### 3.67 $$\int \frac{d+e x^2+f x^4}{x^5 (a+b x^2+c x^4)^2} \, dx$$

Optimal. Leaf size=329 $\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-12 a^3 c^2 e-a b^3 (20 c d-a f)-2 a b^4 e+3 b^5 d\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)-a b^3 e+b^4 d}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{4 a^4}+\frac{\log (x) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{a^4}+\frac{2 b d-a e}{2 a^3 x^2}-\frac{d}{4 a^2 x^4}$

[Out]

-d/(4*a^2*x^4) + (2*b*d - a*e)/(2*a^3*x^2) + (b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*a^2*c*(c*d - a*f) - a*b^2*(4*c
*d - a*f) + c*(b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f))*x^2)/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ ((3*b^5*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + 6*a^2*b*c*(5*c*d - a*f) - a*b^3*(20*c*d - a*f))*Arc
Tanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Lo
g[x])/a^4 - ((3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

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Rubi [A]  time = 1.15731, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.233, Rules used = {1663, 1646, 1628, 634, 618, 206, 628} $\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-12 a^3 c^2 e-a b^3 (20 c d-a f)-2 a b^4 e+3 b^5 d\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)-a b^3 e+b^4 d}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{4 a^4}+\frac{\log (x) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{a^4}+\frac{2 b d-a e}{2 a^3 x^2}-\frac{d}{4 a^2 x^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]

[Out]

-d/(4*a^2*x^4) + (2*b*d - a*e)/(2*a^3*x^2) + (b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*a^2*c*(c*d - a*f) - a*b^2*(4*c
*d - a*f) + c*(b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f))*x^2)/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ ((3*b^5*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + 6*a^2*b*c*(5*c*d - a*f) - a*b^3*(20*c*d - a*f))*Arc
Tanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Lo
g[x])/a^4 - ((3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 634

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

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])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+e x+f x^2}{x^3 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\left (\frac{b^2}{a}-4 c\right ) d+\frac{\left (b^2-4 a c\right ) (b d-a e) x}{a^2}-\frac{\left (b^2-4 a c\right ) \left (b^2 d-a b e-a (c d-a f)\right ) x^2}{a^3}-\frac{c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^3}{a^3}}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{\left (-b^2+4 a c\right ) d}{a^2 x^3}+\frac{\left (-b^2+4 a c\right ) (-2 b d+a e)}{a^3 x^2}+\frac{\left (b^2-4 a c\right ) \left (-3 b^2 d+2 a b e+a (2 c d-a f)\right )}{a^4 x}+\frac{3 b^5 d-2 a b^4 e+10 a^2 b^2 c e-6 a^3 c^2 e+a^2 b c (19 c d-5 a f)-a b^3 (17 c d-a f)+c \left (b^2-4 a c\right ) \left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac{d}{4 a^2 x^4}+\frac{2 b d-a e}{2 a^3 x^2}+\frac{b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac{\operatorname{Subst}\left (\int \frac{3 b^5 d-2 a b^4 e+10 a^2 b^2 c e-6 a^3 c^2 e+a^2 b c (19 c d-5 a f)-a b^3 (17 c d-a f)+c \left (b^2-4 a c\right ) \left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{d}{4 a^2 x^4}+\frac{2 b d-a e}{2 a^3 x^2}+\frac{b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}-\frac{\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{d}{4 a^2 x^4}+\frac{2 b d-a e}{2 a^3 x^2}+\frac{b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac{\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{d}{4 a^2 x^4}+\frac{2 b d-a e}{2 a^3 x^2}+\frac{b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac{\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end{align*}

Mathematica [A]  time = 1.21921, size = 592, normalized size = 1.8 $-\frac{\frac{2 a \left (2 a^2 c \left (a f-c \left (d+e x^2\right )\right )+a b^2 \left (-a f+4 c d+c e x^2\right )+b^3 \left (a e-c d x^2\right )-a b c \left (3 a e+a f x^2-3 c d x^2\right )+b^4 (-d)\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (2 a^2 b c \left (4 e \sqrt{b^2-4 a c}-3 a f+15 c d\right )-4 a^2 c \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right )+a b^3 \left (-2 e \sqrt{b^2-4 a c}+a f-20 c d\right )+a b^2 \left (-14 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+12 a c e\right )+b^4 \left (3 d \sqrt{b^2-4 a c}-2 a e\right )+3 b^5 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (2 a^2 b c \left (4 e \sqrt{b^2-4 a c}+3 a f-15 c d\right )+4 a^2 c \left (2 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+3 a c e\right )-a b^3 \left (2 e \sqrt{b^2-4 a c}+a f-20 c d\right )+a b^2 \left (a f \sqrt{b^2-4 a c}-2 c \left (7 d \sqrt{b^2-4 a c}+6 a e\right )\right )+b^4 \left (3 d \sqrt{b^2-4 a c}+2 a e\right )-3 b^5 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{a^2 d}{x^4}-4 \log (x) \left (-2 a b e+a (a f-2 c d)+3 b^2 d\right )+\frac{2 a (a e-2 b d)}{x^2}}{4 a^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]

