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

Optimal. Leaf size=356 \[ -\frac{x \left (-2 a C+x^2 (2 A c-b C)+A b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (-\frac{4 A b c-C \left (4 a c+b^2\right )}{\sqrt{b^2-4 a c}}+2 A c-b C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{4 A b c-C \left (4 a c+b^2\right )}{\sqrt{b^2-4 a c}}+2 A c-b C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{B \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

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

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Rubi [A]  time = 1.11942, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{x \left (-2 a C+x^2 (2 A c-b C)+A b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (-\frac{4 A b c-C \left (4 a c+b^2\right )}{\sqrt{b^2-4 a c}}+2 A c-b C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{4 A b c-C \left (4 a c+b^2\right )}{\sqrt{b^2-4 a c}}+2 A c-b C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{B \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 0.269288, size = 378, normalized size = 1.06 \[ \frac{1}{4} \left (\frac{4 a (B+C x)+2 x \left (b x (B+C x)-A \left (b+2 c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (C \left (b \sqrt{b^2-4 a c}-4 a c-b^2\right )-2 A c \left (\sqrt{b^2-4 a c}-2 b\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (C \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right )-2 A c \left (\sqrt{b^2-4 a c}+2 b\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{2 b B \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x
^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*S
qrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(S
qrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b
+ Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sq
rt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqr
t[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(
3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4

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Maple [B]  time = 0.007, size = 4063, normalized size = 11.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/(4*a*c-b^2)^3*B*(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*b/(8*a*c^2-
2*b^2*c)*ln(8*x^2*a*c^2-2*x^2*b^2*c+4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2))*a*c^2+(1
/2*(2*A*c-C*b)/(4*a*c-b^2)*x^3-1/2*b*B/(4*a*c-b^2)*x^2+1/2*(A*b-2*C*a)/(4*a*c-b^
2)*x-a*B/(4*a*c-b^2))/(c*x^4+b*x^2+a)-4/(4*a*c-b^2)^3*2^(1/2)/((4*a*c-b^2)*c*(4*
a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/
((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*(-64*a^3*c^3+48*a^
2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*b*a*c^2+1/4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^
2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1
/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^
2)*c)^(1/2))*C*b^7+32/(4*a*c-b^2)^3*c^4*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4
*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c
*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*a^3+c/(4*a*c-b^2)^3*B*(-64*a^3*c
^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*b^3/(8*a*c^2-2*b^2*c)*ln(8*x^2*a*c^2-2*x
^2*b^2*c+4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2))-1/2*c/(4*a*c-b^2)^3*2^(1/2)/((4*a*c
-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)
*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*b^6-1/4
/(4*a*c-b^2)^3*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2
)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^
2)^3)^(1/2)))^(1/2))*C*(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*b^4-4/(
16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1
/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^
3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*C*(-64*a^3*c^3+48*a^2*b^2*c^2-12
*a*b^4*c+b^6)^(1/2)*a^2*c^2-4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*B*(-64*a^3*
c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*b/(-8*a*c^2+2*b^2*c)*ln(-8*x^2*a*c^2+2*
x^2*b^2*c-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*a*c^2-24/(16*a^2*c^2-8*a*b^2*c+b^4
)/(4*a*c-b^2)*c^3*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^
(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(
1/2))*(4*a*c-b^2)*c)^(1/2))*A*a^2*b^2+6/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*c
^2*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1
/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^
2)*c)^(1/2))*A*a*b^4-16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*c^3*2^(1/2)/((-4*
a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b
^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*C*a
^3*b-3*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-
b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-
4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*C*a*b^5-c/(16*a^2*c^2-
8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c
-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-
b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*A*(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^
6)^(1/2)*b^3+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*c^2*2^(1/2)/((-4*a*b*c+b^
3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*
2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*C*a^2*b^3+4
/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^
(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+
b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*A*(-64*a^3*c^3+48*a^2*b^2*c^2-
12*a*b^4*c+b^6)^(1/2)*b*a*c^2+1/4/(4*a*c-b^2)^3*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-
b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*
c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*C*b^7-24/(4*a*c-b^2)^3*c^3
*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(
8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^
(1/2))*A*a^2*b^2+32/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*c^4*2^(1/2)/((-4*a*b*
c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c
)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*A*a^3+1
/4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3
)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*
c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*C*(-64*a^3*c^3+48*a^2*b^2*c^
2-12*a*b^4*c+b^6)^(1/2)*b^4+6/(4*a*c-b^2)^3*c^2*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-
b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*
c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*a*b^4-16/(4*a*c-b^2)^3*c
^3*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2
*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2))
)^(1/2))*C*a^3*b+12/(4*a*c-b^2)^3*c^2*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a
*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(
4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*C*a^2*b^3+4/(4*a*c-b^2)^3*2^(1/2)/((
4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b
^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*C*(-
64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*a^2*c^2+c/(16*a^2*c^2-8*a*b^2*c+
b^4)/(4*a*c-b^2)*B*(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/2)*b^3/(-8*a*c
^2+2*b^2*c)*ln(-8*x^2*a*c^2+2*x^2*b^2*c-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))-1/2*
c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)
^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c
+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*A*b^6-3*c/(4*a*c-b^2)^3*2^(1/
2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^
2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))
*C*a*b^5+c/(4*a*c-b^2)^3*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1
/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(
-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)^(1/
2)*b^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b x^{2} +{\left (C b - 2 \, A c\right )} x^{3} + 2 \, B a +{\left (2 \, C a - A b\right )} x}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}} - \frac{-\int \frac{2 \, B b x +{\left (C b - 2 \, A c\right )} x^{2} - 2 \, C a + A b}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^5 + B*x^4 + A*x^3)/((c*x^4 + b*x^2 + a)^2*x),x, algorithm="maxima")

[Out]

1/2*(B*b*x^2 + (C*b - 2*A*c)*x^3 + 2*B*a + (2*C*a - A*b)*x)/((b^2*c - 4*a*c^2)*x
^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2) - 1/2*integrate(-(2*B*b*x + (C*b - 2
*A*c)*x^2 - 2*C*a + A*b)/(c*x^4 + b*x^2 + a), x)/(b^2 - 4*a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^5 + B*x^4 + A*x^3)/((c*x^4 + b*x^2 + a)^2*x),x, algorithm="giac")

[Out]

Exception raised: TypeError