3.117 \(\int \frac{A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx\)
Optimal. Leaf size=223 \[ \frac{(2 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{-6 a A c-a b B+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{\left (a b B \left (b^2-6 a c\right )-2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{-A \left (b^2-2 a c\right )+c x^2 (-(A b-2 a B))+a b B}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
-(2*A*b^2 - a*b*B - 6*a*A*c)/(2*a^2*(b^2 - 4*a*c)*x^2) - (a*b*B - A*(b^2 - 2*a*c
) - (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)) + ((a*b*B*(
b^2 - 6*a*c) - 2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2
- 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*A*b - a*B)*Log[x])/a^3 + ((2*A*b -
a*B)*Log[a + b*x^2 + c*x^4])/(4*a^3)
_______________________________________________________________________________________
Rubi [A] time = 0.856827, antiderivative size = 223, 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
\[ \frac{(2 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{-6 a A c-a b B+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{\left (a b B \left (b^2-6 a c\right )-2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}+\frac{c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]
[Out]
-(2*A*b^2 - a*b*B - 6*a*A*c)/(2*a^2*(b^2 - 4*a*c)*x^2) + (A*b^2 - a*b*B - 2*a*A*
c + (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)) + ((a*b*B*(
b^2 - 6*a*c) - 2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2
- 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*A*b - a*B)*Log[x])/a^3 + ((2*A*b -
a*B)*Log[a + b*x^2 + c*x^4])/(4*a^3)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 169.834, size = 221, normalized size = 0.99 \[ \frac{- 2 A a c + A b^{2} - B a b + c x^{2} \left (A b - 2 B a\right )}{2 a x^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} - \frac{- 6 A a c + 2 A b^{2} - B a b}{2 a^{2} x^{2} \left (- 4 a c + b^{2}\right )} - \frac{\left (2 A b - B a\right ) \log{\left (x^{2} \right )}}{2 a^{3}} + \frac{\left (2 A b - B a\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{3}} - \frac{\left (12 A a^{2} c^{2} - 12 A a b^{2} c + 2 A b^{4} + 6 B a^{2} b c - B a b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]
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)**2,x)
[Out]
(-2*A*a*c + A*b**2 - B*a*b + c*x**2*(A*b - 2*B*a))/(2*a*x**2*(-4*a*c + b**2)*(a
+ b*x**2 + c*x**4)) - (-6*A*a*c + 2*A*b**2 - B*a*b)/(2*a**2*x**2*(-4*a*c + b**2)
) - (2*A*b - B*a)*log(x**2)/(2*a**3) + (2*A*b - B*a)*log(a + b*x**2 + c*x**4)/(4
*a**3) - (12*A*a**2*c**2 - 12*A*a*b**2*c + 2*A*b**4 + 6*B*a**2*b*c - B*a*b**3)*a
tanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*a**3*(-4*a*c + b**2)**(3/2))
_______________________________________________________________________________________
Mathematica [A] time = 1.06577, size = 379, normalized size = 1.7 \[ \frac{\frac{\left (2 A \left (6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right )+a B \left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right )\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (2 A \left (-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right )+a B \left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{2 a \left (A \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )+a B \left (2 a c-b^2-b c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+4 \log (x) (a B-2 A b)-\frac{2 a A}{x^2}}{4 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]
[Out]
((-2*a*A)/x^2 - (2*a*(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)) + 4*(-2*A*b + a*B)*Log[x]
+ ((a*B*(-b^3 + 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c]) + 2*
A*(b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*
c]))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((a*B*(b^3 - 6*
a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c]) + 2*A*(-b^4 + 6*a*b^2*c
- 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]))*Log[b + Sqrt[
b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^3)
_______________________________________________________________________________________
Maple [B] time = 0.031, size = 991, normalized size = 4.4 \[ \text{result too large to display} \]
