3.124 \(\int \frac{x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=365 \[ -\frac{x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^2 \left (30 a^2 B c^2+7 a A b c^2-21 a b^2 B c-A b^3 c+3 b^4 B\right )}{2 c^3 \left (b^2-4 a c\right )^2}-\frac{\left (-60 a^3 B c^3-30 a^2 A b c^3+90 a^2 b^2 B c^2+10 a A b^3 c^2-30 a b^4 B c-A b^5 c+3 b^6 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]

[Out]

((3*b^4*B - A*b^3*c - 21*a*b^2*B*c + 7*a*A*b*c^2 + 30*a^2*B*c^2)*x^2)/(2*c^3*(b^
2 - 4*a*c)^2) - (x^8*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^
2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x^4*(a*(3*b^3*B - A*b^2*c - 18*a*b*B*c + 16
*a*A*c^2) + (3*b^4*B - A*b^3*c - 20*a*b^2*B*c + 10*a*A*b*c^2 + 20*a^2*B*c^2)*x^2
))/(4*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((3*b^6*B - A*b^5*c - 30*a*b^4*
B*c + 10*a*A*b^3*c^2 + 90*a^2*b^2*B*c^2 - 30*a^2*A*b*c^3 - 60*a^3*B*c^3)*ArcTanh
[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(5/2)) - ((3*b*B - A*c)*
Log[a + b*x^2 + c*x^4])/(4*c^4)

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Rubi [A]  time = 2.7144, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^2 \left (30 a^2 B c^2+7 a A b c^2-21 a b^2 B c-A b^3 c+3 b^4 B\right )}{2 c^3 \left (b^2-4 a c\right )^2}-\frac{\left (-60 a^3 B c^3-30 a^2 A b c^3+90 a^2 b^2 B c^2+10 a A b^3 c^2-30 a b^4 B c-A b^5 c+3 b^6 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^11*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((3*b^4*B - A*b^3*c - 21*a*b^2*B*c + 7*a*A*b*c^2 + 30*a^2*B*c^2)*x^2)/(2*c^3*(b^
2 - 4*a*c)^2) - (x^8*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^
2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x^4*(a*(3*b^3*B - A*b^2*c - 18*a*b*B*c + 16
*a*A*c^2) + (3*b^4*B - A*b^3*c - 20*a*b^2*B*c + 10*a*A*b*c^2 + 20*a^2*B*c^2)*x^2
))/(4*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((3*b^6*B - A*b^5*c - 30*a*b^4*
B*c + 10*a*A*b^3*c^2 + 90*a^2*b^2*B*c^2 - 30*a^2*A*b*c^3 - 60*a^3*B*c^3)*ArcTanh
[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(5/2)) - ((3*b*B - A*c)*
Log[a + b*x^2 + c*x^4])/(4*c^4)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 1.38408, size = 435, normalized size = 1.19 \[ \frac{\frac{a^3 c^2 \left (2 c \left (A+B x^2\right )-5 b B\right )+a^2 b c \left (-b c \left (4 A+9 B x^2\right )+5 A c^2 x^2+5 b^2 B\right )+a b^3 \left (b c \left (A+6 B x^2\right )-5 A c^2 x^2+b^2 (-B)\right )+b^5 x^2 (A c-b B)}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{2 c \left (60 a^3 B c^3+30 a^2 A b c^3-90 a^2 b^2 B c^2-10 a A b^3 c^2+30 a b^4 B c+A b^5 c-3 b^6 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{4 a^3 c^4 \left (8 A+9 B x^2\right )-3 a^2 b^2 c^3 \left (13 A+34 B x^2\right )+2 a^2 b c^3 \left (25 A c x^2-39 a B\right )+2 b^5 c \left (2 A c x^2-7 a B\right )+a b^4 c^2 \left (11 A+48 B x^2\right )+a b^3 c^2 \left (61 a B-30 A c x^2\right )-b^6 c \left (A+6 B x^2\right )+b^7 B}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+c (A c-3 b B) \log \left (a+b x^2+c x^4\right )+2 B c^2 x^2}{4 c^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^11*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

