3.7.54 \(\int \frac {(d f+e f x)^4}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [654]

Optimal. Leaf size=353 \[ \frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {f^4 (d+e x) \left (7 b^2-4 a c+12 b c (d+e x)^2\right )}{8 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 \sqrt {c} \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {3 \sqrt {c} \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

1/4*f^4*(e*x+d)*(2*a+b*(e*x+d)^2)/(-4*a*c+b^2)/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2-1/8*f^4*(e*x+d)*(7*b^2-4*a*c+12
*b*c*(e*x+d)^2)/(-4*a*c+b^2)^2/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)+3/8*f^4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a*c
+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^2+4*a*c-2*b*(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)/e*2^(1/2)/(b-(-4*a*c+b^2)^
(1/2))^(1/2)-3/8*f^4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^2+4*a*c+2*b*(-4
*a*c+b^2)^(1/2))/(-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 = 0.59, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1156, 1134, 1192, 1180, 211} \begin {gather*} \frac {3 \sqrt {c} f^4 \left (-2 b \sqrt {b^2-4 a c}+4 a c+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} e \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} f^4 \left (2 b \sqrt {b^2-4 a c}+4 a c+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{4 \sqrt {2} e \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {f^4 (d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{8 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^4*(d + e*x)*(2*a + b*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) - (f^4*(d + e*
x)*(7*b^2 - 4*a*c + 12*b*c*(d + e*x)^2))/(8*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[c
]*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c])*f^4*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])
/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (3*Sqrt[c]*(3*b^2 + 4*a*c + 2*b*Sqrt[b^2 - 4*
a*c])*f^4*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt
[b + Sqrt[b^2 - 4*a*c]]*e)

Rule 211

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

Rule 1134

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

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1180

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

Rule 1192

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

Rubi steps

\begin {align*} \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {f^4 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e}\\ &=\frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {f^4 \text {Subst}\left (\int \frac {2 a-5 b x^2}{\left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{4 \left (b^2-4 a c\right ) e}\\ &=\frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {f^4 (d+e x) \left (7 b^2-4 a c+12 b c (d+e x)^2\right )}{8 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {f^4 \text {Subst}\left (\int \frac {3 a \left (b^2+4 a c\right )-12 a b c x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{8 a \left (b^2-4 a c\right )^2 e}\\ &=\frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {f^4 (d+e x) \left (7 b^2-4 a c+12 b c (d+e x)^2\right )}{8 \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 (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) f^4\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{8 \left (b^2-4 a c\right )^{5/2} e}-\frac {\left (3 c \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) f^4\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{8 \left (b^2-4 a c\right )^{5/2} e}\\ &=\frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {f^4 (d+e x) \left (7 b^2-4 a c+12 b c (d+e x)^2\right )}{8 \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 (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {3 \sqrt {c} \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {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 = 2.75, size = 331, normalized size = 0.94 \begin {gather*} \frac {f^4 \left (-\frac {2 \left (-2 a (d+e x)-b (d+e x)^3\right )}{\left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (-7 b^2+4 a c-12 b c (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 (3 b^2+4 a c-2 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 )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (3 b^2+4 a c+2 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 )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^4*((-2*(-2*a*(d + e*x) - b*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + ((d + e*x)
*(-7*b^2 + 4*a*c - 12*b*c*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[2]*Sqr
t[c]*(3*b^2 + 4*a*c - 2*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c + 2*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a
*c]])))/(8*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.24, size = 708, normalized size = 2.01