[Out]

-((a^2*d)/x^4 + (2*a*(-2*b*d + a*e))/x^2 + (2*a*(-(b^4*d) + b^3*(a*e - c*d*x^2) + a*b^2*(4*c*d - a*f + c*e*x^2
) - a*b*c*(3*a*e - 3*c*d*x^2 + a*f*x^2) + 2*a^2*c*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
- 4*(3*b^2*d - 2*a*b*e + a*(-2*c*d + a*f))*Log[x] + ((3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c]*d - 2*a*e) + 2*a^2*b*
c*(15*c*d + 4*Sqrt[b^2 - 4*a*c]*e - 3*a*f) + a*b^3*(-20*c*d - 2*Sqrt[b^2 - 4*a*c]*e + a*f) - 4*a^2*c*(-2*c*Sqr
t[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-14*c*Sqrt[b^2 - 4*a*c]*d + 12*a*c*e + a*Sqrt[b^2
- 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c
]*d + 2*a*e) - a*b^3*(-20*c*d + 2*Sqrt[b^2 - 4*a*c]*e + a*f) + 2*a^2*b*c*(-15*c*d + 4*Sqrt[b^2 - 4*a*c]*e + 3*
a*f) + 4*a^2*c*(2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e - a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-2*c*(7*Sqrt[b^2 - 4*a*c]*
d + 6*a*e) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^4)

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Maple [B]  time = 0.028, size = 1078, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x)

[Out]

-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2*f-1/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c^2*d+1/a^3/x^2*b*d-2/a^3*ln(x)*b*e-2
/a^3*ln(x)*c*d+3/a^4*ln(x)*b^2*d+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c*f-1/2/a^3/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b
^3*d+1/2/a^2/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b^2*e+3/2/a^2/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*b*d+1/2/a^2/(
c*x^4+b*x^2+a)/(4*a*c-b^2)*b^3*e-1/2/a^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^4*d+3/4/a^4/(4*a*c-b^2)*ln(c*x^4+b*x^2+
a)*b^4*d+3/2/a^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^5*d-1/a/(4*a*c-b^2)*c*ln(c*x^4+b*x^
2+a)*f+1/4/a^2/(4*a*c-b^2)*ln(c*x^4+b*x^2+a)*b^2*f+2/a^2/(4*a*c-b^2)*c^2*ln(c*x^4+b*x^2+a)*d-1/2/a^3/(4*a*c-b^
2)*ln(c*x^4+b*x^2+a)*b^3*e-6/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c^2*e+1/2/a^2/(4*a*c-b^
2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3*f-1/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1
/2))*b^4*e-1/2/a/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b*f-1/2/a^2/x^2*e+1/a^2*ln(x)*f-1/4*d/a^2/x^4+2/a^2/(4*a*c-
b^2)*c*ln(c*x^4+b*x^2+a)*e*b-1/a/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*e-3/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b*c*e
+2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2*c*d-3/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c*f+6
/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*c*e+15/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+
b)/(4*a*c-b^2)^(1/2))*b*c^2*d-10/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3*c*d-7/2/a^3/(
4*a*c-b^2)*c*ln(c*x^4+b*x^2+a)*b^2*d

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 39.111, size = 5341, normalized size = 16.23 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7*a^3*b^2*c^2 + 12*a^4*c^3)*e + (a^3*b
^3*c - 4*a^4*b*c^2)*f)*x^6 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*a^2*b^5 - 15*a^
3*b^3*c + 28*a^4*b*c^2)*e + 2*(a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*f)*x^4 + (3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*
b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x^2 + (((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*
b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^
2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e + (a^2*b^4 - 6*a^3*b^2*c)*f)*x^6 + ((3*a*b^5 - 20*a^2*b^3*c + 3
0*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c)*f)*x^4)*sqrt(b^2 - 4*a*c)*log
((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - (a^3*b^4 - 8*a
^4*b^2*c + 16*a^5*c^2)*d - (((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3
*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c
^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*f)*
x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e +
(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^
2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3
)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*
e + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d
- 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(x))/((a^4*b^4*
c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*x^8 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^6 + (a^5*b^4 - 8*a^6*b^2*c + 16
*a^7*c^2)*x^4), 1/4*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7*a^3*b^2*c^2 + 12*a^4*
c^3)*e + (a^3*b^3*c - 4*a^4*b*c^2)*f)*x^6 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*
a^2*b^5 - 15*a^3*b^3*c + 28*a^4*b*c^2)*e + 2*(a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*f)*x^4 + (3*(a^2*b^5 - 8*a^3*
b^3*c + 16*a^4*b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x^2 + 2*(((3*b^5*c - 20*a*b^3*c^2 + 30*a^2
*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3*b^6 - 20*a*b^4*
c + 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e + (a^2*b^4 - 6*a^3*b^2*c)*f)*x^6 + ((3*a*b^5 -
20*a^2*b^3*c + 30*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c)*f)*x^4)*sqrt
(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*
d - (((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e
+ (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d -
2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^
2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c
+ 16*a^5*c^2)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d -
2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26
*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3*b^
3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^
3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*
a^6*c^3)*x^8 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x^4)]

