3.7.34 \(\int \frac {1}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [634]

Optimal. Leaf size=437 \[ \frac {\left (\frac {d}{e}+x\right ) \left (b^2-2 a c+b c e^2 \left (\frac {d}{e}+x\right )^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b e^2 \left (\frac {d}{e}+x\right )^2+c e^4 \left (\frac {d}{e}+x\right )^4\right )^2}+\frac {\left (\frac {d}{e}+x\right ) \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) e^2 \left (\frac {d}{e}+x\right )^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b e^2 \left (\frac {d}{e}+x\right )^2+c e^4 \left (\frac {d}{e}+x\right )^4\right )}+\frac {3 \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {3 \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2-b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

1/4*(d/e+x)*(b^2-2*a*c+b*c*e^2*(d/e+x)^2)/a/(-4*a*c+b^2)/(a+b*e^2*(d/e+x)^2+c*e^4*(d/e+x)^4)^2+1/8*(d/e+x)*((-
7*a*c+b^2)*(-4*a*c+3*b^2)+3*b*c*(-8*a*c+b^2)*e^2*(d/e+x)^2)/a^2/(-4*a*c+b^2)^2/(a+b*e^2*(d/e+x)^2+c*e^4*(d/e+x
)^4)+3/16*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^4-10*a*b^2*c+56*a^2*c^2+b*(-
8*a*c+b^2)*(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(5/2)/e*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-3/16*arctan((e*x+
d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^4-10*a*b^2*c+56*a^2*c^2-b*(-8*a*c+b^2)*(-4*a*c+b^2
)^(1/2))/a^2/(-4*a*c+b^2)^(5/2)/e*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 3.65, antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1120, 1106, 1192, 1180, 211} \begin {gather*} \frac {3 \sqrt {c} \left (56 a^2 c^2-10 a b^2 c+b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}+b^4\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^2 e \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (56 a^2 c^2-10 a b^2 c-b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}+b^4\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^2 e \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {\left (\frac {d}{e}+x\right ) \left (3 b c e^2 \left (b^2-8 a c\right ) \left (\frac {d}{e}+x\right )^2+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{8 a^2 \left (b^2-4 a c\right )^2 \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 (\frac {d}{e}+x\right ) \left (-2 a c+b^2+b c e^2 \left (\frac {d}{e}+x\right )^2\right )}{4 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 )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d/e + x)*(b^2 - 2*a*c + b*c*e^2*(d/e + x)^2))/(4*a*(b^2 - 4*a*c)*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)
^2) + ((d/e + x)*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b*c*(b^2 - 8*a*c)*e^2*(d/e + x)^2))/(8*a^2*(b^2 - 4*a*c)^2
*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)) + (3*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b*(b^2 - 8*a*c)*Sq
rt[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^2*(b^2 - 4*a*c)
^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (3*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b*(b^2 - 8*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^2*(b^2 - 4*a*c)^(5/2)*S
qrt[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 1106

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 1120

