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

Optimal. Leaf size=90 \[ -\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}-\frac{c x (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac{b B-2 A c}{b^3 x}-\frac{A}{3 b^2 x^3} \]

[Out]

-A/(3*b^2*x^3) - (b*B - 2*A*c)/(b^3*x) - (c*(b*B - A*c)*x)/(2*b^3*(b + c*x^2)) -
(Sqrt[c]*(3*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.258147, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}-\frac{c x (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac{b B-2 A c}{b^3 x}-\frac{A}{3 b^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*b^2*x^3) - (b*B - 2*A*c)/(b^3*x) - (c*(b*B - A*c)*x)/(2*b^3*(b + c*x^2)) -
(Sqrt[c]*(3*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.3123, size = 80, normalized size = 0.89 \[ - \frac{A}{3 b^{2} x^{3}} + \frac{c x \left (A c - B b\right )}{2 b^{3} \left (b + c x^{2}\right )} + \frac{2 A c - B b}{b^{3} x} + \frac{\sqrt{c} \left (5 A c - 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{2 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(3*b**2*x**3) + c*x*(A*c - B*b)/(2*b**3*(b + c*x**2)) + (2*A*c - B*b)/(b**3*x
) + sqrt(c)*(5*A*c - 3*B*b)*atan(sqrt(c)*x/sqrt(b))/(2*b**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.118447, size = 90, normalized size = 1. \[ -\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}-\frac{c x (b B-A c)}{2 b^3 \left (b+c x^2\right )}+\frac{2 A c-b B}{b^3 x}-\frac{A}{3 b^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*b^2*x^3) + (-(b*B) + 2*A*c)/(b^3*x) - (c*(b*B - A*c)*x)/(2*b^3*(b + c*x^2)
) - (Sqrt[c]*(3*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 110, normalized size = 1.2 \[ -{\frac{A}{3\,{b}^{2}{x}^{3}}}+2\,{\frac{Ac}{{b}^{3}x}}-{\frac{B}{{b}^{2}x}}+{\frac{Ax{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }}-{\frac{Bcx}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}+{\frac{5\,A{c}^{2}}{2\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{3\,Bc}{2\,{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*A/b^2/x^3+2/b^3/x*A*c-1/b^2/x*B+1/2/b^3*c^2*x/(c*x^2+b)*A-1/2/b^2*c*x/(c*x^
2+b)*B+5/2/b^3*c^2/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*A-3/2/b^2*c/(b*c)^(1/2)*a
rctan(c*x/(b*c)^(1/2))*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.218158, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \,{\left (3 \, B b c - 5 \, A c^{2}\right )} x^{4} + 4 \, A b^{2} + 4 \,{\left (3 \, B b^{2} - 5 \, A b c\right )} x^{2} + 3 \,{\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{5} +{\left (3 \, B b^{2} - 5 \, A b c\right )} x^{3}\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} + 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right )}{12 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, -\frac{3 \,{\left (3 \, B b c - 5 \, A c^{2}\right )} x^{4} + 2 \, A b^{2} + 2 \,{\left (3 \, B b^{2} - 5 \, A b c\right )} x^{2} + 3 \,{\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{5} +{\left (3 \, B b^{2} - 5 \, A b c\right )} x^{3}\right )} \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right )}{6 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(6*(3*B*b*c - 5*A*c^2)*x^4 + 4*A*b^2 + 4*(3*B*b^2 - 5*A*b*c)*x^2 + 3*((3*
B*b*c - 5*A*c^2)*x^5 + (3*B*b^2 - 5*A*b*c)*x^3)*sqrt(-c/b)*log((c*x^2 + 2*b*x*sq
rt(-c/b) - b)/(c*x^2 + b)))/(b^3*c*x^5 + b^4*x^3), -1/6*(3*(3*B*b*c - 5*A*c^2)*x
^4 + 2*A*b^2 + 2*(3*B*b^2 - 5*A*b*c)*x^2 + 3*((3*B*b*c - 5*A*c^2)*x^5 + (3*B*b^2
- 5*A*b*c)*x^3)*sqrt(c/b)*arctan(c*x/(b*sqrt(c/b))))/(b^3*c*x^5 + b^4*x^3)]

_______________________________________________________________________________________

Sympy [A]  time = 1.54789, size = 184, normalized size = 2.04 \[ \frac{\sqrt{- \frac{c}{b^{7}}} \left (- 5 A c + 3 B b\right ) \log{\left (- \frac{b^{4} \sqrt{- \frac{c}{b^{7}}} \left (- 5 A c + 3 B b\right )}{- 5 A c^{2} + 3 B b c} + x \right )}}{4} - \frac{\sqrt{- \frac{c}{b^{7}}} \left (- 5 A c + 3 B b\right ) \log{\left (\frac{b^{4} \sqrt{- \frac{c}{b^{7}}} \left (- 5 A c + 3 B b\right )}{- 5 A c^{2} + 3 B b c} + x \right )}}{4} - \frac{2 A b^{2} + x^{4} \left (- 15 A c^{2} + 9 B b c\right ) + x^{2} \left (- 10 A b c + 6 B b^{2}\right )}{6 b^{4} x^{3} + 6 b^{3} c x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-c/b**7)*(-5*A*c + 3*B*b)*log(-b**4*sqrt(-c/b**7)*(-5*A*c + 3*B*b)/(-5*A*c*
*2 + 3*B*b*c) + x)/4 - sqrt(-c/b**7)*(-5*A*c + 3*B*b)*log(b**4*sqrt(-c/b**7)*(-5
*A*c + 3*B*b)/(-5*A*c**2 + 3*B*b*c) + x)/4 - (2*A*b**2 + x**4*(-15*A*c**2 + 9*B*
b*c) + x**2*(-10*A*b*c + 6*B*b**2))/(6*b**4*x**3 + 6*b**3*c*x**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210791, size = 115, normalized size = 1.28 \[ -\frac{{\left (3 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} b^{3}} - \frac{B b c x - A c^{2} x}{2 \,{\left (c x^{2} + b\right )} b^{3}} - \frac{3 \, B b x^{2} - 6 \, A c x^{2} + A b}{3 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(3*B*b*c - 5*A*c^2)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^3) - 1/2*(B*b*c*x -
A*c^2*x)/((c*x^2 + b)*b^3) - 1/3*(3*B*b*x^2 - 6*A*c*x^2 + A*b)/(b^3*x^3)