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)^2,x)
[Out]
-1/a/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*A+1/2/a^2/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)
*x^2*A*b^2-1/2/a/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b*B-3/2/a/(c*x^4+b*x^2+a)/(4*
a*c-b^2)*A*b*c+1/2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A*b^3+1/(c*x^4+b*x^2+a)/(4*a*
c-b^2)*B*c-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*B*b^2+2/a^2/(4*a*c-b^2)*c*ln((4*a*c
-b^2)*(c*x^4+b*x^2+a))*A*b-1/2/a^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*A
*b^3-1/a/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*B+1/4/a^2/(4*a*c-b^2)*ln(
(4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2*B-6/a/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6
)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(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))*A*c^2+6/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2
)*arctan((2*(4*a*c-b^2)*c*x^2+(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))*A*b^2*c-1/a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*ar
ctan((2*(4*a*c-b^2)*c*x^2+(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))*A*b^4-3/a/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*
(4*a*c-b^2)*c*x^2+(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*B*c+1/2/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a
*c-b^2)*c*x^2+(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
*b^3-1/2*A/a^2/x^2-2/a^3*ln(x)*A*b+1/a^2*ln(x)*B
_______________________________________________________________________________________
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)^2*x^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 1.66542, 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)^2*x^3),x, algorithm="fricas")
[Out]
[-1/4*(((12*A*a^2*c^3 + 6*(B*a^2*b - 2*A*a*b^2)*c^2 - (B*a*b^3 - 2*A*b^4)*c)*x^6
- (B*a*b^4 - 2*A*b^5 - 12*A*a^2*b*c^2 - 6*(B*a^2*b^2 - 2*A*a*b^3)*c)*x^4 - (B*a
^2*b^3 - 2*A*a*b^4 - 12*A*a^3*c^2 - 6*(B*a^3*b - 2*A*a^2*b^2)*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*A*a^2*b^2 - 8*A*a^3*c - 2*(6*A*a^2*c^2 +
(B*a^2*b - 2*A*a*b^2)*c)*x^4 - 2*(B*a^2*b^2 - 2*A*a*b^3 - (2*B*a^3 - 7*A*a^2*b)
*c)*x^2 - ((4*(B*a^2 - 2*A*a*b)*c^2 - (B*a*b^2 - 2*A*b^3)*c)*x^6 - (B*a*b^3 - 2*
A*b^4 - 4*(B*a^2*b - 2*A*a*b^2)*c)*x^4 - (B*a^2*b^2 - 2*A*a*b^3 - 4*(B*a^3 - 2*A
*a^2*b)*c)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((4*(B*a^2 - 2*A*a*b)*c^2 - (B*a*b^2
- 2*A*b^3)*c)*x^6 - (B*a*b^3 - 2*A*b^4 - 4*(B*a^2*b - 2*A*a*b^2)*c)*x^4 - (B*a^2
*b^2 - 2*A*a*b^3 - 4*(B*a^3 - 2*A*a^2*b)*c)*x^2)*log(x))*sqrt(b^2 - 4*a*c))/(((a
^3*b^2*c - 4*a^4*c^2)*x^6 + (a^3*b^3 - 4*a^4*b*c)*x^4 + (a^4*b^2 - 4*a^5*c)*x^2)
*sqrt(b^2 - 4*a*c)), 1/4*(2*((12*A*a^2*c^3 + 6*(B*a^2*b - 2*A*a*b^2)*c^2 - (B*a*
b^3 - 2*A*b^4)*c)*x^6 - (B*a*b^4 - 2*A*b^5 - 12*A*a^2*b*c^2 - 6*(B*a^2*b^2 - 2*A
*a*b^3)*c)*x^4 - (B*a^2*b^3 - 2*A*a*b^4 - 12*A*a^3*c^2 - 6*(B*a^3*b - 2*A*a^2*b^
2)*c)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (2*A*a^2*b^
2 - 8*A*a^3*c - 2*(6*A*a^2*c^2 + (B*a^2*b - 2*A*a*b^2)*c)*x^4 - 2*(B*a^2*b^2 - 2
*A*a*b^3 - (2*B*a^3 - 7*A*a^2*b)*c)*x^2 - ((4*(B*a^2 - 2*A*a*b)*c^2 - (B*a*b^2 -
2*A*b^3)*c)*x^6 - (B*a*b^3 - 2*A*b^4 - 4*(B*a^2*b - 2*A*a*b^2)*c)*x^4 - (B*a^2*
b^2 - 2*A*a*b^3 - 4*(B*a^3 - 2*A*a^2*b)*c)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((4*(
B*a^2 - 2*A*a*b)*c^2 - (B*a*b^2 - 2*A*b^3)*c)*x^6 - (B*a*b^3 - 2*A*b^4 - 4*(B*a^
2*b - 2*A*a*b^2)*c)*x^4 - (B*a^2*b^2 - 2*A*a*b^3 - 4*(B*a^3 - 2*A*a^2*b)*c)*x^2)
*log(x))*sqrt(-b^2 + 4*a*c))/(((a^3*b^2*c - 4*a^4*c^2)*x^6 + (a^3*b^3 - 4*a^4*b*
c)*x^4 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(-b^2 + 4*a*c))]
_______________________________________________________________________________________
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)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
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((B*x^2 + A)/((c*x^4 + b*x^2 + a)^2*x^3),x, algorithm="giac")
[Out]
Exception raised: TypeError