(2*B*c^2*x^2 + (b^7*B - b^6*c*(A + 6*B*x^2) + 4*a^3*c^4*(8*A + 9*B*x^2) - 3*a^2*
b^2*c^3*(13*A + 34*B*x^2) + a*b^4*c^2*(11*A + 48*B*x^2) + a*b^3*c^2*(61*a*B - 30
*A*c*x^2) + 2*b^5*c*(-7*a*B + 2*A*c*x^2) + 2*a^2*b*c^3*(-39*a*B + 25*A*c*x^2))/(
(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^5*(-(b*B) + A*c)*x^2 + a^3*c^2*(-5*b*B
+ 2*c*(A + B*x^2)) + a*b^3*(-(b^2*B) - 5*A*c^2*x^2 + b*c*(A + 6*B*x^2)) + a^2*b
*c*(5*b^2*B + 5*A*c^2*x^2 - b*c*(4*A + 9*B*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*
x^4)^2) - (2*c*(-3*b^6*B + A*b^5*c + 30*a*b^4*B*c - 10*a*A*b^3*c^2 - 90*a^2*b^2*
B*c^2 + 30*a^2*A*b*c^3 + 60*a^3*B*c^3)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])
/(-b^2 + 4*a*c)^(5/2) + c*(-3*b*B + A*c)*Log[a + b*x^2 + c*x^4])/(4*c^5)

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Maple [B]  time = 0.034, size = 2916, normalized size = 8. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x)

[Out]

7/c/(c*x^4+b*x^2+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B-21/4/c^2/(c*x^4+b*x^2
+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b^2-29/2/c^2/(c*x^4+b*x^2+a)^2*a^4/(16*a^
2*c^2-8*a*b^2*c+b^4)*B*b-5/4/c^4/(c*x^4+b*x^2+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*B*b^5-3/2/c^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*b^6+9/(c*x^4+
b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*a^3+8/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^4*A*a^3-51/2/c/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6
*B*a^2*b^2+12/c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*a*b^4+1/c^2
/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*b^5-5/4/c^4/(c*x^4+b*x^2+a)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*b^7+11/4/c/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*
b^2*c+b^4)*x^4*A*a^2*b^2-19/4/c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^4*A*a*b^4-21/2/c/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*a^3*b-41/4/
c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*a^2*b^3+31/2/c/(c*x^4+b*x
^2+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b-11/c^2/(c*x^4+b*x^2+a)^2*a^2/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^3-71/2/c^2/(c*x^4+b*x^2+a)^2*a^3/(16*a^2*c^2-8*a*
b^2*c+b^4)*x^2*B*b^2-5/2/c^4/(c*x^4+b*x^2+a)^2*a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*
B*b^6+17/2/c^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*a*b^5+3/2/c^3/
(c*x^4+b*x^2+a)^2*a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^5+3/4/c^3/(c*x^4+b*x^2+a)
^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*b^6+3/4/c^3/(c*x^4+b*x^2+a)^2*a^2/(16*a^2*c^
2-8*a*b^2*c+b^4)*A*b^4+9/c^3/(c*x^4+b*x^2+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*B*
b^3-15/c/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8
*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+
