3.7.57 \(\int \frac {d f+e f x}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [657]

Optimal. Leaf size=153 \[ -\frac {f \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {3 c f \left (b+2 c (d+e x)^2\right )}{2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {6 c^2 f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} e} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1156, 1121, 628, 632, 212} \begin {gather*} -\frac {6 c^2 f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}}+\frac {3 c f \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {f \left (b+2 c (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)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]

[Out]

-1/4*(f*(b + 2*c*(d + e*x)^2))/((b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*c*f*(b + 2*c*(d +
e*x)^2))/(2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (6*c^2*f*ArcTanh[(b + 2*c*(d + e*x)^2)/Sq
rt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*e)

Rule 212

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

Rule 628

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

Rule 632

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

Rule 1121

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

Rule 1156

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

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 148, normalized size = 0.97 \begin {gather*} \frac {f \left (\frac {\left (b^2-4 a c\right ) \left (-b-2 c (d+e x)^2\right )}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {6 c \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}+\frac {24 c^2 \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}\right )}{4 \left (b^2-4 a c\right )^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f*(((b^2 - 4*a*c)*(-b - 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2 + (6*c*(b + 2*c*(d + e*x)^2))
/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + (24*c^2*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4
*a*c]))/(4*(b^2 - 4*a*c)^2*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 543, normalized size = 3.55

method	result	size
default	\(f \left (\frac {\frac {3 c^{3} e^{5} x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {18 e^{4} c^{3} d \,x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 c^{2} e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 c^{2} d \,e^{2} \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 c d \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {12 c^{3} d^{6}+18 b \,c^{2} d^{4}+20 a \,c^{2} d^{2}+4 b^{2} c \,d^{2}+10 a b c -b^{3}}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}+\frac {3 c^{2} \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\right )\)	\(543\)
risch	\(\frac {\frac {3 c^{3} e^{5} f \,x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {18 f \,e^{4} c^{3} d \,x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 c^{2} e^{3} f \left (10 c \,d^{2}+b \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 d \,e^{2} c^{2} f \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c e f \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 d c f \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {f \left (12 c^{3} d^{6}+18 b \,c^{2} d^{4}+20 a \,c^{2} d^{2}+4 b^{2} c \,d^{2}+10 a b c -b^{3}\right )}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}-\frac {3 f \,c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}-16 e^{2} c^{2} a^{2} b +8 a c \,e^{2} b^{3}-e^{2} b^{5}\right ) x^{2}+\left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e -32 a^{2} b \,c^{2} d e +16 a \,b^{3} c d e -2 b^{5} d e \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}-16 a^{2} b \,c^{2} d^{2}+8 a \,b^{3} c \,d^{2}-b^{5} d^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}+\frac {3 f \,c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}+16 e^{2} c^{2} a^{2} b -8 a c \,e^{2} b^{3}+e^{2} b^{5}\right ) x^{2}+\left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e +32 a^{2} b \,c^{2} d e -16 a \,b^{3} c d e +2 b^{5} d e \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}+16 a^{2} b \,c^{2} d^{2}-8 a \,b^{3} c \,d^{2}+b^{5} d^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}\)	\(756\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*((3*c^3*e^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+18*e^4*c^3*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+9/2*c^2*e^3*(10*c*d^2
+b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+6*c^2*d*e^2*(10*c*d^2+3*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+c*e*(45*c^2*d^4+2
7*b*c*d^2+5*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+2*c*d*(9*c^2*d^4+9*b*c*d^2+5*a*c+b^2)/(16*a^2*c^2-8*a*b^2*
c+b^4)*x+1/4/e*(12*c^3*d^6+18*b*c^2*d^4+20*a*c^2*d^2+4*b^2*c*d^2+10*a*b*c-b^3)/(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*c^2/(16*a^2*c^2-8*a*b
^2*c+b^4)/e*sum((_R*e+d)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(e^4*
c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+a)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*c^2*f*integrate((x*e + d)/(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) + 1/4*(12*c^3*f*x^6*e^6 + 72*c^3*d*f*x^5*e^5 + 18*(10*c
^3*d^2*e^4 + b*c^2*e^4)*f*x^4 + 24*(10*c^3*d^3*e^3 + 3*b*c^2*d*e^3)*f*x^3 + 4*(45*c^3*d^4*e^2 + 27*b*c^2*d^2*e
^2 + b^2*c*e^2 + 5*a*c^2*e^2)*f*x^2 + 8*(9*c^3*d^5*e + 9*b*c^2*d^3*e + (b^2*c*e + 5*a*c^2*e)*d)*f*x + (12*c^3*
d^6 + 18*b*c^2*d^4 - b^3 + 10*a*b*c + 4*(b^2*c + 5*a*c^2)*d^2)*f)/((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 + 16*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 + 16*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 + 16*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. 1791 vs. \(2 (149) = 298\).
time = 0.48, size = 3710, normalized size = 24.25 \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)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas")