method	result	size
default	\(f^{4} \left (\frac {-\frac {3 c^{2} e^{6} b \,x^{7}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {21 c^{2} d \,e^{5} b \,x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (-252 b c \,d^{2}+4 a c -19 b^{2}\right ) c \,e^{4} x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {5 c d \,e^{3} \left (-84 b c \,d^{2}+4 a c -19 b^{2}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {e^{2} \left (420 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+16 a b c +5 b^{3}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d e \left (252 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+48 a b c +15 b^{3}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (84 b \,c^{2} d^{6}-20 a \,c^{2} d^{4}+95 b^{2} c \,d^{4}+48 a b c \,d^{2}+15 b^{3} d^{2}+12 a^{2} c +3 a \,b^{2}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d \left (12 b \,c^{2} d^{6}-4 a \,c^{2} d^{4}+19 b^{2} c \,d^{4}+16 a b c \,d^{2}+5 b^{3} d^{2}+12 a^{2} c +3 a \,b^{2}\right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-4 \textit {\_R}^{2} b c \,e^{2}-8 \textit {\_R} b c d e -4 b c \,d^{2}+4 a c +b^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\right )\)	\(708\)
risch	\(\frac {-\frac {3 c^{2} e^{6} b \,f^{4} x^{7}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {21 c^{2} d \,e^{5} b \,f^{4} x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (-252 b c \,d^{2}+4 a c -19 b^{2}\right ) c \,e^{4} f^{4} x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {5 c d \,e^{3} f^{4} \left (-84 b c \,d^{2}+4 a c -19 b^{2}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {e^{2} f^{4} \left (420 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+16 a b c +5 b^{3}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d e \,f^{4} \left (252 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+48 a b c +15 b^{3}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {f^{4} \left (84 b \,c^{2} d^{6}-20 a \,c^{2} d^{4}+95 b^{2} c \,d^{4}+48 a b c \,d^{2}+15 b^{3} d^{2}+12 a^{2} c +3 a \,b^{2}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d \,f^{4} \left (12 b \,c^{2} d^{6}-4 a \,c^{2} d^{4}+19 b^{2} c \,d^{4}+16 a b c \,d^{2}+5 b^{3} d^{2}+12 a^{2} c +3 a \,b^{2}\right )}{8 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 {3 f^{4} \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\frac {4 b c \,e^{2} \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {8 d b c e \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {-4 b c \,d^{2}+4 a c +b^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{16 e}\)	\(775\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^4*((-3/2*c^2*e^6*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7-21/2*c^2*d*e^5*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/8*(-252*
b*c*d^2+4*a*c-19*b^2)*c*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+5/8*c*d*e^3*(-84*b*c*d^2+4*a*c-19*b^2)/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^4-1/8*e^2*(420*b*c^2*d^4-40*a*c^2*d^2+190*b^2*c*d^2+16*a*b*c+5*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^3-1/8*d*e*(252*b*c^2*d^4-40*a*c^2*d^2+190*b^2*c*d^2+48*a*b*c+15*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/8*(8
4*b*c^2*d^6-20*a*c^2*d^4+95*b^2*c*d^4+48*a*b*c*d^2+15*b^3*d^2+12*a^2*c+3*a*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1
/8*d/e*(12*b*c^2*d^6-4*a*c^2*d^4+19*b^2*c*d^4+16*a*b*c*d^2+5*b^3*d^2+12*a^2*c+3*a*b^2)/(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+3/16/(16*a^2*c^