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 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 {1}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\text {Subst}\left (\int \frac {1}{\left (a+b e^2 x^2+c e^4 x^4\right )^3} \, 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 )}{4 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 )^2}-\frac {\text {Subst}\left (\int \frac {b^2 e^4-2 a c e^4-4 \left (b^2 e^4-4 a c e^4\right )-5 b c e^6 x^2}{\left (a+b e^2 x^2+c e^4 x^4\right )^2} \, dx,x,\frac {d}{e}+x\right )}{4 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 )}{4 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 )^2}+\frac {(d+e x) \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) (d+e x)^2\right )}{8 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 {3 \left (b^4-9 a b^2 c+28 a^2 c^2\right ) e^8+3 b c \left (b^2-8 a c\right ) e^{10} x^2}{a+b e^2 x^2+c e^4 x^4} \, dx,x,\frac {d}{e}+x\right )}{8 a^2 \left (b^2-4 a c\right )^2 e^8}\\ &=\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 )}{4 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 )^2}+\frac {(d+e x) \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) (d+e x)^2\right )}{8 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 {\left (3 c \left (b^4-10 a b^2 c+56 a^2 c^2-b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) e^2\right ) \text {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 )}{16 a^2 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 c \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) e^2\right ) \text {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 )}{16 a^2 \left (b^2-4 a c\right )^{5/2}}\\ &=\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 )}{4 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 )^2}+\frac {(d+e x) \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) (d+e x)^2\right )}{8 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 {3 \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 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^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {3 \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2-b \left (b^2-8 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^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}

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Mathematica [A]
time = 4.82, size = 424, normalized size = 0.97 \begin {gather*} \frac {\frac {4 a (d+e x) \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 {2 (d+e x) \left (3 b^4-25 a b^2 c+28 a^2 c^2+3 b^3 c (d+e x)^2-24 a b c^2 (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 a^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a*(d + e*x)*(-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) + (2*
(d + e*x)*(3*b^4 - 25*a*b^2*c + 28*a^2*c^2 + 3*b^3*c*(d + e*x)^2 - 24*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)^2*(
a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c
] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)
^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c
] + 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)
^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*a^2*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 1010, normalized size = 2.31 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3/8*c^2*e^6*b*(8*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/a^2*x^7-21/8*c^2*d*e^5*b*(8*a*c-b^2)/(16*a^2*c^2-8*a*b^
2*c+b^4)/a^2*x^6+1/8*(-504*a*b*c^2*d^2+63*b^3*c*d^2+28*a^2*c^2-49*a*b^2*c+6*b^4)*c*e^4/(16*a^2*c^2-8*a*b^2*c+b
^4)/a^2*x^5+5/8*c*d*e^3*(-168*a*b*c^2*d^2+21*b^3*c*d^2+28*a^2*c^2-49*a*b^2*c+6*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)
/a^2*x^4-1/8*e^2*(840*a*b*c^3*d^4-105*b^3*c^2*d^4-280*a^2*c^3*d^2+490*a*b^2*c^2*d^2-60*b^4*c*d^2+4*a^2*b*c^2+2
0*a*b^3*c-3*b^5)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/8*d*e*(504*a*b*c^3*d^4-63*b^3*c^2*d^4-280*a^2*c^3*d^2+49
0*a*b^2*c^2*d^2-60*b^4*c*d^2+12*a^2*b*c^2+60*a*b^3*c-9*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/a^2*x^2+1/8*(-168*a*b*c
^3*d^6+21*b^3*c^2*d^6+140*a^2*c^3*d^4-245*a*b^2*c^2*d^4+30*b^4*c*d^4-12*a^2*b*c^2*d^2-60*a*b^3*c*d^2+9*b^5*d^2