[Out]

[1/4*(12*(b^2*c^3 - 4*a*c^4)*f*x^6*e^6 + 72*(b^2*c^3 - 4*a*c^4)*d*f*x^5*e^5 + 18*(b^3*c^2 - 4*a*b*c^3 + 10*(b^
2*c^3 - 4*a*c^4)*d^2)*f*x^4*e^4 + 24*(10*(b^2*c^3 - 4*a*c^4)*d^3 + 3*(b^3*c^2 - 4*a*b*c^3)*d)*f*x^3*e^3 + 4*(b
^4*c + a*b^2*c^2 - 20*a^2*c^3 + 45*(b^2*c^3 - 4*a*c^4)*d^4 + 27*(b^3*c^2 - 4*a*b*c^3)*d^2)*f*x^2*e^2 + 8*(9*(b
^2*c^3 - 4*a*c^4)*d^5 + 9*(b^3*c^2 - 4*a*b*c^3)*d^3 + (b^4*c + a*b^2*c^2 - 20*a^2*c^3)*d)*f*x*e + 12*(c^4*f*x^
8*e^8 + 8*c^4*d*f*x^7*e^7 + 2*(14*c^4*d^2 + b*c^3)*f*x^6*e^6 + 4*(14*c^4*d^3 + 3*b*c^3*d)*f*x^5*e^5 + (70*c^4*
d^4 + 30*b*c^3*d^2 + b^2*c^2 + 2*a*c^3)*f*x^4*e^4 + 4*(14*c^4*d^5 + 10*b*c^3*d^3 + (b^2*c^2 + 2*a*c^3)*d)*f*x^
3*e^3 + 2*(14*c^4*d^6 + 15*b*c^3*d^4 + a*b*c^2 + 3*(b^2*c^2 + 2*a*c^3)*d^2)*f*x^2*e^2 + 4*(2*c^4*d^7 + 3*b*c^3
*d^5 + a*b*c^2*d + (b^2*c^2 + 2*a*c^3)*d^3)*f*x*e + (c^4*d^8 + 2*b*c^3*d^6 + 2*a*b*c^2*d^2 + (b^2*c^2 + 2*a*c^
3)*d^4 + a^2*c^2)*f)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4*e^4 + 8*c^2*d*x^3*e^3 + 2*c^2*d^4 + 2*b*c*d^2 + 2*(6*c^2
*d^2 + b*c)*x^2*e^2 + 4*(2*c^2*d^3 + b*c*d)*x*e + b^2 - 2*a*c - (2*c*x^2*e^2 + 4*c*d*x*e + 2*c*d^2 + b)*sqrt(b
^2 - 4*a*c))/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c*d^2 + b)*x^2*e^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a))
+ (12*(b^2*c^3 - 4*a*c^4)*d^6 - b^5 + 14*a*b^3*c - 40*a^2*b*c^2 + 18*(b^3*c^2 - 4*a*b*c^3)*d^4 + 4*(b^4*c + a*
b^2*c^2 - 20*a^2*c^3)*d^2)*f)/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8*e^9 + 8*(b^6*c^2 - 1
2*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*x^7*e^8 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4
+ 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*x^6*e^7 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48
*a^2*b^2*c^4 - 64*a^3*c^5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*x^5*e^6 + (b^8 -
10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*
a^3*c^5)*d^4 + 30*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*x^4*e^5 + 4*(14*(b^6*c^2 - 12*a*
b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (
b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d)*x^3*e^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^
3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^
5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c
^4)*d^2)*x^2*e^3 + 4*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2 +
48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3 +
(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d)*x*e^2 + ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 -
64*a^3*c^5)*d^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3