2-8*a*b^2*c+b^4)/e*sum((-4*_R^2*b*c*e^2-8*_R*b*c*d*e-4*b*c*d^2+4*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(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b
*d*e)*_Z+d^4*c+d^2*b+a)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*f^4*integrate((4*b*c*x^2*e^2 + 8*b*c*d*x*e + 4*b*c*d^2 - b^2 - 4*a*c)/(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)/(b^4 - 8*a*b^2*c + 16*a^2*c^2) - 1/8*(1
2*b*c^2*f^4*x^7*e^7 + 84*b*c^2*d*f^4*x^6*e^6 + (252*b*c^2*d^2*e^5 + 19*b^2*c*e^5 - 4*a*c^2*e^5)*f^4*x^5 + 5*(8
4*b*c^2*d^3*e^4 + (19*b^2*c*e^4 - 4*a*c^2*e^4)*d)*f^4*x^4 + (420*b*c^2*d^4*e^3 + 5*b^3*e^3 + 16*a*b*c*e^3 + 10
*(19*b^2*c*e^3 - 4*a*c^2*e^3)*d^2)*f^4*x^3 + (252*b*c^2*d^5*e^2 + 10*(19*b^2*c*e^2 - 4*a*c^2*e^2)*d^3 + 3*(5*b
^3*e^2 + 16*a*b*c*e^2)*d)*f^4*x^2 + (84*b*c^2*d^6*e + 5*(19*b^2*c*e - 4*a*c^2*e)*d^4 + 3*a*b^2*e + 12*a^2*c*e
+ 3*(5*b^3*e + 16*a*b*c*e)*d^2)*f^4*x + (12*b*c^2*d^7 + (19*b^2*c - 4*a*c^2)*d^5 + (5*b^3 + 16*a*b*c)*d^3 + 3*
(a*b^2 + 4*a^2*c)*d)*f^4)/((b^4*c^2*e - 8*a*b^2*c^3*e + 16*a^2*c^4*e)*d^8 + 8*(b^4*c^2*e^8 - 8*a*b^2*c^3*e^8 +
16*a^2*c^4*e^8)*d*x^7 + (b^4*c^2*e^9 - 8*a*b^2*c^3*e^9 + 16*a^2*c^4*e^9)*x^8 + 2*(b^5*c*e - 8*a*b^3*c^2*e + 1
6*a^2*b*c^3*e)*d^6 + 2*(b^5*c*e^7 - 8*a*b^3*c^2*e^7 + 16*a^2*b*c^3*e^7 + 14*(b^4*c^2*e^7 - 8*a*b^2*c^3*e^7 + 1
6*a^2*c^4*e^7)*d^2)*x^6 + a^2*b^4*e - 8*a^3*b^2*c*e + 16*a^4*c^2*e + 4*(14*(b^4*c^2*e^6 - 8*a*b^2*c^3*e^6 + 16
*a^2*c^4*e^6)*d^3 + 3*(b^5*c*e^6 - 8*a*b^3*c^2*e^6 + 16*a^2*b*c^3*e^6)*d)*x^5 + (b^6*e - 6*a*b^4*c*e + 32*a^3*
c^3*e)*d^4 + (b^6*e^5 - 6*a*b^4*c*e^5 + 32*a^3*c^3*e^5 + 70*(b^4*c^2*e^5 - 8*a*b^2*c^3*e^5 + 16*a^2*c^4*e^5)*d
^4 + 30*(b^5*c*e^5 - 8*a*b^3*c^2*e^5 + 16*a^2*b*c^3*e^5)*d^2)*x^4 + 4*(14*(b^4*c^2*e^4 - 8*a*b^2*c^3*e^4 + 16*
a^2*c^4*e^4)*d^5 + 10*(b^5*c*e^4 - 8*a*b^3*c^2*e^4 + 16*a^2*b*c^3*e^4)*d^3 + (b^6*e^4 - 6*a*b^4*c*e^4 + 32*a^3
*c^3*e^4)*d)*x^3 + 2*(a*b^5*e - 8*a^2*b^3*c*e + 16*a^3*b*c^2*e)*d^2 + 2*(14*(b^4*c^2*e^3 - 8*a*b^2*c^3*e^3 + 1
6*a^2*c^4*e^3)*d^6 + a*b^5*e^3 - 8*a^2*b^3*c*e^3 + 16*a^3*b*c^2*e^3 + 15*(b^5*c*e^3 - 8*a*b^3*c^2*e^3 + 16*a^2
*b*c^3*e^3)*d^4 + 3*(b^6*e^3 - 6*a*b^4*c*e^3 + 32*a^3*c^3*e^3)*d^2)*x^2 + 4*(2*(b^4*c^2*e^2 - 8*a*b^2*c^3*e^2
+ 16*a^2*c^4*e^2)*d^7 + 3*(b^5*c*e^2 - 8*a*b^3*c^2*e^2 + 16*a^2*b*c^3*e^2)*d^5 + (b^6*e^2 - 6*a*b^4*c*e^2 + 32
*a^3*c^3*e^2)*d^3 + (a*b^5*e^2 - 8*a^2*b^3*c*e^2 + 16*a^3*b*c^2*e^2)*d)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6666 vs. \(2 (311) = 622\).
time = 0.56, size = 6666, normalized size = 18.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(24*b*c^2*f^4*x^7*e^7 + 168*b*c^2*d*f^4*x^6*e^6 + 2*(252*b*c^2*d^2 + 19*b^2*c - 4*a*c^2)*f^4*x^5*e^5 + 1
0*(84*b*c^2*d^3 + (19*b^2*c - 4*a*c^2)*d)*f^4*x^4*e^4 + 2*(420*b*c^2*d^4 + 5*b^3 + 16*a*b*c + 10*(19*b^2*c - 4
*a*c^2)*d^2)*f^4*x^3*e^3 + 2*(252*b*c^2*d^5 + 10*(19*b^2*c - 4*a*c^2)*d^3 + 3*(5*b^3 + 16*a*b*c)*d)*f^4*x^2*e^
2 + 2*(84*b*c^2*d^6 + 5*(19*b^2*c - 4*a*c^2)*d^4 + 3*a*b^2 + 12*a^2*c + 3*(5*b^3 + 16*a*b*c)*d^2)*f^4*x*e + 2*
(12*b*c^2*d^7 + (19*b^2*c - 4*a*c^2)*d^5 + (5*b^3 + 16*a*b*c)*d^3 + 3*(a*b^2 + 4*a^2*c)*d)*f^4 + 3*sqrt(1/2)*(
(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8*e^9 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^7*e^8 + 2*(b^5*c - 8
*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*x^6*e^7 + 4*(14*(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^5*e^6 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(
b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*x^4*e^5 + 4*(14*(b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^
