3.127 \(\int \frac{x^5 (A+B x^2)}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=185 \[ -\frac{x^2 \left (4 a A c^2+2 a b B c-4 A b^2 c+b^3 B\right )+a \left (8 a B c-6 A b c+b^2 B\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{x^4 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (3 a b B-A \left (2 a c+b^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

-(x^4*(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) - (a*(b^2*B - 6*A*b*c + 8*a*B
*c) + (b^3*B - 4*A*b^2*c + 2*a*b*B*c + 4*a*A*c^2)*x^2)/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*a*b*B -
A*(b^2 + 2*a*c))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi [A]  time = 0.261968, antiderivative size = 185, 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, 820, 777, 618, 206} \[ -\frac{x^2 \left (4 a A c^2+2 a b B c-4 A b^2 c+b^3 B\right )+a \left (8 a B c-6 A b c+b^2 B\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{x^4 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (3 a b B-A \left (2 a c+b^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(x^4*(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) - (a*(b^2*B - 6*A*b*c + 8*a*B
*c) + (b^3*B - 4*A*b^2*c + 2*a*b*B*c + 4*a*A*c^2)*x^2)/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*a*b*B -
A*(b^2 + 2*a*c))*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 820

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)*(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)*Simp[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, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 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^5 \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^2 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac{x^4 \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{\operatorname{Subst}\left (\int \frac{x (-2 (A b-2 a B)-(b B-2 A c) x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac{x^4 \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{a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (3 a b B-A \left (b^2+2 a c\right )\right ) \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^4 \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{a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (3 a b B-A \left (b^2+2 a c\right )\right ) \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^4 \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{a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (3 a b B-A \left (b^2+2 a c\right )\right ) \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 = 0.271144, size = 233, normalized size = 1.26 \[ \frac{1}{4} \left (\frac{2 a^2 B c+a \left (b c \left (A+3 B x^2\right )-2 A c^2 x^2+b^2 (-B)\right )+b^2 x^2 (A c-b B)}{c^2 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}+\frac{b^2 c \left (5 a B+2 A c x^2\right )+2 a b c^2 \left (A-3 B x^2\right )+4 a c^2 \left (A c x^2-4 a B\right )+A b^3 c+b^4 (-B)}{c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{4 \left (A \left (2 a c+b^2\right )-3 a b B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

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

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Maple [B]  time = 0.014, size = 411, normalized size = 2.2 \begin{align*}{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ({\frac{c \left ( 2\,aAc+A{b}^{2}-3\,abB \right ){x}^{6}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+{\frac{ \left ( 6\,aAb{c}^{2}+3\,A{b}^{3}c-16\,{a}^{2}B{c}^{2}-a{b}^{2}Bc-{b}^{4}B \right ){x}^{4}}{2\,c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}-{\frac{a \left ( 2\,aA{c}^{2}-5\,A{b}^{2}c+5\,abBc+{b}^{3}B \right ){x}^{2}}{c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{{a}^{2} \left ( 6\,Abc-8\,aBc-{b}^{2}B \right ) }{2\,c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }} \right ) }+2\,{\frac{aAc}{ \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 ) }+{\frac{A{b}^{2}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-3\,{\frac{abB}{ \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^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x)

[Out]

