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

Optimal. Leaf size=92 \[ -\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a C)}{a^2}-\frac{A}{2 a x^2}-\frac{B}{a x} \]

[Out]

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

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Rubi [A]  time = 0.237514, antiderivative size = 92, 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 \[ -\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a C)}{a^2}-\frac{A}{2 a x^2}-\frac{B}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.2924, size = 76, normalized size = 0.83 \[ - \frac{A}{2 a x^{2}} - \frac{B}{a x} - \frac{\left (A b - C a\right ) \log{\left (x \right )}}{a^{2}} + \frac{\left (A b - C a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{2}} - \frac{\left (B b - D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.14513, size = 84, normalized size = 0.91 \[ \frac{(A b-a C) \log \left (a+b x^2\right )+2 \log (x) (a C-A b)-\frac{a A}{x^2}+\frac{2 \sqrt{a} (a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{2 a B}{x}}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 102, normalized size = 1.1 \[ -{\frac{A}{2\,a{x}^{2}}}-{\frac{B}{ax}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( x \right ) C}{a}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,a}}-{\frac{Bb}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{D\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A/a/x^2-B/a/x-1/a^2*ln(x)*A*b+1/a*ln(x)*C+1/2/a^2*b*ln(b*x^2+a)*A-1/2/a*ln(
b*x^2+a)*C-1/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*b*B+1/(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282118, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (D a^{2} - B a b\right )} x^{2} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) +{\left ({\left (C a - A b\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (C a - A b\right )} x^{2} \log \left (x\right ) + 2 \, B a x + A a\right )} \sqrt{-a b}}{2 \, \sqrt{-a b} a^{2} x^{2}}, \frac{2 \,{\left (D a^{2} - B a b\right )} x^{2} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left ({\left (C a - A b\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (C a - A b\right )} x^{2} \log \left (x\right ) + 2 \, B a x + A a\right )} \sqrt{a b}}{2 \, \sqrt{a b} a^{2} x^{2}}\right ] \]

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^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 39.1408, size = 1686, normalized size = 18.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))*log(x + (-6*A**
3*b**4 + 18*A**2*C*a*b**3 + 6*A**2*a**2*b**3*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**
5*b)*(-B*b + D*a)/(2*a**4*b)) + 2*A*B**2*a*b**3 - 4*A*B*D*a**2*b**2 - 18*A*C**2*
a**2*b**2 - 12*A*C*a**3*b**2*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a
)/(2*a**4*b)) + 2*A*D**2*a**3*b + 12*A*a**4*b**2*(-(-A*b + C*a)/(2*a**2) - sqrt(
-a**5*b)*(-B*b + D*a)/(2*a**4*b))**2 - 2*B**2*C*a**2*b**2 + 2*B**2*a**3*b**2*(-(
-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 4*B*C*D*a**3*b -
4*B*D*a**4*b*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) +
6*C**3*a**3*b + 6*C**2*a**4*b*(-(-A*b + C*a)/(2*a**2) - sqrt(-a**5*b)*(-B*b + D
*a)/(2*a**4*b)) - 2*C*D**2*a**4 - 12*C*a**5*b*(-(-A*b + C*a)/(2*a**2) - sqrt(-a*
*5*b)*(-B*b + D*a)/(2*a**4*b))**2 + 2*D**2*a**5*(-(-A*b + C*a)/(2*a**2) - sqrt(-
a**5*b)*(-B*b + D*a)/(2*a**4*b)))/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 18*A*B*C*a
*b**3 - 18*A*C*D*a**2*b**2 - B**3*a*b**3 + 3*B**2*D*a**2*b**2 - 9*B*C**2*a**2*b*
*2 - 3*B*D**2*a**3*b + 9*C**2*D*a**3*b + D**3*a**4)) + (-(-A*b + C*a)/(2*a**2) +
sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b))*log(x + (-6*A**3*b**4 + 18*A**2*C*a*b**3
+ 6*A**2*a**2*b**3*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4
*b)) + 2*A*B**2*a*b**3 - 4*A*B*D*a**2*b**2 - 18*A*C**2*a**2*b**2 - 12*A*C*a**3*b
**2*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 2*A*D**2*
a**3*b + 12*A*a**4*b**2*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*
a**4*b))**2 - 2*B**2*C*a**2*b**2 + 2*B**2*a**3*b**2*(-(-A*b + C*a)/(2*a**2) + sq
rt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 4*B*C*D*a**3*b - 4*B*D*a**4*b*(-(-A*b + C
*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) + 6*C**3*a**3*b + 6*C**2*a
**4*b*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**4*b)) - 2*C*D**
2*a**4 - 12*C*a**5*b*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a**
4*b))**2 + 2*D**2*a**5*(-(-A*b + C*a)/(2*a**2) + sqrt(-a**5*b)*(-B*b + D*a)/(2*a
**4*b)))/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 18*A*B*C*a*b**3 - 18*A*C*D*a**2*b**
2 - B**3*a*b**3 + 3*B**2*D*a**2*b**2 - 9*B*C**2*a**2*b**2 - 3*B*D**2*a**3*b + 9*
C**2*D*a**3*b + D**3*a**4)) - (A + 2*B*x)/(2*a*x**2) + (-A*b + C*a)*log(x + (-6*
A**3*b**4 + 18*A**2*C*a*b**3 + 6*A**2*b**3*(-A*b + C*a) + 2*A*B**2*a*b**3 - 4*A*
B*D*a**2*b**2 - 18*A*C**2*a**2*b**2 - 12*A*C*a*b**2*(-A*b + C*a) + 2*A*D**2*a**3
*b + 12*A*b**2*(-A*b + C*a)**2 - 2*B**2*C*a**2*b**2 + 2*B**2*a*b**2*(-A*b + C*a)
+ 4*B*C*D*a**3*b - 4*B*D*a**2*b*(-A*b + C*a) + 6*C**3*a**3*b + 6*C**2*a**2*b*(-
A*b + C*a) - 2*C*D**2*a**4 - 12*C*a*b*(-A*b + C*a)**2 + 2*D**2*a**3*(-A*b + C*a)
)/(-9*A**2*B*b**4 + 9*A**2*D*a*b**3 + 18*A*B*C*a*b**3 - 18*A*C*D*a**2*b**2 - B**
3*a*b**3 + 3*B**2*D*a**2*b**2 - 9*B*C**2*a**2*b**2 - 3*B*D**2*a**3*b + 9*C**2*D*
a**3*b + D**3*a**4))/a**2

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GIAC/XCAS [A]  time = 0.226842, size = 108, normalized size = 1.17 \[ \frac{{\left (D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{{\left (C a - A b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{{\left (C a - A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{2 \, B a x + A a}{2 \, a^{2} x^{2}} \]

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)*x^3),x, algorithm="giac")

[Out]

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