3.126 \(\int \frac{x^7 (A+B x^2)}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=146 \[ -\frac{x^6 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 a (A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

-(x^6*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*(A*b - 2*a*B)*x^2*(2*a +
b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*a*(A*b - 2*a*B)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]]
)/(b^2 - 4*a*c)^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.138687, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1251, 804, 722, 618, 206} \[ -\frac{x^6 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 a (A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(x^6*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*(A*b - 2*a*B)*x^2*(2*a +
b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*a*(A*b - 2*a*B)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]]
)/(b^2 - 4*a*c)^(5/2)

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 804

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(b*f - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(m
*(b*(e*f + d*g) - 2*(c*d*f + a*e*g)))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)
, 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] && EqQ[Sim
plify[m + 2*p + 3], 0] && LtQ[p, -1]

Rule 722

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

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{x^7 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac{x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{(3 (A b-2 a B)) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac{x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{(3 a (A b-2 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac{x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{(3 a (A b-2 a B)) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 a (A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.294007, size = 261, normalized size = 1.79 \[ \frac{1}{4} \left (\frac{-4 a^2 c^3 \left (4 A+5 B x^2\right )+a b^2 c^2 \left (5 A+16 B x^2\right )+2 a b c^2 \left (11 a B-3 A c x^2\right )-8 a b^3 B c-b^4 c \left (A+2 B x^2\right )+b^5 B}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{a^2 c \left (2 c \left (A+B x^2\right )-3 b B\right )+a b \left (-b c \left (A+4 B x^2\right )+3 A c^2 x^2+b^2 B\right )+b^3 x^2 (b B-A c)}{c^3 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}-\frac{12 a (A b-2 a B) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((b^5*B - 8*a*b^3*B*c - b^4*c*(A + 2*B*x^2) - 4*a^2*c^3*(4*A + 5*B*x^2) + a*b^2*c^2*(5*A + 16*B*x^2) + 2*a*b*c
^2*(11*a*B - 3*A*c*x^2))/(c^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (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)))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) - (
12*a*(A*b - 2*a*B)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 398, normalized size = 2.7 \begin{align*}{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 3\,aAb{c}^{2}+10\,{a}^{2}B{c}^{2}-8\,a{b}^{2}Bc+{b}^{4}B \right ){x}^{6}}{c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}-{\frac{ \left ( 16\,A{a}^{2}{c}^{3}+Aa{b}^{2}{c}^{2}+A{b}^{4}c-2\,B{a}^{2}b{c}^{2}-8\,Ba{b}^{3}c+B{b}^{5} \right ){x}^{4}}{ \left ( 32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4} \right ){c}^{2}}}-{\frac{a \left ( 5\,aAb{c}^{2}+A{b}^{3}c+6\,{a}^{2}B{c}^{2}-10\,a{b}^{2}Bc+{b}^{4}B \right ){x}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ){c}^{2}}}-{\frac{{a}^{2} \left ( 8\,aA{c}^{2}+A{b}^{2}c-10\,abBc+{b}^{3}B \right ) }{2\, \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ){c}^{2}}} \right ) }-3\,{\frac{abA}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{B{a}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \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)^3,x)

[Out]

1/2*(-(3*A*a*b*c^2+10*B*a^2*c^2-8*B*a*b^2*c+B*b^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-1/2*(16*A*a^2*c^3+A*a*b^2*
c^2+A*b^4*c-2*B*a^2*b*c^2-8*B*a*b^3*c+B*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^4-a*(5*A*a*b*c^2+A*b^3*c+6*B*a^2
*c^2-10*B*a*b^2*c+B*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2-1/2*a^2/c^2*(8*A*a*c^2+A*b^2*c-10*B*a*b*c+B*b^3)/(
16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2-3*a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+
b)/(4*a*c-b^2)^(1/2))*A*b+6*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1
/2))*B

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.45698, size = 2839, normalized size = 19.45 \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)^3,x, algorithm="fricas")

[Out]

