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

Optimal. Leaf size=93 \[ \frac{(a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}-\frac{a \left (B-\frac{a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}+\frac{D \log \left (a+b x^2\right )}{2 b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.145849, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{(a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}-\frac{a \left (B-\frac{a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}+\frac{D \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.4821, size = 75, normalized size = 0.81 \[ \frac{D \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x \left (A b - C a + x \left (B b - D a\right )\right )}{2 a b \left (a + b x^{2}\right )} + \frac{\left (A b + C a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.164582, size = 83, normalized size = 0.89 \[ \frac{\frac{\sqrt{b} (a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}+\frac{a^2 D-a b (B+C x)+A b^2 x}{a \left (a+b x^2\right )}+D \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 100, normalized size = 1.1 \[{\frac{1}{b{x}^{2}+a} \left ({\frac{ \left ( Ab-aC \right ) x}{2\,ab}}-{\frac{Bb-aD}{2\,{b}^{2}}} \right ) }+{\frac{D\ln \left ( ab \left ( b{x}^{2}+a \right ) \right ) }{2\,{b}^{2}}}+{\frac{A}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{C}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.235445, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (C a^{2} b + A a b^{2} +{\left (C a b^{2} + A b^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (D a^{2} - B a b -{\left (C a b - A b^{2}\right )} x +{\left (D a b x^{2} + D a^{2}\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{4 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{-a b}}, \frac{{\left (C a^{2} b + A a b^{2} +{\left (C a b^{2} + A b^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (D a^{2} - B a b -{\left (C a b - A b^{2}\right )} x +{\left (D a b x^{2} + D a^{2}\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.37742, size = 233, normalized size = 2.51 \[ \left (\frac{D}{2 b^{2}} - \frac{\sqrt{- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log{\left (x + \frac{- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac{D}{2 b^{2}} - \frac{\sqrt{- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \left (\frac{D}{2 b^{2}} + \frac{\sqrt{- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log{\left (x + \frac{- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac{D}{2 b^{2}} + \frac{\sqrt{- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} - \frac{B a b - D a^{2} + x \left (- A b^{2} + C a b\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239666, size = 119, normalized size = 1.28 \[ \frac{D{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{2}} + \frac{{\left (C a + A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b} - \frac{{\left (C a - A b\right )} x - \frac{D a^{2} - B a b}{b}}{2 \,{\left (b x^{2} + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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