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

Optimal. Leaf size=111 \[ \frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{9/2}}+\frac{c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac{c (2 b B-3 A c)}{b^4 x}-\frac{b B-2 A c}{3 b^3 x^3}-\frac{A}{5 b^2 x^5} \]

[Out]

-A/(5*b^2*x^5) - (b*B - 2*A*c)/(3*b^3*x^3) + (c*(2*b*B - 3*A*c))/(b^4*x) + (c^2*
(b*B - A*c)*x)/(2*b^4*(b + c*x^2)) + (c^(3/2)*(5*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)
/Sqrt[b]])/(2*b^(9/2))

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Rubi [A]  time = 0.353048, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{9/2}}+\frac{c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac{c (2 b B-3 A c)}{b^4 x}-\frac{b B-2 A c}{3 b^3 x^3}-\frac{A}{5 b^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-A/(5*b^2*x^5) - (b*B - 2*A*c)/(3*b^3*x^3) + (c*(2*b*B - 3*A*c))/(b^4*x) + (c^2*
(b*B - A*c)*x)/(2*b^4*(b + c*x^2)) + (c^(3/2)*(5*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)
/Sqrt[b]])/(2*b^(9/2))

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Rubi in Sympy [A]  time = 61.7599, size = 102, normalized size = 0.92 \[ - \frac{A}{5 b^{2} x^{5}} + \frac{2 A c - B b}{3 b^{3} x^{3}} - \frac{c^{2} x \left (A c - B b\right )}{2 b^{4} \left (b + c x^{2}\right )} - \frac{c \left (3 A c - 2 B b\right )}{b^{4} x} - \frac{c^{\frac{3}{2}} \left (7 A c - 5 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{2 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(5*b**2*x**5) + (2*A*c - B*b)/(3*b**3*x**3) - c**2*x*(A*c - B*b)/(2*b**4*(b +
c*x**2)) - c*(3*A*c - 2*B*b)/(b**4*x) - c**(3/2)*(7*A*c - 5*B*b)*atan(sqrt(c)*x
/sqrt(b))/(2*b**(9/2))

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Mathematica [A]  time = 0.170314, size = 112, normalized size = 1.01 \[ \frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{9/2}}+\frac{c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac{c (2 b B-3 A c)}{b^4 x}+\frac{2 A c-b B}{3 b^3 x^3}-\frac{A}{5 b^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-A/(5*b^2*x^5) + (-(b*B) + 2*A*c)/(3*b^3*x^3) + (c*(2*b*B - 3*A*c))/(b^4*x) + (c
^2*(b*B - A*c)*x)/(2*b^4*(b + c*x^2)) + (c^(3/2)*(5*b*B - 7*A*c)*ArcTan[(Sqrt[c]
*x)/Sqrt[b]])/(2*b^(9/2))

