### 3.61 $$\int \frac{x^7 (d+e x^2+f x^4)}{(a+b x^2+c x^4)^2} \, dx$$

Optimal. Leaf size=320 $-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (12 a^2 c^3 e-12 a b^2 c^2 e-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)+2 b^4 c e-3 b^5 f\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{x^6 \left (x^2 \left (-\left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x^4 \left (-2 c (4 a f+b e)+3 b^2 f+4 c^2 d\right )}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^2 \left (-b c (c d-11 a f)-6 a c^2 e+2 b^2 c e-3 b^3 f\right )}{2 c^3 \left (b^2-4 a c\right )}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )}{4 c^4}$

[Out]

((2*b^2*c*e - 6*a*c^2*e - 3*b^3*f - b*c*(c*d - 11*a*f))*x^2)/(2*c^3*(b^2 - 4*a*c)) + ((4*c^2*d + 3*b^2*f - 2*c
*(b*e + 4*a*f))*x^4)/(4*c^2*(b^2 - 4*a*c)) + (x^6*(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*
f)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*c*e - 12*a*b^2*c^2*e + 12*a^2*c^3*e - 3*b^5*f - b^3
*c*(c*d - 20*a*f) + 6*a*b*c^2*(c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(3
/2)) + ((c^2*d + 3*b^2*f - 2*c*(b*e + a*f))*Log[a + b*x^2 + c*x^4])/(4*c^4)

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Rubi [A]  time = 1.23253, antiderivative size = 320, 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, 1644, 800, 634, 618, 206, 628} $-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (12 a^2 c^3 e-12 a b^2 c^2 e-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)+2 b^4 c e-3 b^5 f\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{x^6 \left (x^2 \left (-\left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x^4 \left (-2 c (4 a f+b e)+3 b^2 f+4 c^2 d\right )}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^2 \left (-b c (c d-11 a f)-6 a c^2 e+2 b^2 c e-3 b^3 f\right )}{2 c^3 \left (b^2-4 a c\right )}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )}{4 c^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*b^2*c*e - 6*a*c^2*e - 3*b^3*f - b*c*(c*d - 11*a*f))*x^2)/(2*c^3*(b^2 - 4*a*c)) + ((4*c^2*d + 3*b^2*f - 2*c
*(b*e + 4*a*f))*x^4)/(4*c^2*(b^2 - 4*a*c)) + (x^6*(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*
f)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*c*e - 12*a*b^2*c^2*e + 12*a^2*c^3*e - 3*b^5*f - b^3
*c*(c*d - 20*a*f) + 6*a*b*c^2*(c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(3
/2)) + ((c^2*d + 3*b^2*f - 2*c*(b*e + a*f))*Log[a + b*x^2 + c*x^4])/(4*c^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 1644

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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{x^7 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 \left (2 a e-\frac{b (c d+a f)}{c}\right )-\frac{\left (4 c^2 d-2 b c e+3 b^2 f-8 a c f\right ) x}{c}\right )}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)}{c^3}-\frac{\left (4 c^2 d-2 b c e+3 b^2 f-8 a c f\right ) x}{c^2}-\frac{-a \left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right )+\left (b^2-4 a c\right ) \left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac{\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{-a \left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right )+\left (b^2-4 a c\right ) \left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3 \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac{\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4 \left (b^2-4 a c\right )}+\frac{\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac{\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac{\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}-\frac{\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 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 c^4 \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac{\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}\\ \end{align*}

Mathematica [A]  time = 0.551008, size = 309, normalized size = 0.97 $\frac{\frac{2 \left (a^2 c \left (-4 b^2 f+b c \left (3 e+5 f x^2\right )-2 c^2 \left (d+e x^2\right )\right )+2 a^3 c^2 f+a b \left (-b^2 c \left (e+5 f x^2\right )+b^3 f+b c^2 \left (d+4 e x^2\right )-3 c^3 d x^2\right )+b^3 x^2 \left (b^2 f-b c e+c^2 d\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-12 a^2 c^3 e+12 a b^2 c^2 e+b^3 c (c d-20 a f)+6 a b c^2 (5 a f-c d)-2 b^4 c e+3 b^5 f\right )}{\left (4 a c-b^2\right )^{3/2}}+\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )+2 c x^2 (c e-2 b f)+c^2 f x^4}{4 c^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [B]  time = 0.018, size = 1167, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