3)*d)*x^3*e^4 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*
c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*x^2*e^3 + 4*(2*(b^4*c^2 - 8*a*b^2*
c^3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^3 + (a*b
^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*x*e^2 + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2
+ 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*
a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(-((b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*f^8 + sqrt(f^16/(a^2*b^10 - 20*a^3*
b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5))*(a*b^10 - 20*a^2*b^8*c + 160*a^3
*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5))*e^(-2)/(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2
- 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5))*log(27*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*f^12*x*e +
27*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*f^12 + 27/2*sqrt(1/2)*((b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^
4*c^4)*f^8*e - sqrt(f^16/(a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 102
4*a^7*c^5))*(a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5
- 12288*a^7*b*c^6)*e)*sqrt(-((b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*f^8 + sqrt(f^16/(a^2*b^10 - 20*a^3*b^8*c + 160*
a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5))*(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 6
40*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5))*e^(-2)/(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b
^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5))) - 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8*e^9 + 8*(b
^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^7*e^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c
^3 + 16*a^2*c^4)*d^2)*x^6*e^7 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a
^2*b*c^3)*d)*x^5*e^6 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c
- 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*x^4*e^5 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a
*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d)*x^3*e^4 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a
^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*
b^4*c + 32*a^3*c^3)*d^2)*x^2*e^3 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16
*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*x*e^2 + ((b^4*c
^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a
^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(-((b^5 + 40*
a*b^3*c + 80*a^2*b*c^2)*f^8 + sqrt(f^16/(a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^
6*b^2*c^4 - 1024*a^7*c^5))*(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 102
4*a^6*c^5))*e^(-2)/(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^
5))*log(27*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*f^12*x*e + 27*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*f^12 -
27/2*sqrt(1/2)*((b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*f^8*e - sqrt(f^16/(a^2*b^10 - 20*a^3*b^8*c +
160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5))*(a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2
+ 1280*a^4*b^7*c^3 - 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e)*sqrt(-((b^5 + 40*a*b^3*c + 80
*a^2*b*c^2)*f^8 + sqrt(f^16/(a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 -
1024*a^7*c^5))*(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5))*
e^(-2)/(a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2...

