3.86 \(\int \frac{A+B x^2}{x^6 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=113 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.341636, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^6*(a + b*x^2)^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 74.3224, size = 102, normalized size = 0.9 \[ - \frac{A}{5 a^{2} x^{5}} + \frac{2 A b - B a}{3 a^{3} x^{3}} - \frac{b^{2} x \left (A b - B a\right )}{2 a^{4} \left (a + b x^{2}\right )} - \frac{b \left (3 A b - 2 B a\right )}{a^{4} x} - \frac{b^{\frac{3}{2}} \left (7 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**6/(b*x**2+a)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.133975, size = 112, normalized size = 0.99 \[ \frac{b^{3/2} (5 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}+\frac{b^2 x (a B-A b)}{2 a^4 \left (a+b x^2\right )}+\frac{b (2 a B-3 A b)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 136, normalized size = 1.2 \[ -{\frac{A}{5\,{a}^{2}{x}^{5}}}+{\frac{2\,Ab}{3\,{a}^{3}{x}^{3}}}-{\frac{B}{3\,{a}^{2}{x}^{3}}}-3\,{\frac{{b}^{2}A}{{a}^{4}x}}+2\,{\frac{Bb}{{a}^{3}x}}-{\frac{{b}^{3}xA}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}Bx}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,A{b}^{3}}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,B{b}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^6/(b*x^2+a)^2,x)

[Out]

-1/5*A/a^2/x^5+2/3/a^3/x^3*A*b-1/3/a^2/x^3*B-3/a^4*b^2/x*A+2/a^3*b/x*B-1/2/a^4*b
^3*x/(b*x^2+a)*A+1/2/a^3*b^2*x/(b*x^2+a)*B-7/2/a^4*b^3/(a*b)^(1/2)*arctan(x*b/(a
*b)^(1/2))*A+5/2/a^3*b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(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)/((b*x^2 + a)^2*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240682, size = 1, normalized size = 0.01 \[ \left [\frac{30 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 12 \, A a^{3} - 4 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} - 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{60 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, \frac{15 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{30 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^2*x^6),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.05275, size = 218, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log{\left (\frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac{- 6 A a^{3} + x^{6} \left (- 105 A b^{3} + 75 B a b^{2}\right ) + x^{4} \left (- 70 A a b^{2} + 50 B a^{2} b\right ) + x^{2} \left (14 A a^{2} b - 10 B a^{3}\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**6/(b*x**2+a)**2,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229561, size = 151, normalized size = 1.34 \[ \frac{{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} + \frac{B a b^{2} x - A b^{3} x}{2 \,{\left (b x^{2} + a\right )} a^{4}} + \frac{30 \, B a b x^{4} - 45 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^2*x^6),x, algorithm="giac")

[Out]

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