3.657 \(\int \frac{d f+e f x}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\)
Optimal. Leaf size=153 \[ -\frac{6 c^2 f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}}+\frac{3 c f \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{f \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]
[Out]
-(f*(b + 2*c*(d + e*x)^2))/(4*(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)/Sqrt
[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*e)
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Rubi [A] time = 0.191942, antiderivative size = 153, normalized size of antiderivative = 1.,
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 =
{1142, 1107, 614, 618, 206} \[ -\frac{6 c^2 f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}}+\frac{3 c f \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{f \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]
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]
-(f*(b + 2*c*(d + e*x)^2))/(4*(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)/Sqrt
[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*e)
Rule 1142
Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]
Rule 1107
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 614
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 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
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 \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e}\\ &=\frac{f \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.182468, size = 148, normalized size = 0.97 \[ \frac{f \left (\frac{24 c^2 \tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{\left (b^2-4 a c\right ) \left (-b-2 c (d+e x)^2\right )}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac{6 c \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}\right )}{4 e \left (b^2-4 a c\right )^2} \]
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] time = 0.042, size = 2132, normalized size = 13.9 \begin{align*} \text{result too large to display} \end{align*}
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)
[Out]
3*f/(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*c^3*e^5/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^6+18*f/(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*e^4*d*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+45*f/(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*c^3*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*d^2+9/2*f/(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*c^2*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b+60*f/(
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*c^3*d^3*e^2/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^3+18*f/(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*c^2*d*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b+45*f/(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*c^3*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*d^4+27*f/(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*c^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*d^2*b+5*f/(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*c^2*e/(16*a^2*c^2-8*a*
b^2*c+b^4)*x^2*a+f/(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*c
*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^2+18*f/(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*c^3*d^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x+18*f/(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*c^2*d^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b+10*f/(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*c^2*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a+2*f/
(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*c*d/(16*a^2*c^2-8*a*
b^2*c+b^4)*x*b^2+3*f/(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
/e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*d^6+9/2*f/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b*c^2*d^4+5*f/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*a*c^2*d^2+f/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c*d^2+5/2
*f/(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/e/(16*a^2*c^2-8*a
*b^2*c+b^4)*a*b*c-1/4*f/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3+3*f*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(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(
4*c*d^3*e+2*b*d*e)*_Z+c*d^4+b*d^2+a))
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")
[Out]
Timed out
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Fricas [B] time = 3.32785, size = 7892, normalized size = 51.58 \begin{align*} \text{result too large to display} \end{align*}
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)*e^6*f*x^6 + 72*(b^2*c^3 - 4*a*c^4)*d*e^5*f*x^5 + 18*(b^3*c^2 - 4*a*b*c^3 + 10*(b^
2*c^3 - 4*a*c^4)*d^2)*e^4*f*x^4 + 24*(10*(b^2*c^3 - 4*a*c^4)*d^3 + 3*(b^3*c^2 - 4*a*b*c^3)*d)*e^3*f*x^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)*e^2*f*x^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)*e*f*x + 12*(c^4*e^8*
f*x^8 + 8*c^4*d*e^7*f*x^7 + 2*(14*c^4*d^2 + b*c^3)*e^6*f*x^6 + 4*(14*c^4*d^3 + 3*b*c^3*d)*e^5*f*x^5 + (70*c^4*
d^4 + 30*b*c^3*d^2 + b^2*c^2 + 2*a*c^3)*e^4*f*x^4 + 4*(14*c^4*d^5 + 10*b*c^3*d^3 + (b^2*c^2 + 2*a*c^3)*d)*e^3*