+44*a^3*c^2-37*a^2*b^2*c+5*a*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a^2*x+1/8*d/e*(-24*a*b*c^3*d^6+3*b^3*c^2*d^6+28*a
^2*c^3*d^4-49*a*b^2*c^2*d^4+6*b^4*c*d^4-4*a^2*b*c^2*d^2-20*a*b^3*c*d^2+3*b^5*d^2+44*a^3*c^2-37*a^2*b^2*c+5*a*b
^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a^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)^2+3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/a^2/e*sum((b*c*e^2*(-8*a*c+b^2)*_R^2+2*b*c*d*e*(-8*a*c+b^2)*_R
-8*a*b*c^2*d^2+b^3*c*d^2+28*a^2*c^2-9*a*b^2*c+b^4)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*
d)*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+a)
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*(b^3*c^2 - 8*a*b*c^3)*d^7 + 21*(b^3*c^2*e^6 - 8*a*b*c^3*e^6)*d*x^6 + 3*(b^3*c^2*e^7 - 8*a*b*c^3*e^7)*x^
7 + (6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d^5 + (6*b^4*c*e^5 - 49*a*b^2*c^2*e^5 + 28*a^2*c^3*e^5 + 63*(b^3*c^2
*e^5 - 8*a*b*c^3*e^5)*d^2)*x^5 + 5*(21*(b^3*c^2*e^4 - 8*a*b*c^3*e^4)*d^3 + (6*b^4*c*e^4 - 49*a*b^2*c^2*e^4 + 2
8*a^2*c^3*e^4)*d)*x^4 + (3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*d^3 + (3*b^5*e^3 - 20*a*b^3*c*e^3 - 4*a^2*b*c^2*e^3
+ 105*(b^3*c^2*e^3 - 8*a*b*c^3*e^3)*d^4 + 10*(6*b^4*c*e^3 - 49*a*b^2*c^2*e^3 + 28*a^2*c^3*e^3)*d^2)*x^3 + (63
*(b^3*c^2*e^2 - 8*a*b*c^3*e^2)*d^5 + 10*(6*b^4*c*e^2 - 49*a*b^2*c^2*e^2 + 28*a^2*c^3*e^2)*d^3 + 3*(3*b^5*e^2 -
20*a*b^3*c*e^2 - 4*a^2*b*c^2*e^2)*d)*x^2 + (5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^2)*d + (21*(b^3*c^2*e - 8*a*b*c
^3*e)*d^6 + 5*a*b^4*e - 37*a^2*b^2*c*e + 44*a^3*c^2*e + 5*(6*b^4*c*e - 49*a*b^2*c^2*e + 28*a^2*c^3*e)*d^4 + 3*
(3*b^5*e - 20*a*b^3*c*e - 4*a^2*b*c^2*e)*d^2)*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) - 3/8*integrate(-(b^4 - 9*a*b^2*c + 28*a^2*c^2 + (b^3*c - 8*a*b*c^2)*d^2 + 2*(b^3*c*e - 8*a*b*c^2*e)
*d*x + (b^3*c*e^2 - 8*a*b*c^2*e^2)*x^2)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + b*d^2 + (6*c*d^2*e^2 + b*e^2)*x^2
+ 2*(2*c*d^3*e + b*d*e)*x + a), x)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8450 vs. \(2 (375) = 750\).
time = 0.80, size = 8450, normalized size = 19.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(6*(b^3*c^2 - 8*a*b*c^3)*x^7*e^7 + 42*(b^3*c^2 - 8*a*b*c^3)*d*x^6*e^6 + 6*(b^3*c^2 - 8*a*b*c^3)*d^7 + 2*(
6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3 + 63*(b^3*c^2 - 8*a*b*c^3)*d^2)*x^5*e^5 + 2*(6*b^4*c - 49*a*b^2*c^2 + 28*a
^2*c^3)*d^5 + 10*(21*(b^3*c^2 - 8*a*b*c^3)*d^3 + (6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d)*x^4*e^4 + 2*(3*b^5 -
20*a*b^3*c - 4*a^2*b*c^2 + 105*(b^3*c^2 - 8*a*b*c^3)*d^4 + 10*(6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d^2)*x^3*
e^3 + 2*(3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*d^3 + 2*(63*(b^3*c^2 - 8*a*b*c^3)*d^5 + 10*(6*b^4*c - 49*a*b^2*c^2
+ 28*a^2*c^3)*d^3 + 3*(3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*d)*x^2*e^2 + 2*(21*(b^3*c^2 - 8*a*b*c^3)*d^6 + 5*a*b^
4 - 37*a^2*b^2*c + 44*a^3*c^2 + 5*(6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d^4 + 3*(3*b^5 - 20*a*b^3*c - 4*a^2*b*
c^2)*d^2)*x*e + 3*sqrt(1/2)*((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^8*e^9 + 8*(a^2*b^4*c^2 - 8*a^3*b^2*c
^3 + 16*a^4*c^4)*d*x^7*e^8 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3 + 14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 1
6*a^4*c^4)*d^2)*x^6*e^7 + 4*(14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^3 + 3*(a^2*b^5*c - 8*a^3*b^3*c^2
+ 16*a^4*b*c^3)*d)*x^5*e^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3 + 70*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^
4)*d^4 + 30*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^2)*x^4*e^5 + 4*(14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*