b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*
c-b^10)^(1/2))*A*a^2*b+5/c^2/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*
a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(1
6*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a
^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*A*a*b^3+45/c^2/(1024*a^5*c^5-1280*a^4*b^2*c^4
+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*
c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+
640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*B*a^2*b^2-2/c^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*A*a*b^2-12/c^2
/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*B*a^2
*b+6/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a
))*B*a*b^3-15/c^3/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2
+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8
*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+
20*a*b^8*c-b^10)^(1/2))*B*a*b^4+25/2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^6*A*a^2*b+1/2*B*x^2/c^3+19/c^3/(c*x^4+b*x^2+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c+b
^4)*x^2*B*b^4-15/2/c/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*a*b^3+6/
c/(c*x^4+b*x^2+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*A-1/2/c^3/(1024*a^5*c^5-1280*
a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x
^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a
^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*A*b^5-30/c/(1
024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1
/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(10
24*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/
2))*a^3*B+3/2/c^4/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2
+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8
*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+
20*a*b^8*c-b^10)^(1/2))*b^6*B+4/c/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*
b^2*c+b^4)*(c*x^4+b*x^2+a))*A*a^2+1/4/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*
c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*A*b^4-3/4/c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(
(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*B*b^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^11/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.605216, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^11/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[-1/4*((3*B*a^2*b^6 + (3*B*b^6*c^2 - 30*(2*B*a^3 + A*a^2*b)*c^5 + 10*(9*B*a^2*b^
2 + A*a*b^3)*c^4 - (30*B*a*b^4 + A*b^5)*c^3)*x^8 + 2*(3*B*b^7*c - 30*(2*B*a^3*b
+ A*a^2*b^2)*c^4 + 10*(9*B*a^2*b^3 + A*a*b^4)*c^3 - (30*B*a*b^5 + A*b^6)*c^2)*x^
6 + (3*B*b^8 - 60*(2*B*a^4 + A*a^3*b)*c^4 + 10*(12*B*a^3*b^2 - A*a^2*b^3)*c^3 +
2*(15*B*a^2*b^4 + 4*A*a*b^5)*c^2 - (24*B*a*b^6 + A*b^7)*c)*x^4 - 30*(2*B*a^5 + A
*a^4*b)*c^3 + 10*(9*B*a^4*b^2 + A*a^3*b^3)*c^2 + 2*(3*B*a*b^7 - 30*(2*B*a^4*b +
A*a^3*b^2)*c^3 + 10*(9*B*a^3*b^3 + A*a^2*b^4)*c^2 - (30*B*a^2*b^5 + A*a*b^6)*c)*
