Optimal. Leaf size=92 \[ \frac{(A b-a C) \log \left (a+b x^2\right )}{2 b^2}-\frac{\sqrt{a} (b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (b B-a D)}{b^2}+\frac{C x^2}{2 b}+\frac{D x^3}{3 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.210285, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(A b-a C) \log \left (a+b x^2\right )}{2 b^2}-\frac{\sqrt{a} (b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (b B-a D)}{b^2}+\frac{C x^2}{2 b}+\frac{D x^3}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{C \int x\, dx}{b} + \frac{D x^{3}}{3 b} - \frac{\sqrt{a} \left (B b - D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \left (B b - D a\right ) \int \frac{1}{b^{2}}\, dx + \frac{\left (A b - C a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(D*x**3+C*x**2+B*x+A)/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.123848, size = 81, normalized size = 0.88 \[ \frac{3 (A b-a C) \log \left (a+b x^2\right )+x (-6 a D+6 b B+b x (3 C+2 D x))}{6 b^2}+\frac{\sqrt{a} (a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 106, normalized size = 1.2 \[{\frac{D{x}^{3}}{3\,b}}+{\frac{C{x}^{2}}{2\,b}}+{\frac{Bx}{b}}-{\frac{Dxa}{{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) aC}{2\,{b}^{2}}}-{\frac{Ba}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{a}^{2}D}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x)
[Out]
_______________________________________________________________________________________
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)*x/(b*x^2 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.239165, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, D b x^{3} + 3 \, C b x^{2} - 3 \,{\left (D a - B b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \,{\left (D a - B b\right )} x - 3 \,{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{6 \, b^{2}}, \frac{2 \, D b x^{3} + 3 \, C b x^{2} + 6 \,{\left (D a - B b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 6 \,{\left (D a - B b\right )} x - 3 \,{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{6 \, b^{2}}\right ] \]
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, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.80254, size = 211, normalized size = 2.29 \[ \frac{C x^{2}}{2 b} + \frac{D x^{3}}{3 b} + \left (- \frac{- A b + C a}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right ) \log{\left (x + \frac{- A b + C a + 2 b^{2} \left (- \frac{- A b + C a}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right )}{- B b + D a} \right )} + \left (- \frac{- A b + C a}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right ) \log{\left (x + \frac{- A b + C a + 2 b^{2} \left (- \frac{- A b + C a}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right )}{- B b + D a} \right )} - \frac{x \left (- B b + D a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(D*x**3+C*x**2+B*x+A)/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.237712, size = 119, normalized size = 1.29 \[ -\frac{{\left (C a - A b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{2}} + \frac{{\left (D a^{2} - B a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \, D b^{2} x^{3} + 3 \, C b^{2} x^{2} - 6 \, D a b x + 6 \, B b^{2} x}{6 \, b^{3}} \]
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, algorithm="giac")
[Out]