a^4*c^4)*d^5 + 10*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d)*x^3
*e^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + 14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^6 + 15*(a^2*b
^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^4 + 3*(a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^2)*x^2*e^3 + 4*(2*(a^2*b^4
*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^7 + 3*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^5 + (a^2*b^6 - 6*a^3*b
^4*c + 32*a^5*c^3)*d^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d)*x*e^2 + ((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*
a^4*c^4)*d^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^6 + (a^2*b^
6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^2)*e)*sqrt(-(b^9 - 21*a*b^7*c +
189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 + (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4
*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^
4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))
*e^(-2)/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))*log(
27*(21*b^8*c^3 - 447*a*b^6*c^4 + 4189*a^2*b^4*c^5 - 19208*a^3*b^2*c^6 + 38416*a^4*c^7)*x*e + 27*(21*b^8*c^3 -
447*a*b^6*c^4 + 4189*a^2*b^4*c^5 - 19208*a^3*b^2*c^6 + 38416*a^4*c^7)*d + 27/2*sqrt(1/2)*((a^5*b^15 - 31*a^6*b
^13*c + 424*a^7*b^11*c^2 - 3280*a^8*b^9*c^3 + 15360*a^9*b^7*c^4 - 43264*a^10*b^5*c^5 + 67584*a^11*b^3*c^6 - 45
056*a^12*b*c^7)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^
11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5))*e - (b^14 - 32*a*b^12*c +
464*a^2*b^10*c^2 - 3885*a^3*b^8*c^3 + 20088*a^4*b^6*c^4 - 63680*a^5*b^4*c^5 + 113792*a^6*b^2*c^6 - 87808*a^7*
c^7)*e)*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 + (a^5*b^10 - 20*a^6*b^8*
c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4
*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 128
0*a^14*b^2*c^4 - 1024*a^15*c^5)))*e^(-2)/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a
^9*b^2*c^4 - 1024*a^10*c^5))) - 3*sqrt(1/2)*((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^8*e^9 + 8*(a^2*b^4*c
^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d*x^7*e^8 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3 + 14*(a^2*b^4*c^2 - 8
*a^3*b^2*c^3 + 16*a^4*c^4)*d^2)*x^6*e^7 + 4*(14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^3 + 3*(a^2*b^5*c
- 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d)*x^5*e^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3 + 70*(a^2*b^4*c^2 - 8*a^3*b^2
*c^3 + 16*a^4*c^4)*d^4 + 30*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^2)*x^4*e^5 + 4*(14*(a^2*b^4*c^2 - 8*a
^3*b^2*c^3 + 16*a^4*c^4)*d^5 + 10*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3 + (a^2*b^6 - 6*a^3*b^4*c + 32
*a^5*c^3)*d)*x^3*e^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + 14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)
*d^6 + 15*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^4 + 3*(a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^2)*x^2*e^3
+ 4*(2*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^7 + 3*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^5 + (a
^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d)*x*e^2 + ((a^2*b^4*c^2 - 8*a
^3*b^2*c^3 + 16*a^4*c^4)*d^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^
3)*d^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^2)*e)*sqrt(-(b^
9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*...

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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/(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. 2487 vs. \(2 (375) = 750\).