*c^3 - 64*a^3*b*c^4)*d^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^4 + 2*(a*b^7 -
12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2)*e), 1/4*(12*(b^2*c^3 - 4*a*c^4)*f*x^6*e^6 + 72*(b^2*c^3 -
4*a*c^4)*d*f*x^5*e^5 + 18*(b^3*c^2 - 4*a*b*c^3 + 10*(b^2*c^3 - 4*a*c^4)*d^2)*f*x^4*e^4 + 24*(10*(b^2*c^3 - 4*a
*c^4)*d^3 + 3*(b^3*c^2 - 4*a*b*c^3)*d)*f*x^3*e^3 + 4*(b^4*c + a*b^2*c^2 - 20*a^2*c^3 + 45*(b^2*c^3 - 4*a*c^4)*
d^4 + 27*(b^3*c^2 - 4*a*b*c^3)*d^2)*f*x^2*e^2 + 8*(9*(b^2*c^3 - 4*a*c^4)*d^5 + 9*(b^3*c^2 - 4*a*b*c^3)*d^3 + (
b^4*c + a*b^2*c^2 - 20*a^2*c^3)*d)*f*x*e - 24*(c^4*f*x^8*e^8 + 8*c^4*d*f*x^7*e^7 + 2*(14*c^4*d^2 + b*c^3)*f*x^
6*e^6 + 4*(14*c^4*d^3 + 3*b*c^3*d)*f*x^5*e^5 + (70*c^4*d^4 + 30*b*c^3*d^2 + b^2*c^2 + 2*a*c^3)*f*x^4*e^4 + 4*(
14*c^4*d^5 + 10*b*c^3*d^3 + (b^2*c^2 + 2*a*c^3)*d)*f*x^3*e^3 + 2*(14*c^4*d^6 + 15*b*c^3*d^4 + a*b*c^2 + 3*(b^2
*c^2 + 2*a*c^3)*d^2)*f*x^2*e^2 + 4*(2*c^4*d^7 + 3*b*c^3*d^5 + a*b*c^2*d + (b^2*c^2 + 2*a*c^3)*d^3)*f*x*e + (c^
4*d^8 + 2*b*c^3*d^6 + 2*a*b*c^2*d^2 + (b^2*c^2 + 2*a*c^3)*d^4 + a^2*c^2)*f)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^
2*e^2 + 4*c*d*x*e + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (12*(b^2*c^3 - 4*a*c^4)*d^6 - b^5 + 14*a*
b^3*c - 40*a^2*b*c^2 + 18*(b^3*c^2 - 4*a*b*c^3)*d^4 + 4*(b^4*c + a*b^2*c^2 - 20*a^2*c^3)*d^2)*f)/((b^6*c^2 - 1
2*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8*e^9 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*
d*x^7*e^8 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*
c^4 - 64*a^3*c^5)*d^2)*x^6*e^7 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^3 + 3*(b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*x^5*e^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^
3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b^7*c - 12*a*b^5*c^2 + 4
8*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*x^4*e^5 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^5
+ 10*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*
b^2*c^3 - 128*a^4*c^4)*d)*x^3*e^4 + 2*(a*b^7 - ...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1707 vs. \(2 (139) = 278\).
time = 7.70, size = 1707, normalized size = 11.16 \begin {gather*} - \frac {3 c^{2} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 192 a^{3} c^{5} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{2} c^{4} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{4} c^{3} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{6} c^{2} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2} f + 6 c^{3} d^{2} f}{6 c^{3} e^{2} f} \right )}}{e} + \frac {3 c^{2} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {192 