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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((f*x**4+e*x**2+d)/x**5/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 19.6479, size = 722, normalized size = 2.19 \begin{align*} -\frac{{\left (3 \, b^{5} d - 20 \, a b^{3} c d + 30 \, a^{2} b c^{2} d + a^{2} b^{3} f - 6 \, a^{3} b c f - 2 \, a b^{4} e + 12 \, a^{2} b^{2} c e - 12 \, a^{3} c^{2} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, b^{4} c d x^{4} - 14 \, a b^{2} c^{2} d x^{4} + 8 \, a^{2} c^{3} d x^{4} + a^{2} b^{2} c f x^{4} - 4 \, a^{3} c^{2} f x^{4} - 2 \, a b^{3} c x^{4} e + 8 \, a^{2} b c^{2} x^{4} e + 3 \, b^{5} d x^{2} - 12 \, a b^{3} c d x^{2} + 2 \, a^{2} b c^{2} d x^{2} + a^{2} b^{3} f x^{2} - 2 \, a^{3} b c f x^{2} - 2 \, a b^{4} x^{2} e + 6 \, a^{2} b^{2} c x^{2} e + 4 \, a^{3} c^{2} x^{2} e + 5 \, a b^{4} d - 22 \, a^{2} b^{2} c d + 12 \, a^{3} c^{2} d + 3 \, a^{3} b^{2} f - 8 \, a^{4} c f - 4 \, a^{2} b^{3} e + 14 \, a^{3} b c e}{4 \,{\left (a^{4} b^{2} - 4 \, a^{5} c\right )}{\left (c x^{4} + b x^{2} + a\right )}} - \frac{{\left (3 \, b^{2} d - 2 \, a c d + a^{2} f - 2 \, a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac{{\left (3 \, b^{2} d - 2 \, a c d + a^{2} f - 2 \, a b e\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{9 \, b^{2} d x^{4} - 6 \, a c d x^{4} + 3 \, a^{2} f x^{4} - 6 \, a b x^{4} e - 4 \, a b d x^{2} + 2 \, a^{2} x^{2} e + a^{2} d}{4 \, a^{4} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(3*b^5*d - 20*a*b^3*c*d + 30*a^2*b*c^2*d + a^2*b^3*f - 6*a^3*b*c*f - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*
c^2*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*sqrt(-b^2 + 4*a*c)) + 1/4*(3*b^4*c*d*x^4
- 14*a*b^2*c^2*d*x^4 + 8*a^2*c^3*d*x^4 + a^2*b^2*c*f*x^4 - 4*a^3*c^2*f*x^4 - 2*a*b^3*c*x^4*e + 8*a^2*b*c^2*x^4
*e + 3*b^5*d*x^2 - 12*a*b^3*c*d*x^2 + 2*a^2*b*c^2*d*x^2 + a^2*b^3*f*x^2 - 2*a^3*b*c*f*x^2 - 2*a*b^4*x^2*e + 6*
a^2*b^2*c*x^2*e + 4*a^3*c^2*x^2*e + 5*a*b^4*d - 22*a^2*b^2*c*d + 12*a^3*c^2*d + 3*a^3*b^2*f - 8*a^4*c*f - 4*a^
2*b^3*e + 14*a^3*b*c*e)/((a^4*b^2 - 4*a^5*c)*(c*x^4 + b*x^2 + a)) - 1/4*(3*b^2*d - 2*a*c*d + a^2*f - 2*a*b*e)*
log(c*x^4 + b*x^2 + a)/a^4 + 1/2*(3*b^2*d - 2*a*c*d + a^2*f - 2*a*b*e)*log(x^2)/a^4 - 1/4*(9*b^2*d*x^4 - 6*a*c
*d*x^4 + 3*a^2*f*x^4 - 6*a*b*x^4*e - 4*a*b*d*x^2 + 2*a^2*x^2*e + a^2*d)/(a^4*x^4)