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

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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(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.46226, size = 4393, normalized size = 13.73 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*f*x^8 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e - 3*(b^5*c^2 - 8*
a*b^3*c^3 + 16*a^2*b*c^4)*f)*x^6 + (2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e - (4*b^6*c - 33*a*b^4*c^2 + 72*
a^2*b^2*c^3 - 16*a^3*c^4)*f)*x^4 + 2*((b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d - (b^6*c - 9*a*b^4*c^2 + 26*a^2
*b^2*c^3 - 24*a^3*c^4)*e + (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*f)*x^2 - (((b^3*c^3 - 6*a*b*c^4)
*d - 2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*f)*x^4 + ((b^4*c^2 - 6*
a*b^2*c^3)*d - 2*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*f)*x^2 + (a*b^3
*c^2 - 6*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*f)
*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)) + 2*(a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3*c^4)*d - 2*(a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*e + 2*(a*b^6
- 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*f + (((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 2*(b^5*c^2 - 8*a*b^
3*c^3 + 16*a^2*b*c^4)*e + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*f)*x^4 + ((b^5*c^2 - 8*a*b^3*
c^3 + 16*a^2*b*c^4)*d - 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32
*a^3*b*c^3)*f)*x^2 + (a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e
+ (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^4 - 8*a^2*b^2*c^
5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^2), 1/4*(
(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*f*x^8 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e - 3*(b^5*c^2 - 8*a*b^3*
c^3 + 16*a^2*b*c^4)*f)*x^6 + (2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^
2*c^3 - 16*a^3*c^4)*f)*x^4 + 2*((b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d - (b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c
^3 - 24*a^3*c^4)*e + (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*f)*x^2 + 2*(((b^3*c^3 - 6*a*b*c^4)*d -
2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*f)*x^4 + ((b^4*c^2 - 6*a*b^
2*c^3)*d - 2*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*f)*x^2 + (a*b^3*c^2
- 6*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*f)*sqr
t(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + 2*(a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3
*c^4)*d - 2*(a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*e + 2*(a*b^6 - 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*
f + (((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e + (3*b^6*c - 26*a*b^
4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*f)*x^4 + ((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - 2*(b^6*c - 8*a*b^4*c
^2 + 16*a^2*b^2*c^3)*e + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*f)*x^2 + (a*b^4*c^2 - 8*a^2*b^2*
c^3 + 16*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2
- 32*a^4*c^3)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 + 16
*a^2*c^7)*x^4 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^2)]

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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(x**7*(f*x**4+e*x**2+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 19.8252, size = 572, normalized size = 1.79 \begin{align*} -\frac{{\left (b^{3} c^{2} d - 6 \, a b c^{3} d + 3 \, b^{5} f - 20 \, a b^{3} c f + 30 \, a^{2} b c^{2} f - 2 \, b^{4} c e + 12 \, a b^{2} c^{2} e - 12 \, a^{2} c^{3} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{b^{2} c^{3} d x^{4} - 4 \, a c^{4} d x^{4} + 3 \, b^{4} c f x^{4} - 14 \, a b^{2} c^{2} f x^{4} + 8 \, a^{2} c^{3} f x^{4} - 2 \, b^{3} c^{2} x^{4} e + 8 \, a b c^{3} x^{4} e - b^{3} c^{2} d x^{2} + 2 \, a b c^{3} d x^{2} + b^{5} f x^{2} - 4 \, a b^{3} c f x^{2} - 2 \, a^{2} b c^{2} f x^{2} + 4 \, a^{2} c^{3} x^{2} e - a b^{2} c^{2} d + a b^{4} f - 6 \, a^{2} b^{2} c f + 4 \, a^{3} c^{2} f + 2 \, a^{2} b c^{2} e}{4 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}{\left (c x^{4} + b x^{2} + a\right )}} + \frac{{\left (c^{2} d + 3 \, b^{2} f - 2 \, a c f - 2 \, b c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac{c^{2} f x^{4} - 4 \, b c f x^{2} + 2 \, c^{2} x^{2} e}{4 \, c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

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