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

Optimal. Leaf size=73 \[ \frac{(A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{(b B-a D) \log \left (a+b x^2\right )}{2 b^2}+\frac{C x}{b}+\frac{D x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.160775, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{(A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{(b B-a D) \log \left (a+b x^2\right )}{2 b^2}+\frac{C x}{b}+\frac{D x^2}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0754814, size = 68, normalized size = 0.93 \[ \frac{\frac{2 \sqrt{b} (A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}+(b B-a D) \log \left (a+b x^2\right )+b x (2 C+D x)}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 83, normalized size = 1.1 \[{\frac{D{x}^{2}}{2\,b}}+{\frac{Cx}{b}}+{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) aD}{2\,{b}^{2}}}+{A\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{aC}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233294, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (C a b - A b^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) -{\left (D b x^{2} + 2 \, C b x -{\left (D a - B b\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{2 \, \sqrt{-a b} b^{2}}, -\frac{2 \,{\left (C a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (D b x^{2} + 2 \, C b x -{\left (D a - B b\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{2 \, \sqrt{a b} b^{2}}\right ] \]

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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.79122, size = 219, normalized size = 3. \[ \frac{C x}{b} + \frac{D x^{2}}{2 b} + \left (- \frac{- B b + D a}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right ) \log{\left (x + \frac{B a b - D a^{2} - 2 a b^{2} \left (- \frac{- B b + D a}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right )}{- A b^{2} + C a b} \right )} + \left (- \frac{- B b + D a}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right ) \log{\left (x + \frac{B a b - D a^{2} - 2 a b^{2} \left (- \frac{- B b + D a}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right )}{- A b^{2} + C a b} \right )} \]

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)

[Out]

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

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GIAC/XCAS [A]  time = 0.236983, size = 89, normalized size = 1.22 \[ -\frac{{\left (C a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{{\left (D a - B b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{2}} + \frac{D b x^{2} + 2 \, C b x}{2 \, b^{2}} \]

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, algorithm="giac")

[Out]

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