3.112 \(\int \frac{x^7 (A+B x^2)}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=212 \[ -\frac{\left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (-6 a B c-A b c+2 b^2 B\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]

[Out]

((2*b^2*B - A*b*c - 6*a*B*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (x^4*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^
2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)
*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*B - A*c)*Log[a + b*x^2 + c*x^4]
)/(4*c^3)

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Rubi [A]  time = 0.381239, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1251, 818, 773, 634, 618, 206, 628} \[ -\frac{\left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (-6 a B c-A b c+2 b^2 B\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*b^2*B - A*b*c - 6*a*B*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (x^4*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^
2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)
*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*B - A*c)*Log[a + b*x^2 + c*x^4]
)/(4*c^3)

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

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 (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\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 \left (2 a (b B-2 A c)+\left (2 b^2 B-A b c-6 a B c\right ) x\right )}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\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 B-A b c-6 a B c\right )+\left (2 a c (b B-2 A c)-b \left (2 b^2 B-A b c-6 a B c\right )\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^2 \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(2 b B-A c) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}+\frac{\left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac{\left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3 \left (b^2-4 a c\right )}\\ &=\frac{\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\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 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}-\frac{(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end{align*}

Mathematica [A]  time = 0.312656, size = 208, normalized size = 0.98 \[ \frac{-\frac{2 \left (a^2 c \left (2 c \left (A+B x^2\right )-3 b B\right )+a b \left (-b c \left (A+4 B x^2\right )+3 A c^2 x^2+b^2 B\right )+b^3 x^2 (b B-A c)\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{2 \left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+(A c-2 b B) \log \left (a+b x^2+c x^4\right )+2 B c x^2}{4 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*B*c*x^2 - (2*(b^3*(b*B - A*c)*x^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b^2*B + 3*A*c^2*x^2 - b*c*(A +
4*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*
B*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (-2*b*B + A*c)*Log[a + b*x^2 + c*x^4])
/(4*c^3)

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Maple [B]  time = 0.017, size = 689, 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(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x)

[Out]

1/2*B*x^2/c^2+3/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*A*b-1/2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*A*b^3+1/c/(c
*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a^2*B-2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*b^2*B+1/2/c^3/(c*x^4+b*x^2+a)/(4*a
*c-b^2)*x^2*b^4*B+1/c/(c*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*A-1/2/c^2/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*A*b^2-3/2/c^2/(c
*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*b*B+1/2/c^3/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*b^3*B+1/c/(4*a*c-b^2)*ln(c*x^4+b*x^2+a
)*a*A-1/4/c^2/(4*a*c-b^2)*ln(c*x^4+b*x^2+a)*A*b^2-2/c^2/(4*a*c-b^2)*ln(c*x^4+b*x^2+a)*a*b*B+1/2/c^3/(4*a*c-b^2
)*ln(c*x^4+b*x^2+a)*b^3*B-3/c/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*a*b-6/c/(4*a*c-b^2)^(3
/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a^2*B+6/c^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*
B*a*b^2+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*A-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*B

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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*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.24119, size = 2789, normalized size = 13.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

[-1/4*(2*B*a*b^5 - 16*A*a^3*c^3 - 2*(B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*x^6 - 2*(B*b^5*c - 8*B*a*b^3*c^
2 + 16*B*a^2*b*c^3)*x^4 + 12*(2*B*a^3*b + A*a^2*b^2)*c^2 + 2*(B*b^6 - 12*(2*B*a^3 + A*a^2*b)*c^3 + (26*B*a^2*b
^2 + 7*A*a*b^3)*c^2 - (9*B*a*b^4 + A*b^5)*c)*x^2 + (2*B*a*b^4 + (2*B*b^4*c + 6*(2*B*a^2 + A*a*b)*c^3 - (12*B*a
*b^2 + A*b^3)*c^2)*x^4 + 6*(2*B*a^3 + A*a^2*b)*c^2 + (2*B*b^5 + 6*(2*B*a^2*b + A*a*b^2)*c^2 - (12*B*a*b^3 + A*
b^4)*c)*x^2 - (12*B*a^2*b^2 + A*a*b^3)*c)*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*(7*B*a^2*b^3 + A*a*b^4)*c + (2*B*a*b^5 - 16*A*a^3*c^3 + (2*
B*b^5*c - 16*A*a^2*c^4 + 8*(4*B*a^2*b + A*a*b^2)*c^3 - (16*B*a*b^3 + A*b^4)*c^2)*x^4 + 8*(4*B*a^3*b + A*a^2*b^
2)*c^2 + (2*B*b^6 - 16*A*a^2*b*c^3 + 8*(4*B*a^2*b^2 + A*a*b^3)*c^2 - (16*B*a*b^4 + A*b^5)*c)*x^2 - (16*B*a^2*b
^3 + A*a*b^4)*c)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16
*a^2*c^6)*x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^2), -1/4*(2*B*a*b^5 - 16*A*a^3*c^3 - 2*(B*b^4*c^2 - 8
*B*a*b^2*c^3 + 16*B*a^2*c^4)*x^6 - 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*x^4 + 12*(2*B*a^3*b + A*a^2*b^
2)*c^2 + 2*(B*b^6 - 12*(2*B*a^3 + A*a^2*b)*c^3 + (26*B*a^2*b^2 + 7*A*a*b^3)*c^2 - (9*B*a*b^4 + A*b^5)*c)*x^2 +
2*(2*B*a*b^4 + (2*B*b^4*c + 6*(2*B*a^2 + A*a*b)*c^3 - (12*B*a*b^2 + A*b^3)*c^2)*x^4 + 6*(2*B*a^3 + A*a^2*b)*c
^2 + (2*B*b^5 + 6*(2*B*a^2*b + A*a*b^2)*c^2 - (12*B*a*b^3 + A*b^4)*c)*x^2 - (12*B*a^2*b^2 + A*a*b^3)*c)*sqrt(-
b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 2*(7*B*a^2*b^3 + A*a*b^4)*c + (2*B*a*b^
5 - 16*A*a^3*c^3 + (2*B*b^5*c - 16*A*a^2*c^4 + 8*(4*B*a^2*b + A*a*b^2)*c^3 - (16*B*a*b^3 + A*b^4)*c^2)*x^4 + 8
*(4*B*a^3*b + A*a^2*b^2)*c^2 + (2*B*b^6 - 16*A*a^2*b*c^3 + 8*(4*B*a^2*b^2 + A*a*b^3)*c^2 - (16*B*a*b^4 + A*b^5
)*c)*x^2 - (16*B*a^2*b^3 + A*a*b^4)*c)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*
c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^2)]