a^{3} c^{5} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{2} c^{4} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{4} c^{3} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{6} c^{2} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2} f + 6 c^{3} d^{2} f}{6 c^{3} e^{2} f} \right )}}{e} + \frac {10 a b c f + 20 a c^{2} d^{2} f - b^{3} f + 4 b^{2} c d^{2} f + 18 b c^{2} d^{4} f + 12 c^{3} d^{6} f + 72 c^{3} d e^{5} f x^{5} + 12 c^{3} e^{6} f x^{6} + x^{4} \cdot \left (18 b c^{2} e^{4} f + 180 c^{3} d^{2} e^{4} f\right ) + x^{3} \cdot \left (72 b c^{2} d e^{3} f + 240 c^{3} d^{3} e^{3} f\right ) + x^{2} \cdot \left (20 a c^{2} e^{2} f + 4 b^{2} c e^{2} f + 108 b c^{2} d^{2} e^{2} f + 180 c^{3} d^{4} e^{2} f\right ) + x \left (40 a c^{2} d e f + 8 b^{2} c d e f + 72 b c^{2} d^{3} e f + 72 c^{3} d^{5} e f\right )}{64 a^{4} c^{2} e - 32 a^{3} b^{2} c e + 128 a^{3} b c^{2} d^{2} e + 128 a^{3} c^{3} d^{4} e + 4 a^{2} b^{4} e - 64 a^{2} b^{3} c d^{2} e + 128 a^{2} b c^{3} d^{6} e + 64 a^{2} c^{4} d^{8} e + 8 a b^{5} d^{2} e - 24 a b^{4} c d^{4} e - 64 a b^{3} c^{2} d^{6} e - 32 a b^{2} c^{3} d^{8} e + 4 b^{6} d^{4} e + 8 b^{5} c d^{6} e + 4 b^{4} c^{2} d^{8} e + x^{8} \cdot \left (64 a^{2} c^{4} e^{9} - 32 a b^{2} c^{3} e^{9} + 4 b^{4} c^{2} e^{9}\right ) + x^{7} \cdot \left (512 a^{2} c^{4} d e^{8} - 256 a b^{2} c^{3} d e^{8} + 32 b^{4} c^{2} d e^{8}\right ) + x^{6} \cdot \left (128 a^{2} b c^{3} e^{7} + 1792 a^{2} c^{4} d^{2} e^{7} - 64 a b^{3} c^{2} e^{7} - 896 a b^{2} c^{3} d^{2} e^{7} + 8 b^{5} c e^{7} + 112 b^{4} c^{2} d^{2} e^{7}\right ) + x^{5} \cdot \left (768 a^{2} b c^{3} d e^{6} + 3584 a^{2} c^{4} d^{3} e^{6} - 384 a b^{3} c^{2} d e^{6} - 1792 a b^{2} c^{3} d^{3} e^{6} + 48 b^{5} c d e^{6} + 224 b^{4} c^{2} d^{3} e^{6}\right ) + x^{4} \cdot \left (128 a^{3} c^{3} e^{5} + 1920 a^{2} b c^{3} d^{2} e^{5} + 4480 a^{2} c^{4} d^{4} e^{5} - 24 a b^{4} c e^{5} - 960 a b^{3} c^{2} d^{2} e^{5} - 2240 a b^{2} c^{3} d^{4} e^{5} + 4 b^{6} e^{5} + 120 b^{5} c d^{2} e^{5} + 280 b^{4} c^{2} d^{4} e^{5}\right ) + x^{3} \cdot \left (512 a^{3} c^{3} d e^{4} + 2560 a^{2} b c^{3} d^{3} e^{4} + 3584 a^{2} c^{4} d^{5} e^{4} - 96 a b^{4} c d e^{4} - 1280 a b^{3} c^{2} d^{3} e^{4} - 1792 a b^{2} c^{3} d^{5} e^{4} + 16 b^{6} d e^{4} + 160 b^{5} c d^{3} e^{4} + 224 b^{4} c^{2} d^{5} e^{4}\right ) + x^{2} \cdot \left (128 a^{3} b c^{2} e^{3} + 768 a^{3} c^{3} d^{2} e^{3} - 64 a^{2} b^{3} c e^{3} + 1920 a^{2} b c^{3} d^{4} e^{3} + 1792 a^{2} c^{4} d^{6} e^{3} + 8 a b^{5} e^{3} - 144 a b^{4} c d^{2} e^{3} - 960 a b^{3} c^{2} d^{4} e^{3} - 896 a b^{2} c^{3} d^{6} e^{3} + 24 