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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((e*f*x+d*f)**4/(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. 1844 vs. \(2 (311) = 622\).
time = 3.26, size = 1844, normalized size = 5.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*((4*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*c*f^4*e^2 - 8*(d*e^(-1) +
sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*f^4*e + 4*b*c*d^2*f^4 - b^2*f^4 - 4*a*c*f^4)*
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))) + (4*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b
*c*f^4*e^2 - 8*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*f^4*e + 4*b*c*d^2*
f^4 - b^2*f^4 - 4*a*c*f^4)*log(d*e^(-1) + x - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d
*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(
b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sq
rt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))) + (4*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))^2*b*c*f^4*e^2 - 8*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c
))*b*c*d*f^4*e + 4*b*c*d^2*f^4 - b^2*f^4 - 4*a*c*f^4)*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))) + (4*(d*e^(-1) - sqrt(1/2
)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*c*f^4*e^2 - 8*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt
(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*f^4*e + 4*b*c*d^2*f^4 - b^2*f^4 - 4*a*c*f^4)*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))
))/(b^4 - 8*a*b^2*c + 16*a^2*c^2) - 1/8*(12*b*c^2*f^4*x^7*e^7 + 84*b*c^2*d*f^4*x^6*e^6 + 252*b*c^2*d^2*f^4*x^5
*e^5 + 420*b*c^2*d^3*f^4*x^4*e^4 + 420*b*c^2*d^4*f^4*x^3*e^3 + 252*b*c^2*d^5*f^4*x^2*e^2 + 84*b*c^2*d^6*f^4*x*
e + 12*b*c^2*d^7*f^4 + 19*b^2*c*f^4*x^5*e^5 - 4*a*c^2*f^4*x^5*e^5 + 95*b^2*c*d*f^4*x^4*e^4 - 20*a*c^2*d*f^4*x^
4*e^4 + 190*b^2*c*d^2*f^4*x^3*e^3 - 40*a*c^2*d^2*f^4*x^3*e^3 + 190*b^2*c*d^3*f^4*x^2*e^2 - 40*a*c^2*d^3*f^4*x^
2*e^2 + 95*b^2*c*d^4*f^4*x*e - 20*a*c^2*d^4*f^4*x*e + 19*b^2*c*d^5*f^4 - 4*a*c^2*d^5*f^4 + 5*b^3*f^4*x^3*e^3 +
16*a*b*c*f^4*x^3*e^3 + 15*b^3*d*f^4*x^2*e^2 + 48*a*b*c*d*f^4*x^2*e^2 + 15*b^3*d^2*f^4*x*e + 48*a*b*c*d^2*f^4*
x*e + 5*b^3*d^3*f^4 + 16*a*b*c*d^3*f^4 + 3*a*b^2*f^4*x*e + 12*a^2*c*f^4*x*e + 3*a*b^2*d*f^4 + 12*a^2*c*d*f^4)/
((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^
4*e - 8*a*b^2*c*e + 16*a^2*c^2*e))