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Sympy [B]  time = 37.1602, size = 1266, normalized size = 5.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**2+A)/(c*x**4+b*x**2+a)**2,x)

[Out]

B*x**2/(2*c**2) + (-sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b
**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3))*log(x**2 + (8
*A*a**2*c**2 - A*a*b**2*c - 10*B*a**2*b*c + 2*B*a*b**3 - 32*a**2*c**4*(-sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2
- A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4
*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)) + 16*a*b**2*c**3*(-sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c +
12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
(-A*c + 2*B*b)/(4*c**3)) - 2*b**4*c**2*(-sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 -
12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4
*c**3)))/(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)) + (sqrt(-(4*a*c - b**2)**3)*(6
*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3))*log(x**2 + (8*A*a**2*c**2 - A*a*b**2*c - 10*B*a**2*b*c + 2*B
*a*b**3 - 32*a**2*c**4*(sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2
*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)) + 16*a*b*
*2*c**3*(sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**
3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)) - 2*b**4*c**2*(sqrt(-(4*
a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 -
48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)))/(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2
- 12*B*a*b**2*c + 2*B*b**4)) + (2*A*a**2*c**2 - A*a*b**2*c - 3*B*a**2*b*c + B*a*b**3 + x**2*(3*A*a*b*c**2 - A
*b**3*c + 2*B*a**2*c**2 - 4*B*a*b**2*c + B*b**4))/(8*a**2*c**4 - 2*a*b**2*c**3 + x**4*(8*a*c**5 - 2*b**2*c**4)
+ x**2*(8*a*b*c**4 - 2*b**3*c**3))

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Giac [A]  time = 19.541, size = 323, normalized size = 1.52 \begin{align*} \frac{B x^{2}}{2 \, c^{2}} + \frac{{\left (2 \, B b^{4} - 12 \, B a b^{2} c - A b^{3} c + 12 \, B a^{2} c^{2} + 6 \, A a b c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, B b^{3} x^{4} - 8 \, B a b c x^{4} - A b^{2} c x^{4} + 4 \, A a c^{2} x^{4} + A b^{3} x^{2} - 4 \, B a^{2} c x^{2} - 2 \, A a b c x^{2} - 2 \, B a^{2} b + A a b^{2}}{4 \,{\left (c x^{4} + b x^{2} + a\right )}{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} - \frac{{\left (2 \, B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*B*x^2/c^2 + 1/2*(2*B*b^4 - 12*B*a*b^2*c - A*b^3*c + 12*B*a^2*c^2 + 6*A*a*b*c^2)*arctan((2*c*x^2 + b)/sqrt(
-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + 1/4*(2*B*b^3*x^4 - 8*B*a*b*c*x^4 - A*b^2*c*x^4 + 4*A
*a*c^2*x^4 + A*b^3*x^2 - 4*B*a^2*c*x^2 - 2*A*a*b*c*x^2 - 2*B*a^2*b + A*a*b^2)/((c*x^4 + b*x^2 + a)*(b^2*c^2 -
4*a*c^3)) - 1/4*(2*B*b - A*c)*log(c*x^4 + b*x^2 + a)/c^3