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

Optimal. Leaf size=299 \[ \frac{\left (\frac{d}{e}+x\right ) \left (-2 a c+b^2+b c e^2 \left (\frac{d}{e}+x\right )^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b e^2 \left (\frac{d}{e}+x\right )^2+c e^4 \left (\frac{d}{e}+x\right )^4\right )}+\frac{\sqrt{c} \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a e \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-b \sqrt{b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a e \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

((d/e + x)*(b^2 - 2*a*c + b*c*e^2*(d/e + x)^2))/(2*a*(b^2 - 4*a*c)*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)
) + (Sqrt[c]*(b^2 - 12*a*c + b*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*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (Sqrt[c]*(b^2 - 12*a*c - b*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*(b^2 - 4*a*c)^(3/2)*Sqrt[
b + Sqrt[b^2 - 4*a*c]]*e)

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Rubi [A]  time = 0.701916, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1106, 1092, 1166, 205} \[ \frac{\left (\frac{d}{e}+x\right ) \left (-2 a c+b^2+b c e^2 \left (\frac{d}{e}+x\right )^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b e^2 \left (\frac{d}{e}+x\right )^2+c e^4 \left (\frac{d}{e}+x\right )^4\right )}+\frac{\sqrt{c} \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a e \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-b \sqrt{b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a e \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d/e + x)*(b^2 - 2*a*c + b*c*e^2*(d/e + x)^2))/(2*a*(b^2 - 4*a*c)*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)
) + (Sqrt[c]*(b^2 - 12*a*c + b*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*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (Sqrt[c]*(b^2 - 12*a*c - b*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*(b^2 - 4*a*c)^(3/2)*Sqrt[
b + Sqrt[b^2 - 4*a*c]]*e)

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1092

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

Rule 1166

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (a+b e^2 x^2+c e^4 x^4\right )^2} \, dx,x,\frac{d}{e}+x\right )\\ &=\frac{\left (\frac{d}{e}+x\right ) \left (b^2-2 a c+b c e^2 \left (\frac{d}{e}+x\right )^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b e^2 \left (\frac{d}{e}+x\right )^2+c e^4 \left (\frac{d}{e}+x\right )^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{b^2 e^4-2 a c e^4-2 \left (b^2 e^4-4 a c e^4\right )-b c e^6 x^2}{a+b e^2 x^2+c e^4 x^4} \, dx,x,\frac{d}{e}+x\right )}{2 a \left (b^2-4 a c\right ) e^4}\\ &=\frac{\left (\frac{d}{e}+x\right ) \left (b^2-2 a c+b c e^2 \left (\frac{d}{e}+x\right )^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b e^2 \left (\frac{d}{e}+x\right )^2+c e^4 \left (\frac{d}{e}+x\right )^4\right )}-\frac{\left (c \left (b^2-12 a c-b \sqrt{b^2-4 a c}\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e^2}{2}+\frac{1}{2} \sqrt{b^2-4 a c} e^2+c e^4 x^2} \, dx,x,\frac{d}{e}+x\right )}{4 a \left (b^2-4 a c\right )^{3/2}}+\frac{\left (c \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e^2}{2}-\frac{1}{2} \sqrt{b^2-4 a c} e^2+c e^4 x^2} \, dx,x,\frac{d}{e}+x\right )}{4 a \left (b^2-4 a c\right )^{3/2}}\\ &=\frac{\left (\frac{d}{e}+x\right ) \left (b^2-2 a c+b c e^2 \left (\frac{d}{e}+x\right )^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b e^2 \left (\frac{d}{e}+x\right )^2+c e^4 \left (\frac{d}{e}+x\right )^4\right )}+\frac{\sqrt{c} \left (b^2-12 a c+b \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 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} e}-\frac{\sqrt{c} \left (b^2-12 a c-b \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 \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} e}\\ \end{align*}

Mathematica [A]  time = 1.00214, size = 271, normalized size = 0.91 \[ \frac{\frac{2 (d+e x) \left (-2 a c+b^2+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+\frac{\sqrt{2} \sqrt{c} \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (b \sqrt{b^2-4 a c}+12 a c-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(d + e*x)*(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))) + (Sqrt[2
]*Sqrt[c]*(b^2 - 12*a*c + b*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]]) + (Sqrt[2]*Sqrt[c]*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*A
rcTan[(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
]]))/(4*a*e)