b^{6} d^{2} e^{3} + 120 b^{5} c d^{4} e^{3} + 112 b^{4} c^{2} d^{6} e^{3}\right ) + x \left (256 a^{3} b c^{2} d e^{2} + 512 a^{3} c^{3} d^{3} e^{2} - 128 a^{2} b^{3} c d e^{2} + 768 a^{2} b c^{3} d^{5} e^{2} + 512 a^{2} c^{4} d^{7} e^{2} + 16 a b^{5} d e^{2} - 96 a b^{4} c d^{3} e^{2} - 384 a b^{3} c^{2} d^{5} e^{2} - 256 a b^{2} c^{3} d^{7} e^{2} + 16 b^{6} d^{3} e^{2} + 48 b^{5} c d^{5} e^{2} + 32 b^{4} c^{2} d^{7} e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*c**2*f*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/e + x**2 + (-192*a**3*c**5*f*sqrt(-1/(4*a*c - b**2)**5) + 144*a
**2*b**2*c**4*f*sqrt(-1/(4*a*c - b**2)**5) - 36*a*b**4*c**3*f*sqrt(-1/(4*a*c - b**2)**5) + 3*b**6*c**2*f*sqrt(
-1/(4*a*c - b**2)**5) + 3*b*c**2*f + 6*c**3*d**2*f)/(6*c**3*e**2*f))/e + 3*c**2*f*sqrt(-1/(4*a*c - b**2)**5)*l
og(2*d*x/e + x**2 + (192*a**3*c**5*f*sqrt(-1/(4*a*c - b**2)**5) - 144*a**2*b**2*c**4*f*sqrt(-1/(4*a*c - b**2)*
*5) + 36*a*b**4*c**3*f*sqrt(-1/(4*a*c - b**2)**5) - 3*b**6*c**2*f*sqrt(-1/(4*a*c - b**2)**5) + 3*b*c**2*f + 6*
c**3*d**2*f)/(6*c**3*e**2*f))/e + (10*a*b*c*f + 20*a*c**2*d**2*f - b**3*f + 4*b**2*c*d**2*f + 18*b*c**2*d**4*f
+ 12*c**3*d**6*f + 72*c**3*d*e**5*f*x**5 + 12*c**3*e**6*f*x**6 + x**4*(18*b*c**2*e**4*f + 180*c**3*d**2*e**4*
f) + x**3*(72*b*c**2*d*e**3*f + 240*c**3*d**3*e**3*f) + x**2*(20*a*c**2*e**2*f + 4*b**2*c*e**2*f + 108*b*c**2*
d**2*e**2*f + 180*c**3*d**4*e**2*f) + x*(40*a*c**2*d*e*f + 8*b**2*c*d*e*f + 72*b*c**2*d**3*e*f + 72*c**3*d**5*
e*f))/(64*a**4*c**2*e - 32*a**3*b**2*c*e + 128*a**3*b*c**2*d**2*e + 128*a**3*c**3*d**4*e + 4*a**2*b**4*e - 64*
a**2*b**3*c*d**2*e + 128*a**2*b*c**3*d**6*e + 64*a**2*c**4*d**8*e + 8*a*b**5*d**2*e - 24*a*b**4*c*d**4*e - 64*
a*b**3*c**2*d**6*e - 32*a*b**2*c**3*d**8*e + 4*b**6*d**4*e + 8*b**5*c*d**6*e + 4*b**4*c**2*d**8*e + x**8*(64*a
**2*c**4*e**9 - 32*a*b**2*c**3*e**9 + 4*b**4*c**2*e**9) + x**7*(512*a**2*c**4*d*e**8 - 256*a*b**2*c**3*d*e**8
+ 32*b**4*c**2*d*e**8) + x**6*(128*a**2*b*c**3*e**7 + 1792*a**2*c**4*d**2*e**7 - 64*a*b**3*c**2*e**7 - 896*a*b
**2*c**3*d**2*e**7 + 8*b**5*c*e**7 + 112*b**4*c**2*d**2*e**7) + x**5*(768*a**2*b*c**3*d*e**6 + 3584*a**2*c**4*
d**3*e**6 - 384*a*b**3*c**2*d*e**6 - 1792*a*b**2*c**3*d**3*e**6 + 48*b**5*c*d*e**6 + 224*b**4*c**2*d**3*e**6)
+ x**4*(128*a**3*c**3*e**5 + 1920*a**2*b*c**3*d**2*e**5 + 4480*a**2*c**4*d**4*e**5 - 24*a*b**4*c*e**5 - 960*a*
b**3*c**2*d**2*e**5 - 2240*a*b**2*c**3*d**4*e**5 + 4*b**6*e**5 + 120*b**5*c*d**2*e**5 + 280*b**4*c**2*d**4*e**
5) + x**3*(512*a**3*c**3*d*e**4 + 2560*a**2*b*c**3*d**3*e**4 + 3584*a**2*c**4*d**5*e**4 - 96*a*b**4*c*d*e**4 -
