\left (a^{2} b c - \sqrt{b^{2} - 4 \, a c} a^{2} c\right )} x^{2} e^{6} - 4 \,{\left (a^{2} b c - \sqrt{b^{2} - 4 \, a c} a^{2} c\right )} d x e^{5} - 2 \,{\left (a^{2} b c - \sqrt{b^{2} - 4 \, a c} a^{2} c\right )} d^{2} e^{4} \right |}\right )}{4 \,{\left (a^{4} b^{6} e^{2} - 12 \, a^{5} b^{4} c e^{2} + 48 \, a^{6} b^{2} c^{2} e^{2} - 64 \, a^{7} c^{3} e^{2}\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^{2}} + \frac{e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{2}} + \frac{{\left (a b c x^{2} e^{2} + 2 \, a b c d x e + a b c d^{2} + a b^{2} - 2 \, a^{2} c\right )} e^{\left (-1\right )}}{2 \,{\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 )}{\left (b^{2} - 4 \, a c\right )} a^{2}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

-1/4*(a^2*b^5*e - 10*a^3*b^3*c*e + 24*a^4*b*c^2*e)*sqrt(b^2 - 4*a*c)*log(abs(4*a^3*c*e^4 + 2*(a^2*b*c + sqrt(b
^2 - 4*a*c)*a^2*c)*x^2*e^6 + 4*(a^2*b*c + sqrt(b^2 - 4*a*c)*a^2*c)*d*x*e^5 + 2*(a^2*b*c + sqrt(b^2 - 4*a*c)*a^
2*c)*d^2*e^4))/(a^4*b^6*e^2 - 12*a^5*b^4*c*e^2 + 48*a^6*b^2*c^2*e^2 - 64*a^7*c^3*e^2) + 1/4*(a^2*b^5*e - 10*a^
3*b^3*c*e + 24*a^4*b*c^2*e)*sqrt(b^2 - 4*a*c)*log(abs(-4*a^3*c*e^4 - 2*(a^2*b*c - sqrt(b^2 - 4*a*c)*a^2*c)*x^2
*e^6 - 4*(a^2*b*c - sqrt(b^2 - 4*a*c)*a^2*c)*d*x*e^5 - 2*(a^2*b*c - sqrt(b^2 - 4*a*c)*a^2*c)*d^2*e^4))/(a^4*b^
6*e^2 - 12*a^5*b^4*c*e^2 + 48*a^6*b^2*c^2*e^2 - 64*a^7*c^3*e^2) - 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^2 + e^(-1)*log(abs(x*e + d))/
a^2 + 1/2*(a*b*c*x^2*e^2 + 2*a*b*c*d*x*e + a*b*c*d^2 + a*b^2 - 2*a^2*c)*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)*(b^2 - 4*a*c)*a^2)