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

Optimal. Leaf size=110 \[ -\frac{(3 A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{A}{a^2 x}-\frac{B \log \left (a+b x^2\right )}{2 a^2}+\frac{B \log (x)}{a^2}+\frac{-b x \left (\frac{A b}{a}-C\right )-a D+b B}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.282796, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{(3 A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{A}{a^2 x}-\frac{B \log \left (a+b x^2\right )}{2 a^2}+\frac{B \log (x)}{a^2}+\frac{-b x \left (\frac{A b}{a}-C\right )-a D+b B}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.2476, size = 92, normalized size = 0.84 \[ - \frac{C}{a b x} - \frac{C \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} + \frac{D \log{\left (x \right )}}{a b} - \frac{D \log{\left (a + b x^{2} \right )}}{2 a b} + \frac{x \left (\frac{A b}{x^{2}} + \frac{B b}{x} - \frac{C a}{x^{2}} - \frac{D a}{x}\right )}{2 a b \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.134197, size = 110, normalized size = 1. \[ \frac{(a C-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}+\frac{a^2 (-D)+a b B+a b C x-A b^2 x}{2 a^2 b \left (a+b x^2\right )}-\frac{A}{a^2 x}-\frac{B \log \left (a+b x^2\right )}{2 a^2}+\frac{B \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 136, normalized size = 1.2 \[ -{\frac{A}{x{a}^{2}}}+{\frac{\ln \left ( x \right ) B}{{a}^{2}}}-{\frac{Axb}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Cx}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{B}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{D}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,{a}^{2}}}-{\frac{3\,Ab}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{C}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264915, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (C a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (C a^{2} b - 3 \, A a b^{2}\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, A a b -{\left (C a b - 3 \, A b^{2}\right )} x^{2} +{\left (D a^{2} - B a b\right )} x +{\left (B b^{2} x^{3} + B a b x\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (B b^{2} x^{3} + B a b x\right )} \log \left (x\right )\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b^{2} x^{3} + a^{3} b x\right )} \sqrt{-a b}}, \frac{{\left ({\left (C a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (C a^{2} b - 3 \, A a b^{2}\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, A a b -{\left (C a b - 3 \, A b^{2}\right )} x^{2} +{\left (D a^{2} - B a b\right )} x +{\left (B b^{2} x^{3} + B a b x\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (B b^{2} x^{3} + B a b x\right )} \log \left (x\right )\right )} \sqrt{a b}}{2 \,{\left (a^{2} b^{2} x^{3} + a^{3} b x\right )} \sqrt{a b}}\right ] \]

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^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.8858, size = 782, normalized size = 7.11 \[ \frac{B \log{\left (x \right )}}{a^{2}} + \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) \log{\left (x + \frac{- 36 A^{2} B a b^{2} + 36 A^{2} a^{3} b^{2} \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 24 A B C a^{2} b - 24 A C a^{4} b \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 48 B^{3} a^{2} b + 48 B^{2} a^{4} b \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) - 4 B C^{2} a^{3} - 96 B a^{6} b \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )^{2} + 4 C^{2} a^{5} \left (- \frac{B}{2 a^{2}} - \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )}{- 27 A^{3} b^{3} + 27 A^{2} C a b^{2} - 108 A B^{2} a b^{2} - 9 A C^{2} a^{2} b + 36 B^{2} C a^{2} b + C^{3} a^{3}} \right )} + \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) \log{\left (x + \frac{- 36 A^{2} B a b^{2} + 36 A^{2} a^{3} b^{2} \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 24 A B C a^{2} b - 24 A C a^{4} b \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) + 48 B^{3} a^{2} b + 48 B^{2} a^{4} b \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right ) - 4 B C^{2} a^{3} - 96 B a^{6} b \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )^{2} + 4 C^{2} a^{5} \left (- \frac{B}{2 a^{2}} + \frac{\sqrt{- a^{5} b} \left (- 3 A b + C a\right )}{4 a^{5} b}\right )}{- 27 A^{3} b^{3} + 27 A^{2} C a b^{2} - 108 A B^{2} a b^{2} - 9 A C^{2} a^{2} b + 36 B^{2} C a^{2} b + C^{3} a^{3}} \right )} + \frac{- 2 A a b + x^{2} \left (- 3 A b^{2} + C a b\right ) + x \left (B a b - D a^{2}\right )}{2 a^{3} b x + 2 a^{2} b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*log(x)/a**2 + (-B/(2*a**2) - sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))*log(x +
(-36*A**2*B*a*b**2 + 36*A**2*a**3*b**2*(-B/(2*a**2) - sqrt(-a**5*b)*(-3*A*b + C*
a)/(4*a**5*b)) + 24*A*B*C*a**2*b - 24*A*C*a**4*b*(-B/(2*a**2) - sqrt(-a**5*b)*(-
3*A*b + C*a)/(4*a**5*b)) + 48*B**3*a**2*b + 48*B**2*a**4*b*(-B/(2*a**2) - sqrt(-
a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) - 4*B*C**2*a**3 - 96*B*a**6*b*(-B/(2*a**2) -
sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))**2 + 4*C**2*a**5*(-B/(2*a**2) - sqrt(-a
**5*b)*(-3*A*b + C*a)/(4*a**5*b)))/(-27*A**3*b**3 + 27*A**2*C*a*b**2 - 108*A*B**
2*a*b**2 - 9*A*C**2*a**2*b + 36*B**2*C*a**2*b + C**3*a**3)) + (-B/(2*a**2) + sqr
t(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))*log(x + (-36*A**2*B*a*b**2 + 36*A**2*a**3*
b**2*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) + 24*A*B*C*a**2*b -
24*A*C*a**4*b*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)) + 48*B**3
*a**2*b + 48*B**2*a**4*b*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b))
- 4*B*C**2*a**3 - 96*B*a**6*b*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a*
*5*b))**2 + 4*C**2*a**5*(-B/(2*a**2) + sqrt(-a**5*b)*(-3*A*b + C*a)/(4*a**5*b)))
/(-27*A**3*b**3 + 27*A**2*C*a*b**2 - 108*A*B**2*a*b**2 - 9*A*C**2*a**2*b + 36*B*
*2*C*a**2*b + C**3*a**3)) + (-2*A*a*b + x**2*(-3*A*b**2 + C*a*b) + x*(B*a*b - D*
a**2))/(2*a**3*b*x + 2*a**2*b**2*x**3)

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GIAC/XCAS [A]  time = 0.240638, size = 139, normalized size = 1.26 \[ -\frac{B{\rm ln}\left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{B{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (C a - 3 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} + \frac{C a b x^{2} - 3 \, A b^{2} x^{2} - D a^{2} x + B a b x - 2 \, A a b}{2 \,{\left (b x^{3} + a x\right )} a^{2} b} \]

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^2),x, algorithm="giac")

[Out]

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