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

Optimal. Leaf size=67 \[ \frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac{B x}{b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.132068, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac{B x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.3748, size = 60, normalized size = 0.9 \[ \frac{B x}{b^{2}} - \frac{x \left (A b - B a\right )}{2 b^{2} \left (a + b x^{2}\right )} + \frac{\left (A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b**2 - x*(A*b - B*a)/(2*b**2*(a + b*x**2)) + (A*b - 3*B*a)*atan(sqrt(b)*x/sq
rt(a))/(2*sqrt(a)*b**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.114519, size = 68, normalized size = 1.01 \[ -\frac{(3 a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac{B x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 82, normalized size = 1.2 \[{\frac{Bx}{{b}^{2}}}-{\frac{xA}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{Bxa}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{A}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,Ba}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b^2-1/2/b*x/(b*x^2+a)*A+1/2/b^2*x/(b*x^2+a)*B*a+1/2/b/(a*b)^(1/2)*arctan(x*b
/(a*b)^(1/2))*A-3/2/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*B*a

_______________________________________________________________________________________

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)*x^2/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231736, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (2 \, B b x^{3} +{\left (3 \, B a - A b\right )} x\right )} \sqrt{-a b}}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, B b x^{3} +{\left (3 \, B a - A b\right )} x\right )} \sqrt{a b}}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*((3*B*a^2 - A*a*b + (3*B*a*b - A*b^2)*x^2)*log((2*a*b*x + (b*x^2 - a)*sqrt
(-a*b))/(b*x^2 + a)) - 2*(2*B*b*x^3 + (3*B*a - A*b)*x)*sqrt(-a*b))/((b^3*x^2 + a
*b^2)*sqrt(-a*b)), -1/2*((3*B*a^2 - A*a*b + (3*B*a*b - A*b^2)*x^2)*arctan(sqrt(a
*b)*x/a) - (2*B*b*x^3 + (3*B*a - A*b)*x)*sqrt(a*b))/((b^3*x^2 + a*b^2)*sqrt(a*b)
)]

_______________________________________________________________________________________

Sympy [A]  time = 2.63281, size = 114, normalized size = 1.7 \[ \frac{B x}{b^{2}} + \frac{x \left (- A b + B a\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\sqrt{- \frac{1}{a b^{5}}} \left (- A b + 3 B a\right ) \log{\left (- a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a b^{5}}} \left (- A b + 3 B a\right ) \log{\left (a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b**2 + x*(-A*b + B*a)/(2*a*b**2 + 2*b**3*x**2) + sqrt(-1/(a*b**5))*(-A*b + 3
*B*a)*log(-a*b**2*sqrt(-1/(a*b**5)) + x)/4 - sqrt(-1/(a*b**5))*(-A*b + 3*B*a)*lo
g(a*b**2*sqrt(-1/(a*b**5)) + x)/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222282, size = 80, normalized size = 1.19 \[ \frac{B x}{b^{2}} - \frac{{\left (3 \, B a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} + \frac{B a x - A b x}{2 \,{\left (b x^{2} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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