time = 3.65, size = 2487, normalized size = 5.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*(((d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 8*(d*e^(-1) + sqr
t(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*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^3*c*d*e + 16*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*
e^(-4)/c))*a*b*c^2*d*e + b^3*c*d^2 - 8*a*b*c^2*d^2 + b^4 - 9*a*b^2*c + 28*a^2*c^2)*log(d*e^(-1) + x + sqrt(1/2
)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e
^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 -
2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c
))) + ((d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 8*(d*e^(-1) - sqrt
(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 +
sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 16*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e
^(-4)/c))*a*b*c^2*d*e + b^3*c*d^2 - 8*a*b*c^2*d^2 + b^4 - 9*a*b^2*c + 28*a^2*c^2)*log(d*e^(-1) + x - sqrt(1/2)
*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^
2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2
*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c)
)) + ((d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 8*(d*e^(-1) + sqrt(
1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - s
qrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 16*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^
(-4)/c))*a*b*c^2*d*e + b^3*c*d^2 - 8*a*b*c^2*d^2 + b^4 - 9*a*b^2*c + 28*a^2*c^2)*log(d*e^(-1) + x + sqrt(1/2)*
sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2
)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*
c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))
) + ((d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 8*(d*e^(-1) - sqrt(1
/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sq
rt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 16*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(
-4)/c))*a*b*c^2*d*e + b^3*c*d^2 - 8*a*b*c^2*d^2 + b^4 - 9*a*b^2*c + 28*a^2*c^2)*log(d*e^(-1) + x - sqrt(1/2)*s
qrt(-(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^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2) + 1/8*(3*b^3*c^2*x^7*e^7 - 24*a*b*c^3*x^7*e^7 + 21*b^3*c^2*d*x^6*e^6 -
168*a*b*c^3*d*x^6*e^6 + 63*b^3*c^2*d^2*x^5*e^5 - 504*a*b*c^3*d^2*x^5*e^5 + 105*b^3*c^2*d^3*x^4*e^4 - 840*a*b*c
^3*d^3*x^4*e^4 + 105*b^3*c^2*d^4*x^3*e^3 - 840*a*b*c^3*d^4*x^3*e^3 + 63*b^3*c^2*d^5*x^2*e^2 - 504*a*b*c^3*d^5*
x^2*e^2 + 21*b^3*c^2*d^6*x*e - 168*a*b*c^3*d^6*x*e + 3*b^3*c^2*d^7 - 24*a*b*c^3*d^7 + 6*b^4*c*x^5*e^5 - 49*a*b
^2*c^2*x^5*e^5 + 28*a^2*c^3*x^5*e^5 + 30*b^4*c*d*x^4*e^4 - 245*a*b^2*c^2*d*x^4*e^4 + 140*a^2*c^3*d*x^4*e^4 + 6
0*b^4*c*d^2*x^3*e^3 - 490*a*b^2*c^2*d^2*x^3*e^3 + 280*a^2*c^3*d^2*x^3*e^3 + 60*b^4*c*d^3*x^2*e^2 - 490*a*b^2*c