1/2*(c*(2*A*a*c+A*b^2-3*B*a*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*(6*A*a*b*c^2+3*A*b^3*c-16*B*a^2*c^2-B*a*b^2*
c-B*b^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-1/c*a*(2*A*a*c^2-5*A*b^2*c+5*B*a*b*c+B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^
4)*x^2+1/2*a^2*(6*A*b*c-8*B*a*c-B*b^2)/c/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+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))*a*A*c+1/(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^2-3/(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*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^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.46039, size = 2824, normalized size = 15.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/4*(B*a^2*b^4 + 2*(8*A*a^2*c^4 - 2*(6*B*a^2*b - A*a*b^2)*c^3 + (3*B*a*b^3 - A*b^4)*c^2)*x^6 + (B*b^6 - 8*(8
*B*a^3 - 3*A*a^2*b)*c^3 + 6*(2*B*a^2*b^2 + A*a*b^3)*c^2 - 3*(B*a*b^4 + A*b^5)*c)*x^4 - 8*(4*B*a^4 - 3*A*a^3*b)
*c^2 + 2*(B*a*b^5 - 8*A*a^3*c^3 - 2*(10*B*a^3*b - 11*A*a^2*b^2)*c^2 + (B*a^2*b^3 - 5*A*a*b^4)*c)*x^2 - 2*((2*A
*a*c^4 - (3*B*a*b - A*b^2)*c^3)*x^8 + 2*(2*A*a*b*c^3 - (3*B*a*b^2 - A*b^3)*c^2)*x^6 + 2*A*a^3*c^2 + (4*A*a^2*c
^3 - 2*(3*B*a^2*b - 2*A*a*b^2)*c^2 - (3*B*a*b^3 - A*b^4)*c)*x^4 + 2*(2*A*a^2*b*c^2 - (3*B*a^2*b^2 - A*a*b^3)*c
)*x^2 - (3*B*a^3*b - A*a^2*b^2)*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*(2*B*a^3*b^2 - 3*A*a^2*b^3)*c)/(a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^
4*b^2*c^3 - 64*a^5*c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^8 + 2*(b^7*c^2 - 12*a*b^5*c^
3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + (b^8*c - 10*a*b^6*c^2 + 24*a^2*b^4*c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5
)*x^4 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*x^2), -1/4*(B*a^2*b^4 + 2*(8*A*a^2*c^4 -
2*(6*B*a^2*b - A*a*b^2)*c^3 + (3*B*a*b^3 - A*b^4)*c^2)*x^6 + (B*b^6 - 8*(8*B*a^3 - 3*A*a^2*b)*c^3 + 6*(2*B*a^2
*b^2 + A*a*b^3)*c^2 - 3*(B*a*b^4 + A*b^5)*c)*x^4 - 8*(4*B*a^4 - 3*A*a^3*b)*c^2 + 2*(B*a*b^5 - 8*A*a^3*c^3 - 2*
(10*B*a^3*b - 11*A*a^2*b^2)*c^2 + (B*a^2*b^3 - 5*A*a*b^4)*c)*x^2 + 4*((2*A*a*c^4 - (3*B*a*b - A*b^2)*c^3)*x^8
+ 2*(2*A*a*b*c^3 - (3*B*a*b^2 - A*b^3)*c^2)*x^6 + 2*A*a^3*c^2 + (4*A*a^2*c^3 - 2*(3*B*a^2*b - 2*A*a*b^2)*c^2 -
(3*B*a*b^3 - A*b^4)*c)*x^4 + 2*(2*A*a^2*b*c^2 - (3*B*a^2*b^2 - A*a*b^3)*c)*x^2 - (3*B*a^3*b - A*a^2*b^2)*c)*s
qrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + 2*(2*B*a^3*b^2 - 3*A*a^2*b^3)*c)/(
a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5*c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c
^6)*x^8 + 2*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + (b^8*c - 10*a*b^6*c^2 + 24*a^2*b^4*
c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5)*x^4 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*x^2)]