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Maple [A]  time = 0.018, size = 136, normalized size = 1.2 \[ -{\frac{A}{5\,{b}^{2}{x}^{5}}}+{\frac{2\,Ac}{3\,{b}^{3}{x}^{3}}}-{\frac{B}{3\,{b}^{2}{x}^{3}}}-3\,{\frac{A{c}^{2}}{{b}^{4}x}}+2\,{\frac{Bc}{{b}^{3}x}}-{\frac{Ax{c}^{3}}{2\,{b}^{4} \left ( c{x}^{2}+b \right ) }}+{\frac{B{c}^{2}x}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }}-{\frac{7\,A{c}^{3}}{2\,{b}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{5\,B{c}^{2}}{2\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*A/b^2/x^5+2/3/b^3/x^3*A*c-1/3/b^2/x^3*B-3*c^2/b^4/x*A+2*c/b^3/x*B-1/2/b^4*c
^3*x/(c*x^2+b)*A+1/2/b^3*c^2*x/(c*x^2+b)*B-7/2/b^4*c^3/(b*c)^(1/2)*arctan(c*x/(b
*c)^(1/2))*A+5/2/b^3*c^2/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*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)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221481, size = 1, normalized size = 0.01 \[ \left [\frac{30 \,{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 20 \,{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 12 \, A b^{3} - 4 \,{\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} - 15 \,{\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} +{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5}\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right )}{60 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}, \frac{15 \,{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 10 \,{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 6 \, A b^{3} - 2 \,{\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \,{\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} +{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5}\right )} \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right )}{30 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/60*(30*(5*B*b*c^2 - 7*A*c^3)*x^6 + 20*(5*B*b^2*c - 7*A*b*c^2)*x^4 - 12*A*b^3
- 4*(5*B*b^3 - 7*A*b^2*c)*x^2 - 15*((5*B*b*c^2 - 7*A*c^3)*x^7 + (5*B*b^2*c - 7*A
*b*c^2)*x^5)*sqrt(-c/b)*log((c*x^2 - 2*b*x*sqrt(-c/b) - b)/(c*x^2 + b)))/(b^4*c*
x^7 + b^5*x^5), 1/30*(15*(5*B*b*c^2 - 7*A*c^3)*x^6 + 10*(5*B*b^2*c - 7*A*b*c^2)*
x^4 - 6*A*b^3 - 2*(5*B*b^3 - 7*A*b^2*c)*x^2 + 15*((5*B*b*c^2 - 7*A*c^3)*x^7 + (5
*B*b^2*c - 7*A*b*c^2)*x^5)*sqrt(c/b)*arctan(c*x/(b*sqrt(c/b))))/(b^4*c*x^7 + b^5
*x^5)]

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Sympy [A]  time = 1.96827, size = 218, normalized size = 1.96 \[ - \frac{\sqrt{- \frac{c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right ) \log{\left (- \frac{b^{5} \sqrt{- \frac{c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right )}{- 7 A c^{3} + 5 B b c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right ) \log{\left (\frac{b^{5} \sqrt{- \frac{c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right )}{- 7 A c^{3} + 5 B b c^{2}} + x \right )}}{4} + \frac{- 6 A b^{3} + x^{6} \left (- 105 A c^{3} + 75 B b c^{2}\right ) + x^{4} \left (- 70 A b c^{2} + 50 B b^{2} c\right ) + x^{2} \left (14 A b^{2} c - 10 B b^{3}\right )}{30 b^{5} x^{5} + 30 b^{4} c x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-c**3/b**9)*(-7*A*c + 5*B*b)*log(-b**5*sqrt(-c**3/b**9)*(-7*A*c + 5*B*b)/(
-7*A*c**3 + 5*B*b*c**2) + x)/4 + sqrt(-c**3/b**9)*(-7*A*c + 5*B*b)*log(b**5*sqrt
(-c**3/b**9)*(-7*A*c + 5*B*b)/(-7*A*c**3 + 5*B*b*c**2) + x)/4 + (-6*A*b**3 + x**
6*(-105*A*c**3 + 75*B*b*c**2) + x**4*(-70*A*b*c**2 + 50*B*b**2*c) + x**2*(14*A*b
**2*c - 10*B*b**3))/(30*b**5*x**5 + 30*b**4*c*x**7)

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GIAC/XCAS [A]  time = 0.217009, size = 151, normalized size = 1.36 \[ \frac{{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} b^{4}} + \frac{B b c^{2} x - A c^{3} x}{2 \,{\left (c x^{2} + b\right )} b^{4}} + \frac{30 \, B b c x^{4} - 45 \, A c^{2} x^{4} - 5 \, B b^{2} x^{2} + 10 \, A b c x^{2} - 3 \, A b^{2}}{15 \, b^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(5*B*b*c^2 - 7*A*c^3)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^4) + 1/2*(B*b*c^2*x
- A*c^3*x)/((c*x^2 + b)*b^4) + 1/15*(30*B*b*c*x^4 - 45*A*c^2*x^4 - 5*B*b^2*x^2
+ 10*A*b*c*x^2 - 3*A*b^2)/(b^4*x^5)