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Maple [C]  time = 0.017, size = 364, normalized size = 1.2 \begin{align*}{\frac{1}{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} \left ( -{\frac{bc{e}^{2}{x}^{3}}{2\,a \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{3\,bcde{x}^{2}}{2\,a \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{ \left ( -3\,c{d}^{2}b+2\,ac-{b}^{2} \right ) x}{2\,a \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{d \left ( -c{d}^{2}b+2\,ac-{b}^{2} \right ) }{2\,ea \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+{\frac{1}{4\,ea \left ( 4\,ac-{b}^{2} \right ) }\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 ( -{{\it \_R}}^{2}bc{e}^{2}-2\,{\it \_R}\,bcde-c{d}^{2}b+6\,ac-{b}^{2} \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/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b c e^{3} x^{3} + 3 \, b c d e^{2} x^{2} + b c d^{3} +{\left (3 \, b c d^{2} + b^{2} - 2 \, a c\right )} e x +{\left (b^{2} - 2 \, a c\right )} d}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{5} x^{4} + 4 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d e^{4} x^{3} +{\left (a b^{3} - 4 \, a^{2} b c + 6 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{2}\right )} e^{3} x^{2} + 2 \,{\left (2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3} +{\left (a b^{3} - 4 \, a^{2} b c\right )} d\right )} e^{2} x +{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} d^{2}\right )} e\right )}} - \frac{-\int \frac{b c e^{2} x^{2} + 2 \, b c d e x + b c d^{2} + b^{2} - 6 \, a c}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{4} x^{4} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} x^{3} +{\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} +{\left (b^{3} - 4 \, a b c + 6 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{2} x^{2} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} d^{2} + 2 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 4 \, a b c\right )} d\right )} e x}\,{d x}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*c*e^3*x^3 + 3*b*c*d*e^2*x^2 + b*c*d^3 + (3*b*c*d^2 + b^2 - 2*a*c)*e*x + (b^2 - 2*a*c)*d)/((a*b^2*c - 4*
a^2*c^2)*e^5*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*x
^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*
a^3*c + (a*b^3 - 4*a^2*b*c)*d^2)*e) - 1/2*integrate(-(b*c*e^2*x^2 + 2*b*c*d*e*x + b*c*d^2 + b^2 - 6*a*c)/((b^2
*c - 4*a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*d^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*
a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*
e*x), x)/a

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Fricas [B]  time = 2.62106, size = 6835, normalized size = 22.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b*c*e^3*x^3 + 6*b*c*d*e^2*x^2 + 2*b*c*d^3 + 2*(3*b*c*d^2 + b^2 - 2*a*c)*e*x - sqrt(1/2)*((a*b^2*c - 4*a
^2*c^2)*e^5*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*x^
2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a
^3*c + (a*b^3 - 4*a^2*b*c)*d^2)*e)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b
^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a
^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))*log((5*b^4*c^2 - 81*a*b^2*c^3 + 3
24*a^2*c^4)*e*x + (5*b^4*c^2 - 81*a*b^2*c^3 + 324*a^2*c^4)*d + 1/2*sqrt(1/2)*((a^3*b^9 - 20*a^4*b^7*c + 144*a^
5*b^5*c^2 - 448*a^6*b^3*c^3 + 512*a^7*b*c^4)*e^3*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c
+ 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)) - (b^8 - 23*a*b^6*c + 190*a^2*b^4*c^2 - 672*a^3*b^2*c^3 + 864*a^4*c^4)*e
)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^
4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4
*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))) + sqrt(1/2)*((a*b^2*c - 4*a^2*c^2)*e^5*x^4 + 4*(a*b^2*c - 4*a^2*c
^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*x^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (
a*b^3 - 4*a^2*b*c)*d)*e^2*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*d^2)*e)*sqr
t(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^4 - 1
8*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4*
c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))*log((5*b^4*c^2 - 81*a*b^2*c^3 + 324*a^2*c^4)*e*x + (5*b^4*c^2 - 81*a*b^
2*c^3 + 324*a^2*c^4)*d - 1/2*sqrt(1/2)*((a^3*b^9 - 20*a^4*b^7*c + 144*a^5*b^5*c^2 - 448*a^6*b^3*c^3 + 512*a^7*
b*c^4)*e^3*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4))
- (b^8 - 23*a*b^6*c + 190*a^2*b^4*c^2 - 672*a^3*b^2*c^3 + 864*a^4*c^4)*e)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c
^2 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6
- 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*
e^2))) + sqrt(1/2)*((a*b^2*c - 4*a^2*c^2)*e^5*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6
*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*x^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*x + ((a*b^2*
c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*d^2)*e)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 -
(a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12
*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))
*log((5*b^4*c^2 - 81*a*b^2*c^3 + 324*a^2*c^4)*e*x + (5*b^4*c^2 - 81*a*b^2*c^3 + 324*a^2*c^4)*d + 1/2*sqrt(1/2)
*((a^3*b^9 - 20*a^4*b^7*c + 144*a^5*b^5*c^2 - 448*a^6*b^3*c^3 + 512*a^7*b*c^4)*e^3*sqrt((b^4 - 18*a*b^2*c + 81
*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)) + (b^8 - 23*a*b^6*c + 190*a^2*b^4*c^2
- 672*a^3*b^2*c^3 + 864*a^4*c^4)*e)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 - (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*
b^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*
a^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))) - sqrt(1/2)*((a*b^2*c - 4*a^2*c
^2)*e^5*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*x^2 +
2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c
+ (a*b^3 - 4*a^2*b*c)*d^2)*e)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 - (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c
^2 - 64*a^6*c^3)*e^2*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c
^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))*log((5*b^4*c^2 - 81*a*b^2*c^3 + 324*a
^2*c^4)*e*x + (5*b^4*c^2 - 81*a*b^2*c^3 + 324*a^2*c^4)*d - 1/2*sqrt(1/2)*((a^3*b^9 - 20*a^4*b^7*c + 144*a^5*b^
5*c^2 - 448*a^6*b^3*c^3 + 512*a^7*b*c^4)*e^3*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 4
8*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)) + (b^8 - 23*a*b^6*c + 190*a^2*b^4*c^2 - 672*a^3*b^2*c^3 + 864*a^4*c^4)*e)*sq
rt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 - (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2*sqrt((b^4 -
18*a*b^2*c + 81*a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^4)))/((a^3*b^6 - 12*a^4*b^4
*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^2))) + 2*(b^2 - 2*a*c)*d)/((a*b^2*c - 4*a^2*c^2)*e^5*x^4 + 4*(a*b^2*c - 4*
a^2*c^2)*d*e^4*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*x^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^
3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*d^2)*e
)