^2*d^3*x^2*e^2 + 280*a^2*c^3*d^3*x^2*e^2 + 30*b^4*c*d^4*x*e - 245*a*b^2*c^2*d^4*x*e + 140*a^2*c^3*d^4*x*e + 6*
b^4*c*d^5 - 49*a*b^2*c^2*d^5 + 28*a^2*c^3*d^5 + 3*b^5*x^3*e^3 - 20*a*b^3*c*x^3*e^3 - 4*a^2*b*c^2*x^3*e^3 + 9*b
^5*d*x^2*e^2 - 60*a*b^3*c*d*x^2*e^2 - 12*a^2*b*c^2*d*x^2*e^2 + 9*b^5*d^2*x*e - 60*a*b^3*c*d^2*x*e - 12*a^2*b*c
^2*d^2*x*e + 3*b^5*d^3 - 20*a*b^3*c*d^3 - 4*a^2*b*c^2*d^3 + 5*a*b^4*x*e - 37*a^2*b^2*c*x*e + 44*a^3*c^2*x*e +
5*a*b^4*d - 37*a^2*b^2*c*d + 44*a^3*c^2*d)/((a^2*b^4*e - 8*a^3*b^2*c*e + 16*a^4*c^2*e)*(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)

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Mupad [B]
time = 7.80, size = 2500, normalized size = 5.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*b^5*d^3 + 44*a^3*c^2*d + 6*b^4*c*d^5 + 28*a^2*c^3*d^5 + 3*b^3*c^2*d^7 + 5*a*b^4*d - 4*a^2*b*c^2*d^3 - 49*a
*b^2*c^2*d^5 - 37*a^2*b^2*c*d - 20*a*b^3*c*d^3 - 24*a*b*c^3*d^7)/(8*a^2*e*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x
^3*(3*b^5*e^2 - 4*a^2*b*c^2*e^2 + 60*b^4*c*d^2*e^2 + 280*a^2*c^3*d^2*e^2 + 105*b^3*c^2*d^4*e^2 - 20*a*b^3*c*e^
2 - 840*a*b*c^3*d^4*e^2 - 490*a*b^2*c^2*d^2*e^2))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^5*(6*b^4*c*e^4 +
28*a^2*c^3*e^4 - 49*a*b^2*c^2*e^4 + 63*b^3*c^2*d^2*e^4 - 504*a*b*c^3*d^2*e^4))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a
*b^2*c)) + (x^2*(9*b^5*d*e + 280*a^2*c^3*d^3*e + 63*b^3*c^2*d^5*e + 60*b^4*c*d^3*e - 12*a^2*b*c^2*d*e - 504*a*
b*c^3*d^5*e - 490*a*b^2*c^2*d^3*e - 60*a*b^3*c*d*e))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (5*x^4*(28*a^2*c
^3*d*e^3 + 21*b^3*c^2*d^3*e^3 + 6*b^4*c*d*e^3 - 49*a*b^2*c^2*d*e^3 - 168*a*b*c^3*d^3*e^3))/(8*a^2*(b^4 + 16*a^
2*c^2 - 8*a*b^2*c)) + (21*x^6*(b^3*c^2*d*e^5 - 8*a*b*c^3*d*e^5))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(
5*a*b^4 + 44*a^3*c^2 + 9*b^5*d^2 - 37*a^2*b^2*c + 30*b^4*c*d^4 + 140*a^2*c^3*d^4 + 21*b^3*c^2*d^6 - 12*a^2*b*c
^2*d^2 - 245*a*b^2*c^2*d^4 - 60*a*b^3*c*d^2 - 168*a*b*c^3*d^6))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^
7*(b^3*c^2*e^6 - 8*a*b*c^3*e^6))/(8*a^2*(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((((3612672*a^6*
c^9*d*e^11 + 144*b^12*c^3*d*e^11 - 4032*a*b^10*c^4*d*e^11 + 49824*a^2*b^8*c^5*d*e^11 - 340992*a^3*b^6*c^6*d*e^
11 + 1410048*a^4*b^4*c^7*d*e^11 - 3391488*a^5*b^2*c^8*d*e^11)/(512*(a^4*b^12 + 4096*a^10*c^6 - 24*a^5*b^10*c +
240*a^6*b^8*c^2 - 1280*a^7*b^6*c^3 + 3840*a^8*b^4*c^4 - 6144*a^9*b^2*c^5)) - (((67108864*a^11*b*c^9*d*e^13 -
4096*a^4*b^15*c^2*d*e^13 + 114688*a^5*b^13*c^3*d*e^13 - 1376256*a^6*b^11*c^4*d*e^13 + 9175040*a^7*b^9*c^5*d*e^
13 - 36700160*a^8*b^7*c^6*d*e^13 + 88080384*a^9*b^5*c^7*d*e^13 - 117440512*a^10*b^3*c^8*d*e^13)/(512*(a^4*b^12
+ 4096*a^10*c^6 - 24*a^5*b^10*c + 240*a^6*b^8*c^2 - 1280*a^7*b^6*c^3 + 3840*a^8*b^4*c^4 - 6144*a^9*b^2*c^5))
