3.97 \(\int \frac{x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=101 \[ -\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-3 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}+\frac{D x}{b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.254629, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-3 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}+\frac{D x}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.4923, size = 83, normalized size = 0.82 \[ \frac{C \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{3 D x}{2 b^{2}} - \frac{A + B x + C x^{2} + D x^{3}}{2 b \left (a + b x^{2}\right )} + \frac{\left (B b - 3 D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

C*log(a + b*x**2)/(2*b**2) + 3*D*x/(2*b**2) - (A + B*x + C*x**2 + D*x**3)/(2*b*(
a + b*x**2)) + (B*b - 3*D*a)*atan(sqrt(b)*x/sqrt(a))/(2*sqrt(a)*b**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0998942, size = 92, normalized size = 0.91 \[ \frac{a C+a D x-A b-b B x}{2 b^2 \left (a+b x^2\right )}-\frac{(3 a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}+\frac{D x}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 127, normalized size = 1.3 \[{\frac{Dx}{{b}^{2}}}-{\frac{Bx}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{Dxa}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{A}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{aC}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{C\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}+{\frac{B}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,aD}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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((D*x^3 + C*x^2 + B*x + A)*x/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240564, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, D a^{2} - B a b +{\left (3 \, D a b - B b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (2 \, D b x^{3} + C a - A b +{\left (3 \, D a - B b\right )} x +{\left (C b x^{2} + C a\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (3 \, D a^{2} - B a b +{\left (3 \, D a b - B b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, D b x^{3} + C a - A b +{\left (3 \, D a - B b\right )} x +{\left (C b x^{2} + C a\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.39716, size = 212, normalized size = 2.1 \[ \frac{D x}{b^{2}} + \left (\frac{C}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log{\left (x + \frac{2 C a - 4 a b^{2} \left (\frac{C}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \left (\frac{C}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log{\left (x + \frac{2 C a - 4 a b^{2} \left (\frac{C}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \frac{- A b + C a + x \left (- B b + D a\right )}{2 a b^{2} + 2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

D*x/b**2 + (C/(2*b**2) - sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5))*log(x + (2*C*a
- 4*a*b**2*(C/(2*b**2) - sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5)))/(-B*b + 3*D*
a)) + (C/(2*b**2) + sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5))*log(x + (2*C*a - 4*
a*b**2*(C/(2*b**2) + sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5)))/(-B*b + 3*D*a)) +
(-A*b + C*a + x*(-B*b + D*a))/(2*a*b**2 + 2*b**3*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22131, size = 109, normalized size = 1.08 \[ \frac{D x}{b^{2}} + \frac{C{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{2}} - \frac{{\left (3 \, D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} + \frac{C a - A b +{\left (D a - B b\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

D*x/b^2 + 1/2*C*ln(b*x^2 + a)/b^2 - 1/2*(3*D*a - B*b)*arctan(b*x/sqrt(a*b))/(sqr
t(a*b)*b^2) + 1/2*(C*a - A*b + (D*a - B*b)*x)/((b*x^2 + a)*b^2)