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Sympy [B]  time = 63.2358, size = 833, normalized size = 4.5 \begin{align*} \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) \log{\left (x^{2} + \frac{- 2 A a b c - A b^{3} + 3 B a b^{2} - 64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) + 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) - 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) + b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right )}{- 4 A a c^{2} - 2 A b^{2} c + 6 B a b c} \right )}}{2} - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) \log{\left (x^{2} + \frac{- 2 A a b c - A b^{3} + 3 B a b^{2} + 64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) - 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) + 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right ) - b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A a c - A b^{2} + 3 B a b\right )}{- 4 A a c^{2} - 2 A b^{2} c + 6 B a b c} \right )}}{2} - \frac{- 6 A a^{2} b c + 8 B a^{3} c + B a^{2} b^{2} + x^{6} \left (- 4 A a c^{3} - 2 A b^{2} c^{2} + 6 B a b c^{2}\right ) + x^{4} \left (- 6 A a b c^{2} - 3 A b^{3} c + 16 B a^{2} c^{2} + B a b^{2} c + B b^{4}\right ) + x^{2} \left (4 A a^{2} c^{2} - 10 A a b^{2} c + 10 B a^{2} b c + 2 B a b^{3}\right )}{64 a^{4} c^{3} - 32 a^{3} b^{2} c^{2} + 4 a^{2} b^{4} c + x^{8} \left (64 a^{2} c^{5} - 32 a b^{2} c^{4} + 4 b^{4} c^{3}\right ) + x^{6} \left (128 a^{2} b c^{4} - 64 a b^{3} c^{3} + 8 b^{5} c^{2}\right ) + x^{4} \left (128 a^{3} c^{4} - 24 a b^{4} c^{2} + 4 b^{6} c\right ) + x^{2} \left (128 a^{3} b c^{3} - 64 a^{2} b^{3} c^{2} + 8 a b^{5} c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

sqrt(-1/(4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b)*log(x**2 + (-2*A*a*b*c - A*b**3 + 3*B*a*b**2 - 64*a**
3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b) + 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)
*(-2*A*a*c - A*b**2 + 3*B*a*b) - 12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b) + b**6*s
qrt(-1/(4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b))/(-4*A*a*c**2 - 2*A*b**2*c + 6*B*a*b*c))/2 - sqrt(-1/(
4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b)*log(x**2 + (-2*A*a*b*c - A*b**3 + 3*B*a*b**2 + 64*a**3*c**3*sq
rt(-1/(4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b) - 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*a*
c - A*b**2 + 3*B*a*b) + 12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b) - b**6*sqrt(-1/(4
*a*c - b**2)**5)*(-2*A*a*c - A*b**2 + 3*B*a*b))/(-4*A*a*c**2 - 2*A*b**2*c + 6*B*a*b*c))/2 - (-6*A*a**2*b*c + 8
*B*a**3*c + B*a**2*b**2 + x**6*(-4*A*a*c**3 - 2*A*b**2*c**2 + 6*B*a*b*c**2) + x**4*(-6*A*a*b*c**2 - 3*A*b**3*c
+ 16*B*a**2*c**2 + B*a*b**2*c + B*b**4) + x**2*(4*A*a**2*c**2 - 10*A*a*b**2*c + 10*B*a**2*b*c + 2*B*a*b**3))/
(64*a**4*c**3 - 32*a**3*b**2*c**2 + 4*a**2*b**4*c + x**8*(64*a**2*c**5 - 32*a*b**2*c**4 + 4*b**4*c**3) + x**6*
(128*a**2*b*c**4 - 64*a*b**3*c**3 + 8*b**5*c**2) + x**4*(128*a**3*c**4 - 24*a*b**4*c**2 + 4*b**6*c) + x**2*(12
8*a**3*b*c**3 - 64*a**2*b**3*c**2 + 8*a*b**5*c))

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Giac [A]  time = 31.8293, size = 362, normalized size = 1.96 \begin{align*} -\frac{{\left (3 \, B a b - A b^{2} - 2 \, A a c\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{6 \, B a b c^{2} x^{6} - 2 \, A b^{2} c^{2} x^{6} - 4 \, A a c^{3} x^{6} + B b^{4} x^{4} + B a b^{2} c x^{4} - 3 \, A b^{3} c x^{4} + 16 \, B a^{2} c^{2} x^{4} - 6 \, A a b c^{2} x^{4} + 2 \, B a b^{3} x^{2} + 10 \, B a^{2} b c x^{2} - 10 \, A a b^{2} c x^{2} + 4 \, A a^{2} c^{2} x^{2} + B a^{2} b^{2} + 8 \, B a^{3} c - 6 \, A a^{2} b c}{4 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\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^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

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