x^2 - (30*B*a^3*b^4 + A*a^2*b^5)*c)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2
+ (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a))
- (2*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*x^10 + 4*(B*b^5*c^2 - 8*B*a*b^3
*c^3 + 16*B*a^2*b*c^4)*x^8 - 5*B*a^2*b^5 + 24*A*a^4*c^3 - 2*(2*B*b^6*c - 25*(2*B
*a^3 + A*a^2*b)*c^4 + 3*(17*B*a^2*b^2 + 5*A*a*b^3)*c^3 - 2*(9*B*a*b^4 + A*b^5)*c
^2)*x^6 - (5*B*b^7 - 32*A*a^3*c^4 - 11*(2*B*a^3*b + A*a^2*b^2)*c^3 + (73*B*a^2*b
^3 + 19*A*a*b^4)*c^2 - (38*B*a*b^5 + 3*A*b^6)*c)*x^4 - (58*B*a^4*b + 21*A*a^3*b^
2)*c^2 - 2*(5*B*a*b^6 - (30*B*a^4 + 31*A*a^3*b)*c^3 + (79*B*a^3*b^2 + 22*A*a^2*b
^3)*c^2 - 3*(13*B*a^2*b^4 + A*a*b^5)*c)*x^2 + 3*(12*B*a^3*b^3 + A*a^2*b^4)*c - (
(3*B*b^5*c^2 - 16*A*a^2*c^5 + 8*(6*B*a^2*b + A*a*b^2)*c^4 - (24*B*a*b^3 + A*b^4)
*c^3)*x^8 + 3*B*a^2*b^5 - 16*A*a^4*c^3 + 2*(3*B*b^6*c - 16*A*a^2*b*c^4 + 8*(6*B*
a^2*b^2 + A*a*b^3)*c^3 - (24*B*a*b^4 + A*b^5)*c^2)*x^6 + (3*B*b^7 + 6*A*a*b^4*c^
2 + 96*B*a^3*b*c^3 - 32*A*a^3*c^4 - (18*B*a*b^5 + A*b^6)*c)*x^4 + 8*(6*B*a^4*b +
A*a^3*b^2)*c^2 + 2*(3*B*a*b^6 - 16*A*a^3*b*c^3 + 8*(6*B*a^3*b^2 + A*a^2*b^3)*c^
2 - (24*B*a^2*b^4 + A*a*b^5)*c)*x^2 - (24*B*a^3*b^3 + A*a^2*b^4)*c)*log(c*x^4 +
b*x^2 + a))*sqrt(b^2 - 4*a*c))/((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6 + (b^4
*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^8 + 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*
x^6 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^4 + 2*(a*b^5*c^4 - 8*a^2*b^3*c^5 +
16*a^3*b*c^6)*x^2)*sqrt(b^2 - 4*a*c)), 1/4*(2*(3*B*a^2*b^6 + (3*B*b^6*c^2 - 30*(
2*B*a^3 + A*a^2*b)*c^5 + 10*(9*B*a^2*b^2 + A*a*b^3)*c^4 - (30*B*a*b^4 + A*b^5)*c
^3)*x^8 + 2*(3*B*b^7*c - 30*(2*B*a^3*b + A*a^2*b^2)*c^4 + 10*(9*B*a^2*b^3 + A*a*
b^4)*c^3 - (30*B*a*b^5 + A*b^6)*c^2)*x^6 + (3*B*b^8 - 60*(2*B*a^4 + A*a^3*b)*c^4
+ 10*(12*B*a^3*b^2 - A*a^2*b^3)*c^3 + 2*(15*B*a^2*b^4 + 4*A*a*b^5)*c^2 - (24*B*
a*b^6 + A*b^7)*c)*x^4 - 30*(2*B*a^5 + A*a^4*b)*c^3 + 10*(9*B*a^4*b^2 + A*a^3*b^3
)*c^2 + 2*(3*B*a*b^7 - 30*(2*B*a^4*b + A*a^3*b^2)*c^3 + 10*(9*B*a^3*b^3 + A*a^2*
b^4)*c^2 - (30*B*a^2*b^5 + A*a*b^6)*c)*x^2 - (30*B*a^3*b^4 + A*a^2*b^5)*c)*arcta
n(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (2*(B*b^4*c^3 - 8*B*a*b^2*c
^4 + 16*B*a^2*c^5)*x^10 + 4*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*x^8 - 5
*B*a^2*b^5 + 24*A*a^4*c^3 - 2*(2*B*b^6*c - 25*(2*B*a^3 + A*a^2*b)*c^4 + 3*(17*B*
a^2*b^2 + 5*A*a*b^3)*c^3 - 2*(9*B*a*b^4 + A*b^5)*c^2)*x^6 - (5*B*b^7 - 32*A*a^3*
c^4 - 11*(2*B*a^3*b + A*a^2*b^2)*c^3 + (73*B*a^2*b^3 + 19*A*a*b^4)*c^2 - (38*B*a
*b^5 + 3*A*b^6)*c)*x^4 - (58*B*a^4*b + 21*A*a^3*b^2)*c^2 - 2*(5*B*a*b^6 - (30*B*
a^4 + 31*A*a^3*b)*c^3 + (79*B*a^3*b^2 + 22*A*a^2*b^3)*c^2 - 3*(13*B*a^2*b^4 + A*
a*b^5)*c)*x^2 + 3*(12*B*a^3*b^3 + A*a^2*b^4)*c - ((3*B*b^5*c^2 - 16*A*a^2*c^5 +
8*(6*B*a^2*b + A*a*b^2)*c^4 - (24*B*a*b^3 + A*b^4)*c^3)*x^8 + 3*B*a^2*b^5 - 16*A
*a^4*c^3 + 2*(3*B*b^6*c - 16*A*a^2*b*c^4 + 8*(6*B*a^2*b^2 + A*a*b^3)*c^3 - (24*B
