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

Optimal. Leaf size=363 \[ \frac{(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{a b B \left (b^2-7 a c\right )-3 A \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{\left (a b B \left (30 a^2 c^2-10 a b^2 c+b^4\right )-3 A \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac{-A \left (b^2-2 a c\right )+c x^2 (-(A b-2 a B))+a b B}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.51923, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{a b B \left (b^2-7 a c\right )-3 A \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{\left (a b B \left (30 a^2 c^2-10 a b^2 c+b^4\right )-3 A \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}+\frac{c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(a*b*B*(b^2 - 7*a*c) - 3*A*(b^4 - 7*a*b^2*c + 10*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^
2*x^2) + (A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*
(a + b*x^2 + c*x^4)^2) - (a*b*B*(b^2 - 10*a*c) - A*(3*b^4 - 20*a*b^2*c + 20*a^2*
c^2) + c*(a*B*(b^2 - 16*a*c) - 3*A*(b^3 - 6*a*b*c))*x^2)/(4*a^2*(b^2 - 4*a*c)^2*
x^2*(a + b*x^2 + c*x^4)) + ((a*b*B*(b^4 - 10*a*b^2*c + 30*a^2*c^2) - 3*A*(b^6 -
10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*
c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - ((3*A*b - a*B)*Log[x])/a^4 + ((3*A*b - a*B)*L
og[a + b*x^2 + c*x^4])/(4*a^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((B*x**2+A)/x**3/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.97907, size = 642, normalized size = 1.77 \[ \frac{-\frac{a^2 \left (A \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )+a B \left (2 a c-b^2-b c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{a \left (a B \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 x^2+2 b^4+2 b^3 c x^2\right )-A \left (46 a^2 b c^2+28 a^2 c^3 x^2-29 a b^3 c-26 a b^2 c^2 x^2+4 b^5+4 b^4 c x^2\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (3 A \left (-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right )-a B \left (16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right )\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{\left (a B \left (-16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}+b^5\right )+3 A \left (20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+4 \log (x) (a B-3 A b)-\frac{2 a A}{x^2}}{4 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a*A)/x^2 - (a^2*(a*B*(-b^2 + 2*a*c - b*c*x^2) + A*(b^3 - 3*a*b*c + b^2*c*x^
2 - 2*a*c^2*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a*(a*B*(2*b^4 - 15*a
*b^2*c + 16*a^2*c^2 + 2*b^3*c*x^2 - 14*a*b*c^2*x^2) - A*(4*b^5 - 29*a*b^3*c + 46
*a^2*b*c^2 + 4*b^4*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3*x^2)))/((b^2 - 4*a*c)^2
*(a + b*x^2 + c*x^4)) + 4*(-3*A*b + a*B)*Log[x] + ((-(a*B*(b^5 - 10*a*b^3*c + 30
