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

Optimal. Leaf size=95 \[ \frac{(a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}-\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{x (b B-a D)-a C+A b}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.268661, antiderivative size = 95, 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 \[ \frac{(a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}-\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{x (b B-a D)-a C+A b}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.913, size = 85, normalized size = 0.89 \[ \frac{C \log{\left (x \right )}}{a b} - \frac{C \log{\left (a + b x^{2} \right )}}{2 a b} + \frac{x \left (\frac{A b}{x} + B b - \frac{C a}{x} - D a\right )}{2 a b \left (a + b x^{2}\right )} + \frac{\left (B b + D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.144014, size = 85, normalized size = 0.89 \[ \frac{\frac{a (-a (C+D x)+A b+b B x)}{b \left (a+b x^2\right )}-A \log \left (a+b x^2\right )+\frac{\sqrt{a} (a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}+2 A \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 127, normalized size = 1.3 \[{\frac{A\ln \left ( x \right ) }{{a}^{2}}}+{\frac{Bx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{xD}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}+{\frac{A}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{C}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}-{\frac{A\ln \left ( b \left ( b{x}^{2}+a \right ) \right ) }{2\,{a}^{2}}}+{\frac{B}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{D}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26292, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (D a^{3} + B a^{2} b +{\left (D a^{2} b + B a b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (C a^{2} - A a b +{\left (D a^{2} - B a b\right )} x +{\left (A b^{2} x^{2} + A a b\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (A b^{2} x^{2} + A a b\right )} \log \left (x\right )\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )} \sqrt{-a b}}, \frac{{\left (D a^{3} + B a^{2} b +{\left (D a^{2} b + B a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (C a^{2} - A a b +{\left (D a^{2} - B a b\right )} x +{\left (A b^{2} x^{2} + A a b\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (A b^{2} x^{2} + A a b\right )} \log \left (x\right )\right )} \sqrt{a b}}{2 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )} \sqrt{a b}}\right ] \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 14.5026, size = 797, normalized size = 8.39 \[ \frac{A \log{\left (x \right )}}{a^{2}} + \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) \log{\left (x + \frac{48 A^{3} b^{4} + 48 A^{2} a^{2} b^{4} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) - 4 A B^{2} a b^{3} - 8 A B D a^{2} b^{2} - 4 A D^{2} a^{3} b - 96 A a^{4} b^{4} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )^{2} + 4 B^{2} a^{3} b^{3} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 8 B D a^{4} b^{2} \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 4 D^{2} a^{5} b \left (- \frac{A}{2 a^{2}} - \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )}{36 A^{2} B b^{4} + 36 A^{2} D a b^{3} + B^{3} a b^{3} + 3 B^{2} D a^{2} b^{2} + 3 B D^{2} a^{3} b + D^{3} a^{4}} \right )} + \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) \log{\left (x + \frac{48 A^{3} b^{4} + 48 A^{2} a^{2} b^{4} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) - 4 A B^{2} a b^{3} - 8 A B D a^{2} b^{2} - 4 A D^{2} a^{3} b - 96 A a^{4} b^{4} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )^{2} + 4 B^{2} a^{3} b^{3} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 8 B D a^{4} b^{2} \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right ) + 4 D^{2} a^{5} b \left (- \frac{A}{2 a^{2}} + \frac{\sqrt{- a^{5} b^{3}} \left (B b + D a\right )}{4 a^{4} b^{3}}\right )}{36 A^{2} B b^{4} + 36 A^{2} D a b^{3} + B^{3} a b^{3} + 3 B^{2} D a^{2} b^{2} + 3 B D^{2} a^{3} b + D^{3} a^{4}} \right )} - \frac{- A b + C a + x \left (- B b + D a\right )}{2 a^{2} b + 2 a b^{2} x^{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)

[Out]

A*log(x)/a**2 + (-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))*log(x
+ (48*A**3*b**4 + 48*A**2*a**2*b**4*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)
/(4*a**4*b**3)) - 4*A*B**2*a*b**3 - 8*A*B*D*a**2*b**2 - 4*A*D**2*a**3*b - 96*A*a
**4*b**4*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))**2 + 4*B**2*
a**3*b**3*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) + 8*B*D*a**
4*b**2*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) + 4*D**2*a**5*
b*(-A/(2*a**2) - sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)))/(36*A**2*B*b**4 +
36*A**2*D*a*b**3 + B**3*a*b**3 + 3*B**2*D*a**2*b**2 + 3*B*D**2*a**3*b + D**3*a**
4)) + (-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))*log(x + (48*A**
3*b**4 + 48*A**2*a**2*b**4*(-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b
**3)) - 4*A*B**2*a*b**3 - 8*A*B*D*a**2*b**2 - 4*A*D**2*a**3*b - 96*A*a**4*b**4*(
-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3))**2 + 4*B**2*a**3*b**3*
(-A/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) + 8*B*D*a**4*b**2*(-A
/(2*a**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)) + 4*D**2*a**5*b*(-A/(2*a
**2) + sqrt(-a**5*b**3)*(B*b + D*a)/(4*a**4*b**3)))/(36*A**2*B*b**4 + 36*A**2*D*
a*b**3 + B**3*a*b**3 + 3*B**2*D*a**2*b**2 + 3*B*D**2*a**3*b + D**3*a**4)) - (-A*
b + C*a + x*(-B*b + D*a))/(2*a**2*b + 2*a*b**2*x**2)

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GIAC/XCAS [A]  time = 0.23593, size = 126, normalized size = 1.33 \[ -\frac{A{\rm ln}\left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (D a + B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b} - \frac{C a^{2} - A a b +{\left (D a^{2} - B a b\right )} x}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \]

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),x, algorithm="giac")

[Out]

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