[-1/4*(B*a^2*b^5 - 32*A*a^4*c^3 + 2*(B*b^6*c - 12*B*a*b^4*c^2 - 4*(10*B*a^3 + 3*A*a^2*b)*c^4 + 3*(14*B*a^2*b^2
+ A*a*b^3)*c^3)*x^6 + (B*b^7 - 64*A*a^3*c^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 3*(10*B*a^2*b^3 - A*a*b^4)*c^
2 - (12*B*a*b^5 - A*b^6)*c)*x^4 + 4*(10*B*a^4*b + A*a^3*b^2)*c^2 + 2*(B*a*b^6 - 4*(6*B*a^4 + 5*A*a^3*b)*c^3 +
(46*B*a^3*b^2 + A*a^2*b^3)*c^2 - (14*B*a^2*b^4 - A*a*b^5)*c)*x^2 + 6*((2*B*a^2 - A*a*b)*c^4*x^8 + 2*(2*B*a^2*b
- A*a*b^2)*c^3*x^6 + 2*(2*B*a^3*b - A*a^2*b^2)*c^2*x^2 + (2*(2*B*a^3 - A*a^2*b)*c^3 + (2*B*a^2*b^2 - A*a*b^3)
*c^2)*x^4 + (2*B*a^4 - A*a^3*b)*c^2)*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)) - (14*B*a^3*b^3 - A*a^2*b^4)*c)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a
^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^8 + 2*(b^7*c^3 - 12*a*b^5*c
^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*
c^6)*x^4 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c^5)*x^2), -1/4*(B*a^2*b^5 - 32*A*a^4*c^3
+ 2*(B*b^6*c - 12*B*a*b^4*c^2 - 4*(10*B*a^3 + 3*A*a^2*b)*c^4 + 3*(14*B*a^2*b^2 + A*a*b^3)*c^3)*x^6 + (B*b^7 -
64*A*a^3*c^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 3*(10*B*a^2*b^3 - A*a*b^4)*c^2 - (12*B*a*b^5 - A*b^6)*c)*x^4
+ 4*(10*B*a^4*b + A*a^3*b^2)*c^2 + 2*(B*a*b^6 - 4*(6*B*a^4 + 5*A*a^3*b)*c^3 + (46*B*a^3*b^2 + A*a^2*b^3)*c^2
- (14*B*a^2*b^4 - A*a*b^5)*c)*x^2 + 12*((2*B*a^2 - A*a*b)*c^4*x^8 + 2*(2*B*a^2*b - A*a*b^2)*c^3*x^6 + 2*(2*B*a
^3*b - A*a^2*b^2)*c^2*x^2 + (2*(2*B*a^3 - A*a^2*b)*c^3 + (2*B*a^2*b^2 - A*a*b^3)*c^2)*x^4 + (2*B*a^4 - A*a^3*b
)*c^2)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (14*B*a^3*b^3 - A*a^2*b^4)
*c)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 6
4*a^3*c^7)*x^8 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + (b^8*c^2 - 10*a*b^6*c^3 + 24
*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^4 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c
^5)*x^2)]

________________________________________________________________________________________

Sympy [B]  time = 149.303, size = 775, normalized size = 5.31 \begin{align*} - \frac{3 a \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) \log{\left (x^{2} + \frac{- 3 A a b^{2} + 6 B a^{2} b - 192 a^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) + 144 a^{3} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) - 36 a^{2} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) + 3 a b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right )}{- 6 A a b c + 12 B a^{2} c} \right )}}{2} + \frac{3 a \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) \log{\left (x^{2} + \frac{- 3 A a b^{2} + 6 B a^{2} b + 192 a^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) - 144 a^{3} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) + 36 a^{2} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right ) - 3 a b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- A b + 2 B a\right )}{- 6 A a b c + 12 B a^{2} c} \right )}}{2} - \frac{8 A a^{3} c^{2} + A a^{2} b^{2} c - 10 B a^{3} b c + B a^{2} b^{3} + x^{6} \left (6 A a b c^{3} + 20 B a^{2} c^{3} - 16 B a b^{2} c^{2} + 2 B b^{4} c\right ) + x^{4} \left (16 A a^{2} c^{3} + A a b^{2} c^{2} + A b^{4} c - 2 B a^{2} b c^{2} - 8 B a b^{3} c + B b^{5}\right ) + x^{2} \left (10 A a^{2} b c^{2} + 2 A a b^{3} c + 12 B a^{3} c^{2} - 20 B a^{2} b^{2} c + 2 B a b^{4}\right )}{64 a^{4} c^{4} - 32 a^{3} b^{2} c^{3} + 4 a^{2} b^{4} c^{2} + x^{8} \left (64 a^{2} c^{6} - 32 a b^{2} c^{5} + 4 b^{4} c^{4}\right ) + x^{6} \left (128 a^{2} b c^{5} - 64 a b^{3} c^{4} + 8 b^{5} c^{3}\right ) + x^{4} \left (128 a^{3} c^{5} - 24 a b^{4} c^{3} + 4 b^{6} c^{2}\right ) + x^{2} \left (128 a^{3} b c^{4} - 64 a^{2} b^{3} c^{3} + 8 a b^{5} c^{2}\right )} \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)**3,x)