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Mupad [B]
time = 7.52, size = 2500, normalized size = 7.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((-(9*(b^15*f^8 + f^8*(-(4*a*c - b^2)^15)^(1/2) - 81920*a^7*b*c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b
^9*c^3*f^8 - 11520*a^4*b^7*c^4*f^8 - 1024*a^5*b^5*c^5*f^8 + 61440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(512*(a*
b^20*e^2 + 1048576*a^11*c^10*e^2 - 40*a^2*b^18*c*e^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^
5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*
c^8*e^2 - 2621440*a^10*b^2*c^9*e^2)))^(1/2)*((((1024*b^15*c^2*d*e^13 - 28672*a*b^13*c^3*d*e^13 - 16777216*a^7*
b*c^9*d*e^13 + 344064*a^2*b^11*c^4*d*e^13 - 2293760*a^3*b^9*c^5*d*e^13 + 9175040*a^4*b^7*c^6*d*e^13 - 22020096
*a^5*b^5*c^7*d*e^13 + 29360128*a^6*b^3*c^8*d*e^13)/(128*(b^12 + 4096*a^6*c^6 + 240*a^2*b^8*c^2 - 1280*a^3*b^6*
c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10*c)) + (x*(128*b^11*c^2*e^14 - 2560*a*b^9*c^3*e^14 - 1310
72*a^5*b*c^7*e^14 + 20480*a^2*b^7*c^4*e^14 - 81920*a^3*b^5*c^5*e^14 + 163840*a^4*b^3*c^6*e^14))/(16*(b^8 + 256
*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))*(-(9*(b^15*f^8 + f^8*(-(4*a*c - b^2)^15)^(1/2) - 8
1920*a^7*b*c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^7*c^4*f^8 - 1024*a^5*b^5*c^5*f^
8 + 61440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(512*(a*b^20*e^2 + 1048576*a^11*c^10*e^2 - 40*a^2*b^18*c*e^2 + 7
20*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^
8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2)))^(1/2) - (786432*a^
6*c^8*e^12*f^4 - 192*b^12*c^2*e^12*f^4 - 15360*a^2*b^8*c^4*e^12*f^4 + 245760*a^4*b^4*c^6*e^12*f^4 - 786432*a^5
*b^2*c^7*e^12*f^4 + 3072*a*b^10*c^3*e^12*f^4)/(128*(b^12 + 4096*a^6*c^6 + 240*a^2*b^8*c^2 - 1280*a^3*b^6*c^3 +
3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10*c)))*(-(9*(b^15*f^8 + f^8*(-(4*a*c - b^2)^15)^(1/2) - 81920*a
^7*b*c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^7*c^4*f^8 - 1024*a^5*b^5*c^5*f^8 + 61
440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(512*(a*b^20*e^2 + 1048576*a^11*c^10*e^2 - 40*a^2*b^18*c*e^2 + 720*a^3
*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*
e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2)))^(1/2) + (18432*a^4*c^7*d
*e^11*f^8 + 936*b^8*c^3*d*e^11*f^8 - 6912*a*b^6*c^4*d*e^11*f^8 + 11520*a^2*b^4*c^5*d*e^11*f^8)/(128*(b^12 + 40
96*a^6*c^6 + 240*a^2*b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10*c)) + (x*(14
4*a^2*c^5*e^12*f^8 + 117*b^4*c^3*e^12*f^8 + 72*a*b^2*c^4*e^12*f^8))/(16*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 -
256*a^3*b^2*c^3 - 16*a*b^6*c)))*1i + (-(9*(b^15*f^8 + f^8*(-(4*a*c - b^2)^15)^(1/2) - 81920*a^7*b*c^7*f^8 - 56
0*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^7*c^4*f^8 - 1024*a^5*b^5*c^5*f^8 + 61440*a^6*b^3*c^6*f
^8 + 20*a*b^13*c*f^8))/(512*(a*b^20*e^2 + 1048576*a^11*c^10*e^2 - 40*a^2*b^18*c*e^2 + 720*a^3*b^16*c^2*e^2 - 7
680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8
*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2)))^(1/2)*((((1024*b^15*c^2*d*e^13 - 28672*a*
b^13*c^3*d*e^13 - 16777216*a^7*b*c^9*d*e^13 + 344064*a^2*b^11*c^4*d*e^13 - 2293760*a^3*b^9*c^5*d*e^13 + 917504
0*a^4*b^7*c^6*d*e^13 - 22020096*a^5*b^5*c^7*d*e^13 + 29360128*a^6*b^3*c^8*d*e^13)/(128*(b^12 + 4096*a^6*c^6 +
240*a^2*b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10*c)) + (x*(128*b^11*c^2*e^
14 - 2560*a*b^9*c^3*e^14 - 131072*a^5*b*c^7*e^14 + 20480*a^2*b^7*c^4*e^14 - 81920*a^3*b^5*c^5*e^14 + 163840*a^
4*b^3*c^6*e^14))/(16*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))*(-(9*(b^15*f^8 + f^
8*(-(4*a*c - b^2)^15)^(1/2) - 81920*a^7*b*c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^
7*c^4*f^8 - 1024*a^5*b^5*c^5*f^8 + 61440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(512*(a*b^20*e^2 + 1048576*a^11*c
^10*e^2 - 40*a^2*b^18*c*e^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a
^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^
2*c^9*e^2)))^(1/2) + (786432*a^6*c^8*e^12*f^4 - 192*b^12*c^2*e^12*f^4 - 15360*a^2*b^8*c^4*e^12*f^4 + 245760*a^
4*b^4*c^6*e^12*f^4 - 786432*a^5*b^2*c^7*e^12*f^4 + 3072*a*b^10*c^3*e^12*f^4)/(128*(b^12 + 4096*a^6*c^6 + 240*a
^2*b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10*c)))*(-(9*(b^15*f^8 + f^8*(-(4
*a*c - b^2)^15)^(1/2) - 81920*a^7*b*c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^7*c^4*
f^8 - 1024*a^5*b^5*c^5*f^8 + 61440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(512*(a*b^20*e^2 + 1048576*a^11*c^10*e^
2 - 40*a^2*b^18*c*e^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^1
0*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*
e^2)))^(1/2) + (18432*a^4*c^7*d*e^11*f^8 + 936*...

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