+ (x*(262144*a^9*b*c^7*e^14 - 256*a^4*b^11*c^2*e^14 + 5120*a^5*b^9*c^3*e^14 - 40960*a^6*b^7*c^4*e^14 + 163840*
a^7*b^5*c^5*e^14 - 327680*a^8*b^3*c^6*e^14))/(32*(a^4*b^8 + 256*a^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*
a^7*b^2*c^3)))*(-(9*(b^19 + b^4*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^9*b*c^9 + 769*a^2*b^15*c^2 - 8620*a^3*b^
13*c^3 + 63440*a^4*b^11*c^4 - 316864*a^5*b^9*c^5 + 1069824*a^6*b^7*c^6 - 2343936*a^7*b^5*c^7 + 3010560*a^8*b^3
*c^8 + 49*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 41*a*b^17*c - 11*a*b^2*c*(-(4*a*c - b^2)^15)^(1/2)))/(512*(a^5*b
^20*e^2 + 1048576*a^15*c^10*e^2 - 40*a^6*b^18*c*e^2 + 720*a^7*b^16*c^2*e^2 - 7680*a^8*b^14*c^3*e^2 + 53760*a^9
*b^12*c^4*e^2 - 258048*a^10*b^10*c^5*e^2 + 860160*a^11*b^8*c^6*e^2 - 1966080*a^12*b^6*c^7*e^2 + 2949120*a^13*b
^4*c^8*e^2 - 2621440*a^14*b^2*c^9*e^2)))^(1/2) - (22020096*a^9*c^9*e^12 - 768*a^2*b^14*c^2*e^12 + 22272*a^3*b^
12*c^3*e^12 - 282624*a^4*b^10*c^4*e^12 + 2027520*a^5*b^8*c^5*e^12 - 8847360*a^6*b^6*c^6*e^12 + 23396352*a^7*b^
4*c^7*e^12 - 34603008*a^8*b^2*c^8*e^12)/(512*(a^4*b^12 + 4096*a^10*c^6 - 24*a^5*b^10*c + 240*a^6*b^8*c^2 - 128
0*a^7*b^6*c^3 + 3840*a^8*b^4*c^4 - 6144*a^9*b^2*c^5)))*(-(9*(b^19 + b^4*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^
9*b*c^9 + 769*a^2*b^15*c^2 - 8620*a^3*b^13*c^3 + 63440*a^4*b^11*c^4 - 316864*a^5*b^9*c^5 + 1069824*a^6*b^7*c^6
- 2343936*a^7*b^5*c^7 + 3010560*a^8*b^3*c^8 + 49*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 41*a*b^17*c - 11*a*b^2*c
*(-(4*a*c - b^2)^15)^(1/2)))/(512*(a^5*b^20*e^2 + 1048576*a^15*c^10*e^2 - 40*a^6*b^18*c*e^2 + 720*a^7*b^16*c^2
*e^2 - 7680*a^8*b^14*c^3*e^2 + 53760*a^9*b^12*c^4*e^2 - 258048*a^10*b^10*c^5*e^2 + 860160*a^11*b^8*c^6*e^2 - 1
966080*a^12*b^6*c^7*e^2 + 2949120*a^13*b^4*c^8*e^2 - 2621440*a^14*b^2*c^9*e^2)))^(1/2) + (x*(14112*a^4*c^7*e^1
2 + 9*b^8*c^3*e^12 - 180*a*b^6*c^4*e^12 + 1530*a^2*b^4*c^5*e^12 - 6192*a^3*b^2*c^6*e^12))/(32*(a^4*b^8 + 256*a
^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3)))*(-(9*(b^19 + b^4*(-(4*a*c - b^2)^15)^(1/2) - 17203
20*a^9*b*c^9 + 769*a^2*b^15*c^2 - 8620*a^3*b^13*c^3 + 63440*a^4*b^11*c^4 - 316864*a^5*b^9*c^5 + 1069824*a^6*b^
7*c^6 - 2343936*a^7*b^5*c^7 + 3010560*a^8*b^3*c^8 + 49*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 41*a*b^17*c - 11*a*
b^2*c*(-(4*a*c - b^2)^15)^(1/2)))/(512*(a^5*b^20*e^2 + 1048576*a^15*c^10*e^2 - 40*a^6*b^18*c*e^2 + 720*a^7*b^1
6*c^2*e^2 - 7680*a^8*b^14*c^3*e^2 + 53760*a^9*b^12*c^4*e^2 - 258048*a^10*b^10*c^5*e^2 + 860160*a^11*b^8*c^6*e^
2 - 1966080*a^12*b^6*c^7*e^2 + 2949120*a^13*b^4*c^8*e^2 - 2621440*a^14*b^2*c^9*e^2)))^(1/2)*1i + ((3612672*a^6
*c^9*d*e^11 + 144*b^12*c^3*d*e^11 - 4032*a*b^10...

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