*a*b^4 + A*b^5)*c^2)*x^6 + (3*B*b^7 + 6*A*a*b^4*c^2 + 96*B*a^3*b*c^3 - 32*A*a^3*
c^4 - (18*B*a*b^5 + A*b^6)*c)*x^4 + 8*(6*B*a^4*b + A*a^3*b^2)*c^2 + 2*(3*B*a*b^6
- 16*A*a^3*b*c^3 + 8*(6*B*a^3*b^2 + A*a^2*b^3)*c^2 - (24*B*a^2*b^4 + A*a*b^5)*c
)*x^2 - (24*B*a^3*b^3 + A*a^2*b^4)*c)*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4*a*c)
)/((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c
^8)*x^8 + 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^6 + (b^6*c^4 - 6*a*b^4*c^5
+ 32*a^3*c^7)*x^4 + 2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x^2)*sqrt(-b^2
+ 4*a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 15.9764, size = 807, normalized size = 2.21 \[ \frac{{\left (3 \, B b^{6} - 30 \, B a b^{4} c - A b^{5} c + 90 \, B a^{2} b^{2} c^{2} + 10 \, A a b^{3} c^{2} - 60 \, B a^{3} c^{3} - 30 \, A a^{2} b c^{3}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{B x^{2}}{2 \, c^{3}} + \frac{9 \, B b^{5} c^{2} x^{8} - 72 \, B a b^{3} c^{3} x^{8} - 3 \, A b^{4} c^{3} x^{8} + 144 \, B a^{2} b c^{4} x^{8} + 24 \, A a b^{2} c^{4} x^{8} - 48 \, A a^{2} c^{5} x^{8} + 6 \, B b^{6} c x^{6} - 48 \, B a b^{4} c^{2} x^{6} + 2 \, A b^{5} c^{2} x^{6} + 84 \, B a^{2} b^{2} c^{3} x^{6} - 12 \, A a b^{3} c^{3} x^{6} + 72 \, B a^{3} c^{4} x^{6} + 4 \, A a^{2} b c^{4} x^{6} - B b^{7} x^{4} + 14 \, B a b^{5} c x^{4} + 3 \, A b^{6} c x^{4} - 82 \, B a^{2} b^{3} c^{2} x^{4} - 20 \, A a b^{4} c^{2} x^{4} + 204 \, B a^{3} b c^{3} x^{4} + 22 \, A a^{2} b^{2} c^{3} x^{4} - 32 \, A a^{3} c^{4} x^{4} - 2 \, B a b^{6} x^{2} + 8 \, B a^{2} b^{4} c x^{2} + 6 \, A a b^{5} c x^{2} + 4 \, B a^{3} b^{2} c^{2} x^{2} - 40 \, A a^{2} b^{3} c^{2} x^{2} + 56 \, B a^{4} c^{3} x^{2} + 28 \, A a^{3} b c^{3} x^{2} - B a^{2} b^{5} + 3 \, A a^{2} b^{4} c + 28 \, B a^{4} b c^{2} - 18 \, A a^{3} b^{2} c^{2}}{8 \,{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac{{\left (3 \, B b - A c\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^11/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

1/2*(3*B*b^6 - 30*B*a*b^4*c - A*b^5*c + 90*B*a^2*b^2*c^2 + 10*A*a*b^3*c^2 - 60*B
*a^3*c^3 - 30*A*a^2*b*c^3)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^4 -
8*a*b^2*c^5 + 16*a^2*c^6)*sqrt(-b^2 + 4*a*c)) + 1/2*B*x^2/c^3 + 1/8*(9*B*b^5*c^2
*x^8 - 72*B*a*b^3*c^3*x^8 - 3*A*b^4*c^3*x^8 + 144*B*a^2*b*c^4*x^8 + 24*A*a*b^2*c
^4*x^8 - 48*A*a^2*c^5*x^8 + 6*B*b^6*c*x^6 - 48*B*a*b^4*c^2*x^6 + 2*A*b^5*c^2*x^6
+ 84*B*a^2*b^2*c^3*x^6 - 12*A*a*b^3*c^3*x^6 + 72*B*a^3*c^4*x^6 + 4*A*a^2*b*c^4*
x^6 - B*b^7*x^4 + 14*B*a*b^5*c*x^4 + 3*A*b^6*c*x^4 - 82*B*a^2*b^3*c^2*x^4 - 20*A
*a*b^4*c^2*x^4 + 204*B*a^3*b*c^3*x^4 + 22*A*a^2*b^2*c^3*x^4 - 32*A*a^3*c^4*x^4 -
2*B*a*b^6*x^2 + 8*B*a^2*b^4*c*x^2 + 6*A*a*b^5*c*x^2 + 4*B*a^3*b^2*c^2*x^2 - 40*
A*a^2*b^3*c^2*x^2 + 56*B*a^4*c^3*x^2 + 28*A*a^3*b*c^3*x^2 - B*a^2*b^5 + 3*A*a^2*
b^4*c + 28*B*a^4*b*c^2 - 18*A*a^3*b^2*c^2)/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)
*(c*x^4 + b*x^2 + a)^2) - 1/4*(3*B*b - A*c)*ln(c*x^4 + b*x^2 + a)/c^4