*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sq
rt[b^2 - 4*a*c])) + 3*A*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sq
rt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c]))
*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + ((a*B*(b^5 - 10*a*b
^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a
^2*c^2*Sqrt[b^2 - 4*a*c]) + 3*A*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3
+ b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 -
4*a*c]))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^4)

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Maple [B]  time = 0.045, size = 2724, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30/a/(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*c^3-1/2/a^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*b^5+3/4/a^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))*A*b^5-4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2
*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*B+3/2/a^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))*A*b^6-1/4/a^3/(1
6*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^4-5/
4/a^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b^5+6*a/(c*x^4+b*x^2+a)^2/(
16*a^2*c^2-8*a*b^2*c+b^4)*B*c^2+3/4/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^
4)*B*b^4-3/a^4*ln(x)*A*b-29/4/a/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)
*x^4*B*b^2+1/a^2/(c*x^4+b*x^2+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*b^4-7/2/a/
(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^2*c^2+6/a^2/(c*x^4+b*x^2+a)
^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^4*c-3/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*
b^2*c+b^4)*x^2*B*b^3*c+13/2/a^2/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)
*x^6*A*b^2-1/a^3/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*b^4-7/2/
a/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*b*B+5/a^2/(1024*a^5*c^5-1
280*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^5-12
80*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*b^3*c-1
/2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B*b*c^2+1/2/a^2/(c*x^4+b*x^2
+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B*b^5+9/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a
*b^2*c+b^4)*A*b^3*c-6/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln((16*a^2*c^2-8*a*b^2*c+
b^4)*(c*x^4+b*x^2+a))*A*b^3+2/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln((16*a^2*c^2-8*
a*b^2*c+b^4)*(c*x^4+b*x^2+a))*B*b^2+45/a^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))*A*b^2*c^2+12/a^2/(16*a^2*c^2-8*
a*b^2*c+b^4)*c^2*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*A*b-15/a^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^4*c-15/a/(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*b*c^2-7/a/(c*x^4+b*x^2+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)