1280*a*b**3*c**2*d**3*e**4 - 1792*a*b**2*c**3*d**5*e**4 + 16*b**6*d*e**4 + 160*b**5*c*d**3*e**4 + 224*b**4*c*
*2*d**5*e**4) + x**2*(128*a**3*b*c**2*e**3 + 768*a**3*c**3*d**2*e**3 - 64*a**2*b**3*c*e**3 + 1920*a**2*b*c**3*
d**4*e**3 + 1792*a**2*c**4*d**6*e**3 + 8*a*b**5*e**3 - 144*a*b**4*c*d**2*e**3 - 960*a*b**3*c**2*d**4*e**3 - 89
6*a*b**2*c**3*d**6*e**3 + 24*b**6*d**2*e**3 + 120*b**5*c*d**4*e**3 + 112*b**4*c**2*d**6*e**3) + x*(256*a**3*b*
c**2*d*e**2 + 512*a**3*c**3*d**3*e**2 - 128*a**2*b**3*c*d*e**2 + 768*a**2*b*c**3*d**5*e**2 + 512*a**2*c**4*d**
7*e**2 + 16*a*b**5*d*e**2 - 96*a*b**4*c*d**3*e**2 - 384*a*b**3*c**2*d**5*e**2 - 256*a*b**2*c**3*d**7*e**2 + 16
*b**6*d**3*e**2 + 48*b**5*c*d**5*e**2 + 32*b**4*c**2*d**7*e**2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (149) = 298\).
time = 3.10, size = 445, normalized size = 2.91 \begin {gather*} \frac {6 \, c^{2} f \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right ) e^{\left (-1\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{3} d^{6} f^{5} + 36 \, {\left (f x^{2} e + 2 \, d f x\right )} c^{3} d^{4} f^{4} e + 18 \, b c^{2} d^{4} f^{5} + 36 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} c^{3} d^{2} f^{3} e^{2} + 36 \, {\left (f x^{2} e + 2 \, d f x\right )} b c^{2} d^{2} f^{4} e + 4 \, b^{2} c d^{2} f^{5} + 20 \, a c^{2} d^{2} f^{5} + 12 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} c^{3} f^{2} e^{3} + 18 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} b c^{2} f^{3} e^{2} + 4 \, {\left (f x^{2} e + 2 \, d f x\right )} b^{2} c f^{4} e + 20 \, {\left (f x^{2} e + 2 \, d f x\right )} a c^{2} f^{4} e - b^{3} f^{5} + 10 \, a b c f^{5}}{4 \, {\left (c d^{4} f^{2} + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c d^{2} f e + b d^{2} f^{2} + {\left (f x^{2} e + 2 \, d f x\right )}^{2} c e^{2} + {\left (f x^{2} e + 2 \, d f x\right )} b f e + a f^{2}\right )}^{2} {\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*c^2*f*arctan((2*c*d^2*f + 2*(f*x^2*e + 2*d*f*x)*c*e + b*f)/(sqrt(-b^2 + 4*a*c)*f))*e^(-1)/((b^4 - 8*a*b^2*c
+ 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) + 1/4*(12*c^3*d^6*f^5 + 36*(f*x^2*e + 2*d*f*x)*c^3*d^4*f^4*e + 18*b*c^2*d^4*
f^5 + 36*(f*x^2*e + 2*d*f*x)^2*c^3*d^2*f^3*e^2 + 36*(f*x^2*e + 2*d*f*x)*b*c^2*d^2*f^4*e + 4*b^2*c*d^2*f^5 + 20
*a*c^2*d^2*f^5 + 12*(f*x^2*e + 2*d*f*x)^3*c^3*f^2*e^3 + 18*(f*x^2*e + 2*d*f*x)^2*b*c^2*f^3*e^2 + 4*(f*x^2*e +
2*d*f*x)*b^2*c*f^4*e + 20*(f*x^2*e + 2*d*f*x)*a*c^2*f^4*e - b^3*f^5 + 10*a*b*c*f^5)/((c*d^4*f^2 + 2*(f*x^2*e +
2*d*f*x)*c*d^2*f*e + b*d^2*f^2 + (f*x^2*e + 2*d*f*x)^2*c*e^2 + (f*x^2*e + 2*d*f*x)*b*f*e + a*f^2)^2*(b^4*e -
8*a*b^2*c*e + 16*a^2*c^2*e))