[Out]

-3*a*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a)*log(x**2 + (-3*A*a*b**2 + 6*B*a**2*b - 192*a**4*c**3*sqrt(-1/(4
*a*c - b**2)**5)*(-A*b + 2*B*a) + 144*a**3*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a) - 36*a**2*b**4*
c*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a) + 3*a*b**6*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a))/(-6*A*a*b*c
+ 12*B*a**2*c))/2 + 3*a*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a)*log(x**2 + (-3*A*a*b**2 + 6*B*a**2*b + 192*a
**4*c**3*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a) - 144*a**3*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B
*a) + 36*a**2*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2*B*a) - 3*a*b**6*sqrt(-1/(4*a*c - b**2)**5)*(-A*b + 2
*B*a))/(-6*A*a*b*c + 12*B*a**2*c))/2 - (8*A*a**3*c**2 + A*a**2*b**2*c - 10*B*a**3*b*c + B*a**2*b**3 + x**6*(6*
A*a*b*c**3 + 20*B*a**2*c**3 - 16*B*a*b**2*c**2 + 2*B*b**4*c) + x**4*(16*A*a**2*c**3 + A*a*b**2*c**2 + A*b**4*c
- 2*B*a**2*b*c**2 - 8*B*a*b**3*c + B*b**5) + x**2*(10*A*a**2*b*c**2 + 2*A*a*b**3*c + 12*B*a**3*c**2 - 20*B*a*
*2*b**2*c + 2*B*a*b**4))/(64*a**4*c**4 - 32*a**3*b**2*c**3 + 4*a**2*b**4*c**2 + x**8*(64*a**2*c**6 - 32*a*b**2
*c**5 + 4*b**4*c**4) + x**6*(128*a**2*b*c**5 - 64*a*b**3*c**4 + 8*b**5*c**3) + x**4*(128*a**3*c**5 - 24*a*b**4
*c**3 + 4*b**6*c**2) + x**2*(128*a**3*b*c**4 - 64*a**2*b**3*c**3 + 8*a*b**5*c**2))

________________________________________________________________________________________

Giac [B]  time = 35.4841, size = 429, normalized size = 2.94 \begin{align*} \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, B b^{4} c x^{6} - 16 \, B a b^{2} c^{2} x^{6} + 20 \, B a^{2} c^{3} x^{6} + 6 \, A a b c^{3} x^{6} + B b^{5} x^{4} - 8 \, B a b^{3} c x^{4} + A b^{4} c x^{4} - 2 \, B a^{2} b c^{2} x^{4} + A a b^{2} c^{2} x^{4} + 16 \, A a^{2} c^{3} x^{4} + 2 \, B a b^{4} x^{2} - 20 \, B a^{2} b^{2} c x^{2} + 2 \, A a b^{3} c x^{2} + 12 \, B a^{3} c^{2} x^{2} + 10 \, A a^{2} b c^{2} x^{2} + B a^{2} b^{3} - 10 \, B a^{3} b c + A a^{2} b^{2} c + 8 \, A a^{3} c^{2}}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

3*(2*B*a^2 - A*a*b)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c
)) - 1/4*(2*B*b^4*c*x^6 - 16*B*a*b^2*c^2*x^6 + 20*B*a^2*c^3*x^6 + 6*A*a*b*c^3*x^6 + B*b^5*x^4 - 8*B*a*b^3*c*x^
4 + A*b^4*c*x^4 - 2*B*a^2*b*c^2*x^4 + A*a*b^2*c^2*x^4 + 16*A*a^2*c^3*x^4 + 2*B*a*b^4*x^2 - 20*B*a^2*b^2*c*x^2
+ 2*A*a*b^3*c*x^2 + 12*B*a^3*c^2*x^2 + 10*A*a^2*b*c^2*x^2 + B*a^2*b^3 - 10*B*a^3*b*c + A*a^2*b^2*c + 8*A*a^3*c
^2)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^4 + b*x^2 + a)^2)