*x^6*A-1/a^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^6+1/2/a^2/(c*x
^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*b^3-37/2/a/(c*x^4+b*x^2+a)^2*
c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*b+55/4/a^2/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^4*A*b^3-2/a^3/(c*x^4+b*x^2+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*
x^4*A*b^5-29/2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b*c^2-21/4/(c*x^4+
b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*B*b^2*c+4/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^4*B-9/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*c^3-
1/2*A/a^3/x^2+1/a^3*ln(x)*B

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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)/((c*x^4 + b*x^2 + a)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.15119, 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)/((c*x^4 + b*x^2 + a)^3*x^3),x, algorithm="fricas")

[Out]

[1/4*(((60*A*a^3*c^5 + 30*(B*a^3*b - 3*A*a^2*b^2)*c^4 - 10*(B*a^2*b^3 - 3*A*a*b^
4)*c^3 + (B*a*b^5 - 3*A*b^6)*c^2)*x^10 + 2*(60*A*a^3*b*c^4 + 30*(B*a^3*b^2 - 3*A
*a^2*b^3)*c^3 - 10*(B*a^2*b^4 - 3*A*a*b^5)*c^2 + (B*a*b^6 - 3*A*b^7)*c)*x^8 + (B
*a*b^7 - 3*A*b^8 + 120*A*a^4*c^4 + 60*(B*a^4*b - 2*A*a^3*b^2)*c^3 + 10*(B*a^3*b^
3 - 3*A*a^2*b^4)*c^2 - 8*(B*a^2*b^5 - 3*A*a*b^6)*c)*x^6 + 2*(B*a^2*b^6 - 3*A*a*b
^7 + 60*A*a^4*b*c^3 + 30*(B*a^4*b^2 - 3*A*a^3*b^3)*c^2 - 10*(B*a^3*b^4 - 3*A*a^2
*b^5)*c)*x^4 + (B*a^3*b^5 - 3*A*a^2*b^6 + 60*A*a^5*c^3 + 30*(B*a^5*b - 3*A*a^4*b
^2)*c^2 - 10*(B*a^4*b^3 - 3*A*a^3*b^4)*c)*x^2)*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*(30*A*a^3*c^4 + 7*(B*a^3*b - 3*A*a^2*b^2)*c^3 - (B*a^2*b^3 - 3*
A*a*b^4)*c^2)*x^8 + 2*A*a^3*b^4 - 16*A*a^4*b^2*c + 32*A*a^5*c^2 - (2*(8*B*a^4 -
69*A*a^3*b)*c^3 - 29*(B*a^3*b^2 - 3*A*a^2*b^3)*c^2 + 4*(B*a^2*b^4 - 3*A*a*b^5)*c
)*x^6 - 2*(B*a^2*b^5 - 3*A*a*b^6 - 50*A*a^4*c^3 - (B*a^4*b + 7*A*a^3*b^2)*c^2 -
6*(B*a^3*b^3 - 3*A*a^2*b^4)*c)*x^4 - (3*B*a^3*b^4 - 9*A*a^2*b^5 + 2*(12*B*a^5 -
61*A*a^4*b)*c^2 - (21*B*a^4*b^2 - 68*A*a^3*b^3)*c)*x^2 + ((16*(B*a^3 - 3*A*a^2*b
)*c^4 - 8*(B*a^2*b^2 - 3*A*a*b^3)*c^3 + (B*a*b^4 - 3*A*b^5)*c^2)*x^10 + 2*(16*(B
*a^3*b - 3*A*a^2*b^2)*c^3 - 8*(B*a^2*b^3 - 3*A*a*b^4)*c^2 + (B*a*b^5 - 3*A*b^6)*
c)*x^8 + (B*a*b^6 - 3*A*b^7 + 32*(B*a^4 - 3*A*a^3*b)*c^3 - 6*(B*a^2*b^4 - 3*A*a*
b^5)*c)*x^6 + 2*(B*a^2*b^5 - 3*A*a*b^6 + 16*(B*a^4*b - 3*A*a^3*b^2)*c^2 - 8*(B*a
^3*b^3 - 3*A*a^2*b^4)*c)*x^4 + (B*a^3*b^4 - 3*A*a^2*b^5 + 16*(B*a^5 - 3*A*a^4*b)
*c^2 - 8*(B*a^4*b^2 - 3*A*a^3*b^3)*c)*x^2)*log(c*x^4 + b*x^2 + a) - 4*((16*(B*a^
3 - 3*A*a^2*b)*c^4 - 8*(B*a^2*b^2 - 3*A*a*b^3)*c^3 + (B*a*b^4 - 3*A*b^5)*c^2)*x^
10 + 2*(16*(B*a^3*b - 3*A*a^2*b^2)*c^3 - 8*(B*a^2*b^3 - 3*A*a*b^4)*c^2 + (B*a*b^