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Sympy [B]  time = 24.5303, size = 740, normalized size = 2.47 \begin{align*} - \frac{- 2 a c d + b^{2} d + b c d^{3} + 3 b c d e^{2} x^{2} + b c e^{3} x^{3} + x \left (- 2 a c e + b^{2} e + 3 b c d^{2} e\right )}{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} \left (8 a^{2} c^{2} e^{5} - 2 a b^{2} c e^{5}\right ) + x^{3} \left (32 a^{2} c^{2} d e^{4} - 8 a b^{2} c d e^{4}\right ) + x^{2} \left (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}\right ) + x \left (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}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{9} c^{6} e^{4} - 1572864 a^{8} b^{2} c^{5} e^{4} + 983040 a^{7} b^{4} c^{4} e^{4} - 327680 a^{6} b^{6} c^{3} e^{4} + 61440 a^{5} b^{8} c^{2} e^{4} - 6144 a^{4} b^{10} c e^{4} + 256 a^{3} b^{12} e^{4}\right ) + t^{2} \left (- 61440 a^{5} b c^{5} e^{2} + 61440 a^{4} b^{3} c^{4} e^{2} - 24064 a^{3} b^{5} c^{3} e^{2} + 4608 a^{2} b^{7} c^{2} e^{2} - 432 a b^{9} c e^{2} + 16 b^{11} e^{2}\right ) + 1296 a^{2} c^{5} - 360 a b^{2} c^{4} + 25 b^{4} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{32768 t^{3} a^{7} b c^{4} e^{3} - 28672 t^{3} a^{6} b^{3} c^{3} e^{3} + 9216 t^{3} a^{5} b^{5} c^{2} e^{3} - 1280 t^{3} a^{4} b^{7} c e^{3} + 64 t^{3} a^{3} b^{9} e^{3} + 1728 t a^{4} c^{4} e - 2304 t a^{3} b^{2} c^{3} e + 740 t a^{2} b^{4} c^{2} e - 92 t a b^{6} c e + 4 t b^{8} e + 324 a^{2} c^{4} d - 81 a b^{2} c^{3} d + 5 b^{4} c^{2} d}{324 a^{2} c^{4} e - 81 a b^{2} c^{3} e + 5 b^{4} c^{2} e} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-2*a*c*d + b**2*d + b*c*d**3 + 3*b*c*d*e**2*x**2 + b*c*e**3*x**3 + x*(-2*a*c*e + b**2*e + 3*b*c*d**2*e))/(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)) + RootSum(_t**4*(1048576*a**9*c**6*e**4 - 1572864*a**8*b**2*c
**5*e**4 + 983040*a**7*b**4*c**4*e**4 - 327680*a**6*b**6*c**3*e**4 + 61440*a**5*b**8*c**2*e**4 - 6144*a**4*b**
10*c*e**4 + 256*a**3*b**12*e**4) + _t**2*(-61440*a**5*b*c**5*e**2 + 61440*a**4*b**3*c**4*e**2 - 24064*a**3*b**
5*c**3*e**2 + 4608*a**2*b**7*c**2*e**2 - 432*a*b**9*c*e**2 + 16*b**11*e**2) + 1296*a**2*c**5 - 360*a*b**2*c**4
+ 25*b**4*c**3, Lambda(_t, _t*log(x + (32768*_t**3*a**7*b*c**4*e**3 - 28672*_t**3*a**6*b**3*c**3*e**3 + 9216*
_t**3*a**5*b**5*c**2*e**3 - 1280*_t**3*a**4*b**7*c*e**3 + 64*_t**3*a**3*b**9*e**3 + 1728*_t*a**4*c**4*e - 2304
*_t*a**3*b**2*c**3*e + 740*_t*a**2*b**4*c**2*e - 92*_t*a*b**6*c*e + 4*_t*b**8*e + 324*a**2*c**4*d - 81*a*b**2*
c**3*d + 5*b**4*c**2*d)/(324*a**2*c**4*e - 81*a*b**2*c**3*e + 5*b**4*c**2*e))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)^(-2), x)