f*x^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)*e^2*f*x^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)*e*f*x + (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*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*
e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*a*c - (2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(b
^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a))
+ (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)*e^9*x^8 + 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*e^8*x^7 + 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)*e^7*x^6 + 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)*e^6*x^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 + 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)*e^5*x^4 + 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)*e^4*x^3 + 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)*e^3*x^2 + 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)*e^2*x + ((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)*e^6*f*x^6 + 72*(b^2*c^3 -
4*a*c^4)*d*e^5*f*x^5 + 18*(b^3*c^2 - 4*a*b*c^3 + 10*(b^2*c^3 - 4*a*c^4)*d^2)*e^4*f*x^4 + 24*(10*(b^2*c^3 - 4*a
*c^4)*d^3 + 3*(b^3*c^2 - 4*a*b*c^3)*d)*e^3*f*x^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)*e^2*f*x^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)*e*f*x - 24*(c^4*e^8*f*x^8 + 8*c^4*d*e^7*f*x^7 + 2*(14*c^4*d^2 + b*c^3)*e^6*
f*x^6 + 4*(14*c^4*d^3 + 3*b*c^3*d)*e^5*f*x^5 + (70*c^4*d^4 + 30*b*c^3*d^2 + b^2*c^2 + 2*a*c^3)*e^4*f*x^4 + 4*(
14*c^4*d^5 + 10*b*c^3*d^3 + (b^2*c^2 + 2*a*c^3)*d)*e^3*f*x^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)*e^2*f*x^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)*e*f*x + (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*e^
2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (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)*e^9*x^8 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*
d*e^8*x^7 + 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)*e^7*x^6 + 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)*e^6*x^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 + 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)*e^5*x^4 + 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)*e^4*x^3 + 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)*e^3*x^2 + 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 - 6
4*a^4*b*c^3)*d)*e^2*x + ((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)]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 11.7623, size = 909, normalized size = 5.94 \begin{align*} \frac{3 \,{\left (b^{4} c^{2} f e - 8 \, a b^{2} c^{3} f e + 16 \, a^{2} c^{4} f e\right )} \sqrt{b^{2} - 4 \, a c} \log \left ({\left |{\left (b + \sqrt{b^{2} - 4 \, a c}\right )} x^{2} e^{2} + 2 \,{\left (b + \sqrt{b^{2} - 4 \, a c}\right )} d x e +{\left (b + \sqrt{b^{2} - 4 \, a c}\right )} d^{2} + 2 \, a \right |}\right )}{b^{10} e^{2} - 20 \, a b^{8} c e^{2} + 160 \, a^{2} b^{6} c^{2} e^{2} - 640 \, a^{3} b^{4} c^{3} e^{2} + 1280 \, a^{4} b^{2} c^{4} e^{2} - 1024 \, a^{5} c^{5} e^{2}} - \frac{3 \,{\left (b^{4} c^{2} f e - 8 \, a b^{2} c^{3} f e + 16 \, a^{2} c^{4} f e\right )} \sqrt{b^{2} - 4 \, a c} \log \left ({\left | -{\left (b - \sqrt{b^{2} - 4 \, a c}\right )} x^{2} e^{2} - 2 \,{\left (b - \sqrt{b^{2} - 4 \, a c}\right )} d x e -{\left (b - \sqrt{b^{2} - 4 \, a c}\right )} d^{2} - 2 \, a \right |}\right )}{b^{10} e^{2} - 20 \, a b^{8} c e^{2} + 160 \, a^{2} b^{6} c^{2} e^{2} - 640 \, a^{3} b^{4} c^{3} e^{2} + 1280 \, a^{4} b^{2} c^{4} e^{2} - 1024 \, a^{5} c^{5} e^{2}} + \frac{12 \, c^{3} f x^{6} e^{6} + 72 \, c^{3} d f x^{5} e^{5} + 180 \, c^{3} d^{2} f x^{4} e^{4} + 240 \, c^{3} d^{3} f x^{3} e^{3} + 180 \, c^{3} d^{4} f x^{2} e^{2} + 72 \, c^{3} d^{5} f x e + 12 \, c^{3} d^{6} f + 18 \, b c^{2} f x^{4} e^{4} + 72 \, b c^{2} d f x^{3} e^{3} + 108 \, b c^{2} d^{2} f x^{2} e^{2} + 72 \, b c^{2} d^{3} f x e + 18 \, b c^{2} d^{4} f + 4 \, b^{2} c f x^{2} e^{2} + 20 \, a c^{2} f x^{2} e^{2} + 8 \, b^{2} c d f x e + 40 \, a c^{2} d f x e + 4 \, b^{2} c d^{2} f + 20 \, a c^{2} d^{2} f - b^{3} f + 10 \, a b c f}{4 \,{\left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a\right )}^{2}{\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}} \end{align*}
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]
3*(b^4*c^2*f*e - 8*a*b^2*c^3*f*e + 16*a^2*c^4*f*e)*sqrt(b^2 - 4*a*c)*log(abs((b + sqrt(b^2 - 4*a*c))*x^2*e^2 +
2*(b + sqrt(b^2 - 4*a*c))*d*x*e + (b + sqrt(b^2 - 4*a*c))*d^2 + 2*a))/(b^10*e^2 - 20*a*b^8*c*e^2 + 160*a^2*b^
6*c^2*e^2 - 640*a^3*b^4*c^3*e^2 + 1280*a^4*b^2*c^4*e^2 - 1024*a^5*c^5*e^2) - 3*(b^4*c^2*f*e - 8*a*b^2*c^3*f*e
+ 16*a^2*c^4*f*e)*sqrt(b^2 - 4*a*c)*log(abs(-(b - sqrt(b^2 - 4*a*c))*x^2*e^2 - 2*(b - sqrt(b^2 - 4*a*c))*d*x*e
- (b - sqrt(b^2 - 4*a*c))*d^2 - 2*a))/(b^10*e^2 - 20*a*b^8*c*e^2 + 160*a^2*b^6*c^2*e^2 - 640*a^3*b^4*c^3*e^2
+ 1280*a^4*b^2*c^4*e^2 - 1024*a^5*c^5*e^2) + 1/4*(12*c^3*f*x^6*e^6 + 72*c^3*d*f*x^5*e^5 + 180*c^3*d^2*f*x^4*e^
4 + 240*c^3*d^3*f*x^3*e^3 + 180*c^3*d^4*f*x^2*e^2 + 72*c^3*d^5*f*x*e + 12*c^3*d^6*f + 18*b*c^2*f*x^4*e^4 + 72*
b*c^2*d*f*x^3*e^3 + 108*b*c^2*d^2*f*x^2*e^2 + 72*b*c^2*d^3*f*x*e + 18*b*c^2*d^4*f + 4*b^2*c*f*x^2*e^2 + 20*a*c
^2*f*x^2*e^2 + 8*b^2*c*d*f*x*e + 40*a*c^2*d*f*x*e + 4*b^2*c*d^2*f + 20*a*c^2*d^2*f - b^3*f + 10*a*b*c*f)/((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))