5 - 3*A*b^6)*c)*x^8 + (B*a*b^6 - 3*A*b^7 + 32*(B*a^4 - 3*A*a^3*b)*c^3 - 6*(B*a^2
*b^4 - 3*A*a*b^5)*c)*x^6 + 2*(B*a^2*b^5 - 3*A*a*b^6 + 16*(B*a^4*b - 3*A*a^3*b^2)
*c^2 - 8*(B*a^3*b^3 - 3*A*a^2*b^4)*c)*x^4 + (B*a^3*b^4 - 3*A*a^2*b^5 + 16*(B*a^5
- 3*A*a^4*b)*c^2 - 8*(B*a^4*b^2 - 3*A*a^3*b^3)*c)*x^2)*log(x))*sqrt(b^2 - 4*a*c
))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*x^10 + 2*(a^4*b^5*c - 8*a^5*b^3*
c^2 + 16*a^6*b*c^3)*x^8 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^6 + 2*(a^5*b^5
- 8*a^6*b^3*c + 16*a^7*b*c^2)*x^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x^2)*sq
rt(b^2 - 4*a*c)), -1/4*(2*((60*A*a^3*c^5 + 30*(B*a^3*b - 3*A*a^2*b^2)*c^4 - 10*(
B*a^2*b^3 - 3*A*a*b^4)*c^3 + (B*a*b^5 - 3*A*b^6)*c^2)*x^10 + 2*(60*A*a^3*b*c^4 +
30*(B*a^3*b^2 - 3*A*a^2*b^3)*c^3 - 10*(B*a^2*b^4 - 3*A*a*b^5)*c^2 + (B*a*b^6 -
3*A*b^7)*c)*x^8 + (B*a*b^7 - 3*A*b^8 + 120*A*a^4*c^4 + 60*(B*a^4*b - 2*A*a^3*b^2
)*c^3 + 10*(B*a^3*b^3 - 3*A*a^2*b^4)*c^2 - 8*(B*a^2*b^5 - 3*A*a*b^6)*c)*x^6 + 2*
(B*a^2*b^6 - 3*A*a*b^7 + 60*A*a^4*b*c^3 + 30*(B*a^4*b^2 - 3*A*a^3*b^3)*c^2 - 10*
(B*a^3*b^4 - 3*A*a^2*b^5)*c)*x^4 + (B*a^3*b^5 - 3*A*a^2*b^6 + 60*A*a^5*c^3 + 30*
(B*a^5*b - 3*A*a^4*b^2)*c^2 - 10*(B*a^4*b^3 - 3*A*a^3*b^4)*c)*x^2)*arctan(-(2*c*
x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (2*(30*A*a^3*c^4 + 7*(B*a^3*b - 3*A
*a^2*b^2)*c^3 - (B*a^2*b^3 - 3*A*a*b^4)*c^2)*x^8 + 2*A*a^3*b^4 - 16*A*a^4*b^2*c
+ 32*A*a^5*c^2 - (2*(8*B*a^4 - 69*A*a^3*b)*c^3 - 29*(B*a^3*b^2 - 3*A*a^2*b^3)*c^
2 + 4*(B*a^2*b^4 - 3*A*a*b^5)*c)*x^6 - 2*(B*a^2*b^5 - 3*A*a*b^6 - 50*A*a^4*c^3 -
(B*a^4*b + 7*A*a^3*b^2)*c^2 - 6*(B*a^3*b^3 - 3*A*a^2*b^4)*c)*x^4 - (3*B*a^3*b^4
- 9*A*a^2*b^5 + 2*(12*B*a^5 - 61*A*a^4*b)*c^2 - (21*B*a^4*b^2 - 68*A*a^3*b^3)*c
)*x^2 + ((16*(B*a^3 - 3*A*a^2*b)*c^4 - 8*(B*a^2*b^2 - 3*A*a*b^3)*c^3 + (B*a*b^4
- 3*A*b^5)*c^2)*x^10 + 2*(16*(B*a^3*b - 3*A*a^2*b^2)*c^3 - 8*(B*a^2*b^3 - 3*A*a*
b^4)*c^2 + (B*a*b^5 - 3*A*b^6)*c)*x^8 + (B*a*b^6 - 3*A*b^7 + 32*(B*a^4 - 3*A*a^3
*b)*c^3 - 6*(B*a^2*b^4 - 3*A*a*b^5)*c)*x^6 + 2*(B*a^2*b^5 - 3*A*a*b^6 + 16*(B*a^
4*b - 3*A*a^3*b^2)*c^2 - 8*(B*a^3*b^3 - 3*A*a^2*b^4)*c)*x^4 + (B*a^3*b^4 - 3*A*a
^2*b^5 + 16*(B*a^5 - 3*A*a^4*b)*c^2 - 8*(B*a^4*b^2 - 3*A*a^3*b^3)*c)*x^2)*log(c*
x^4 + b*x^2 + a) - 4*((16*(B*a^3 - 3*A*a^2*b)*c^4 - 8*(B*a^2*b^2 - 3*A*a*b^3)*c^
3 + (B*a*b^4 - 3*A*b^5)*c^2)*x^10 + 2*(16*(B*a^3*b - 3*A*a^2*b^2)*c^3 - 8*(B*a^2
*b^3 - 3*A*a*b^4)*c^2 + (B*a*b^5 - 3*A*b^6)*c)*x^8 + (B*a*b^6 - 3*A*b^7 + 32*(B*
a^4 - 3*A*a^3*b)*c^3 - 6*(B*a^2*b^4 - 3*A*a*b^5)*c)*x^6 + 2*(B*a^2*b^5 - 3*A*a*b
^6 + 16*(B*a^4*b - 3*A*a^3*b^2)*c^2 - 8*(B*a^3*b^3 - 3*A*a^2*b^4)*c)*x^4 + (B*a^
3*b^4 - 3*A*a^2*b^5 + 16*(B*a^5 - 3*A*a^4*b)*c^2 - 8*(B*a^4*b^2 - 3*A*a^3*b^3)*c
)*x^2)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*
x^10 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^8 + (a^4*b^6 - 6*a^5*b^4*c