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Mupad [B]
time = 3.99, size = 1199, normalized size = 7.84 \begin {gather*} \frac {\frac {x^2\,\left (e\,f\,b^2\,c+27\,e\,f\,b\,c^2\,d^2+45\,e\,f\,c^3\,d^4+5\,a\,e\,f\,c^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {-f\,b^3+4\,f\,b^2\,c\,d^2+18\,f\,b\,c^2\,d^4+10\,a\,f\,b\,c+12\,f\,c^3\,d^6+20\,a\,f\,c^2\,d^2}{4\,e\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,x^4\,\left (10\,f\,c^3\,d^2\,e^3+b\,f\,c^2\,e^3\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {2\,d\,x\,\left (f\,b^2\,c+9\,f\,b\,c^2\,d^2+9\,f\,c^3\,d^4+5\,a\,f\,c^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {6\,d\,x^3\,\left (10\,f\,c^3\,d^2\,e^2+3\,b\,f\,c^2\,e^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {3\,c^3\,e^5\,f\,x^6}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {18\,c^3\,d\,e^4\,f\,x^5}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (6\,b^2\,d^2\,e^2+30\,b\,c\,d^4\,e^2+2\,a\,b\,e^2+28\,c^2\,d^6\,e^2+12\,a\,c\,d^2\,e^2\right )+x^6\,\left (28\,c^2\,d^2\,e^6+2\,b\,c\,e^6\right )+x\,\left (4\,e\,b^2\,d^3+12\,e\,b\,c\,d^5+4\,a\,e\,b\,d+8\,e\,c^2\,d^7+8\,a\,e\,c\,d^3\right )+x^3\,\left (4\,b^2\,d\,e^3+40\,b\,c\,d^3\,e^3+56\,c^2\,d^5\,e^3+8\,a\,c\,d\,e^3\right )+x^5\,\left (56\,c^2\,d^3\,e^5+12\,b\,c\,d\,e^5\right )+x^4\,\left (b^2\,e^4+30\,b\,c\,d^2\,e^4+70\,c^2\,d^4\,e^4+2\,a\,c\,e^4\right )+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}+\frac {6\,c^2\,f\,\mathrm {atan}\left (\frac {\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )\,\left (x^2\,\left (\frac {36\,c^6\,e^8\,f^2}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {36\,b\,c^4\,f^2\,\left (16\,a^2\,b\,c^4\,e^{10}-8\,a\,b^3\,c^3\,e^{10}+b^5\,c^2\,e^{10}\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+x\,\left (\frac {72\,c^6\,d\,e^7\,f^2}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {72\,b\,c^4\,f^2\,\left (16\,d\,a^2\,b\,c^4\,e^9-8\,d\,a\,b^3\,c^3\,e^9+d\,b^5\,c^2\,e^9\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {36\,c^6\,d^2\,e^6\,f^2}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {36\,b\,c^4\,f^2\,\left (32\,a^3\,c^4\,e^8-16\,a^2\,b^2\,c^3\,e^8+16\,a^2\,b\,c^4\,d^2\,e^8+2\,a\,b^4\,c^2\,e^8-8\,a\,b^3\,c^3\,d^2\,e^8+b^5\,c^2\,d^2\,e^8\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )}{72\,c^6\,e^6\,f^2}\right )}{e\,{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2*(5*a*c^2*e*f + b^2*c*e*f + 45*c^3*d^4*e*f + 27*b*c^2*d^2*e*f))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (12*c^3*
d^6*f - b^3*f + 20*a*c^2*d^2*f + 4*b^2*c*d^2*f + 18*b*c^2*d^4*f + 10*a*b*c*f)/(4*e*(b^4 + 16*a^2*c^2 - 8*a*b^2
*c)) + (9*x^4*(10*c^3*d^2*e^3*f + b*c^2*e^3*f))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (2*d*x*(9*c^3*d^4*f + 5*a
*c^2*f + b^2*c*f + 9*b*c^2*d^2*f))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (6*d*x^3*(10*c^3*d^2*e^2*f + 3*b*c^2*e^2*f
))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (3*c^3*e^5*f*x^6)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (18*c^3*d*e^4*f*x^5)/(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) + (6*c^2*f*atan(((b^4*(4*a*c - b^2)^5 + 16*a^2*c^2*(4*a*c - b^2)^
5 - 8*a*b^2*c*(4*a*c - b^2)^5)*(x^2*((36*c^6*e^8*f^2)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) +
(36*b*c^4*f^2*(b^5*c^2*e^10 - 8*a*b^3*c^3*e^10 + 16*a^2*b*c^4*e^10))/(a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^
2*c^2 - 8*a*b^2*c))) + x*((72*c^6*d*e^7*f^2)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (72*b*c^
4*f^2*(b^5*c^2*d*e^9 - 8*a*b^3*c^3*d*e^9 + 16*a^2*b*c^4*d*e^9))/(a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2
- 8*a*b^2*c))) + (36*c^6*d^2*e^6*f^2)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (36*b*c^4*f^2*(
32*a^3*c^4*e^8 + 2*a*b^4*c^2*e^8 - 16*a^2*b^2*c^3*e^8 + b^5*c^2*d^2*e^8 - 8*a*b^3*c^3*d^2*e^8 + 16*a^2*b*c^4*d
^2*e^8))/(a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))))/(72*c^6*e^6*f^2)))/(e*(4*a*c - b^2)^(5/
2))

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