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

Optimal. Leaf size=72 \[ -\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]

[Out]

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

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Rubi [A]  time = 0.209655, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A \log{\left (x \right )}}{a} + \frac{\int D\, dx}{b} - \frac{\left (A b - C a\right ) \log{\left (a + b x^{2} \right )}}{2 a b} + \frac{\left (B b - D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0994818, size = 73, normalized size = 1.01 \[ \frac{(a C-A b) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}-\frac{(a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 80, normalized size = 1.1 \[{\frac{Dx}{b}}+{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,b}}+{B\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{aD}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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),x)

[Out]

D*x/b+A*ln(x)/a-1/2/a*ln(b*x^2+a)*A+1/2/b*ln(b*x^2+a)*C+1/(a*b)^(1/2)*arctan(x*b
/(a*b)^(1/2))*B-a/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)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270926, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (D a^{2} - B a b\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) -{\left (2 \, D a x + 2 \, A b \log \left (x\right ) +{\left (C a - A b\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{2 \, \sqrt{-a b} a b}, -\frac{2 \,{\left (D a^{2} - B a b\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, D a x + 2 \, A b \log \left (x\right ) +{\left (C a - A b\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{2 \, \sqrt{a b} a b}\right ] \]

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)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 39.1185, size = 1268, normalized size = 17.61 \[ \text{result too large to display} \]

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),x)

[Out]

A*log(x)/a + D*x/b + ((-A*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a*
*2*b**3))*log(x + (-6*A**3*b**4 + 8*A**2*C*a*b**3 - 6*A**2*a*b**4*((-A*b + C*a)/
(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3)) + 2*A*B**2*a*b**3 - 4*A*B
*D*a**2*b**2 - 2*A*C**2*a**2*b**2 - 4*A*C*a**2*b**3*((-A*b + C*a)/(2*a*b) - sqrt
(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3)) + 2*A*D**2*a**3*b + 12*A*a**2*b**4*((-A
*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3))**2 - 2*B**2*a**
2*b**3*((-A*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3)) + 4*
B*D*a**3*b**2*((-A*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3
)) + 2*C**2*a**3*b**2*((-A*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a
**2*b**3)) - 4*C*a**3*b**3*((-A*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*b + D*a)
/(2*a**2*b**3))**2 - 2*D**2*a**4*b*((-A*b + C*a)/(2*a*b) - sqrt(-a**3*b**3)*(-B*
b + D*a)/(2*a**2*b**3)))/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 6*A*B*C*a*b**3 - 6*
A*C*D*a**2*b**2 - B**3*a*b**3 + 3*B**2*D*a**2*b**2 - B*C**2*a**2*b**2 - 3*B*D**2
*a**3*b + C**2*D*a**3*b + D**3*a**4)) + ((-A*b + C*a)/(2*a*b) + sqrt(-a**3*b**3)
*(-B*b + D*a)/(2*a**2*b**3))*log(x + (-6*A**3*b**4 + 8*A**2*C*a*b**3 - 6*A**2*a*
b**4*((-A*b + C*a)/(2*a*b) + sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3)) + 2*A*
B**2*a*b**3 - 4*A*B*D*a**2*b**2 - 2*A*C**2*a**2*b**2 - 4*A*C*a**2*b**3*((-A*b +
C*a)/(2*a*b) + sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3)) + 2*A*D**2*a**3*b +
12*A*a**2*b**4*((-A*b + C*a)/(2*a*b) + sqrt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**
3))**2 - 2*B**2*a**2*b**3*((-A*b + C*a)/(2*a*b) + sqrt(-a**3*b**3)*(-B*b + D*a)/
(2*a**2*b**3)) + 4*B*D*a**3*b**2*((-A*b + C*a)/(2*a*b) + sqrt(-a**3*b**3)*(-B*b
+ D*a)/(2*a**2*b**3)) + 2*C**2*a**3*b**2*((-A*b + C*a)/(2*a*b) + sqrt(-a**3*b**3
)*(-B*b + D*a)/(2*a**2*b**3)) - 4*C*a**3*b**3*((-A*b + C*a)/(2*a*b) + sqrt(-a**3
*b**3)*(-B*b + D*a)/(2*a**2*b**3))**2 - 2*D**2*a**4*b*((-A*b + C*a)/(2*a*b) + sq
rt(-a**3*b**3)*(-B*b + D*a)/(2*a**2*b**3)))/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 +
6*A*B*C*a*b**3 - 6*A*C*D*a**2*b**2 - B**3*a*b**3 + 3*B**2*D*a**2*b**2 - B*C**2*a
**2*b**2 - 3*B*D**2*a**3*b + C**2*D*a**3*b + D**3*a**4))

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GIAC/XCAS [A]  time = 0.224664, size = 89, normalized size = 1.24 \[ \frac{D x}{b} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{{\left (D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{{\left (C a - A b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, a b} \]

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)*x),x, algorithm="giac")

[Out]

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