+ 32*a^7*c^3)*x^6 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^4 + (a^6*b^4 - 8
*a^7*b^2*c + 16*a^8*c^2)*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((B*x**2+A)/x**3/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 15.7322, size = 875, normalized size = 2.41 \[ -\frac{{\left (B a b^{5} - 3 \, A b^{6} - 10 \, B a^{2} b^{3} c + 30 \, A a b^{4} c + 30 \, B a^{3} b c^{2} - 90 \, A a^{2} b^{2} c^{2} + 60 \, A a^{3} c^{3}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, B a b^{4} c^{2} x^{8} - 9 \, A b^{5} c^{2} x^{8} - 24 \, B a^{2} b^{2} c^{3} x^{8} + 72 \, A a b^{3} c^{3} x^{8} + 48 \, B a^{3} c^{4} x^{8} - 144 \, A a^{2} b c^{4} x^{8} + 6 \, B a b^{5} c x^{6} - 18 \, A b^{6} c x^{6} - 44 \, B a^{2} b^{3} c^{2} x^{6} + 136 \, A a b^{4} c^{2} x^{6} + 68 \, B a^{3} b c^{3} x^{6} - 236 \, A a^{2} b^{2} c^{3} x^{6} - 56 \, A a^{3} c^{4} x^{6} + 3 \, B a b^{6} x^{4} - 9 \, A b^{7} x^{4} - 10 \, B a^{2} b^{4} c x^{4} + 38 \, A a b^{5} c x^{4} - 58 \, B a^{3} b^{2} c^{2} x^{4} + 110 \, A a^{2} b^{3} c^{2} x^{4} + 128 \, B a^{4} c^{3} x^{4} - 436 \, A a^{3} b c^{3} x^{4} + 10 \, B a^{2} b^{5} x^{2} - 26 \, A a b^{6} x^{2} - 72 \, B a^{3} b^{3} c x^{2} + 192 \, A a^{2} b^{4} c x^{2} + 92 \, B a^{4} b c^{2} x^{2} - 316 \, A a^{3} b^{2} c^{2} x^{2} - 72 \, A a^{4} c^{3} x^{2} + 9 \, B a^{3} b^{4} - 19 \, A a^{2} b^{5} - 66 \, B a^{4} b^{2} c + 144 \, A a^{3} b^{3} c + 96 \, B a^{5} c^{2} - 260 \, A a^{4} b c^{2}}{8 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac{{\left (B a - 3 \, A b\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac{{\left (B a - 3 \, A b\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} - \frac{B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(B*a*b^5 - 3*A*b^6 - 10*B*a^2*b^3*c + 30*A*a*b^4*c + 30*B*a^3*b*c^2 - 90*A*
a^2*b^2*c^2 + 60*A*a^3*c^3)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^4 -
8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) + 1/8*(3*B*a*b^4*c^2*x^8 - 9*A*b^
5*c^2*x^8 - 24*B*a^2*b^2*c^3*x^8 + 72*A*a*b^3*c^3*x^8 + 48*B*a^3*c^4*x^8 - 144*A
*a^2*b*c^4*x^8 + 6*B*a*b^5*c*x^6 - 18*A*b^6*c*x^6 - 44*B*a^2*b^3*c^2*x^6 + 136*A
*a*b^4*c^2*x^6 + 68*B*a^3*b*c^3*x^6 - 236*A*a^2*b^2*c^3*x^6 - 56*A*a^3*c^4*x^6 +
3*B*a*b^6*x^4 - 9*A*b^7*x^4 - 10*B*a^2*b^4*c*x^4 + 38*A*a*b^5*c*x^4 - 58*B*a^3*
b^2*c^2*x^4 + 110*A*a^2*b^3*c^2*x^4 + 128*B*a^4*c^3*x^4 - 436*A*a^3*b*c^3*x^4 +
10*B*a^2*b^5*x^2 - 26*A*a*b^6*x^2 - 72*B*a^3*b^3*c*x^2 + 192*A*a^2*b^4*c*x^2 + 9
2*B*a^4*b*c^2*x^2 - 316*A*a^3*b^2*c^2*x^2 - 72*A*a^4*c^3*x^2 + 9*B*a^3*b^4 - 19*
A*a^2*b^5 - 66*B*a^4*b^2*c + 144*A*a^3*b^3*c + 96*B*a^5*c^2 - 260*A*a^4*b*c^2)/(
(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^4 + b*x^2 + a)^2) - 1/4*(B*a - 3*A*b)*
ln(c*x^4 + b*x^2 + a)/a^4 + 1/2*(B*a - 3*A*b)*ln(x^2)/a^4 - 1/2*(B*a*x^2 - 3*A*b
*x^2 + A*a)/(a^4*x^2)