Optimal. Leaf size=151 \[ \frac{a^{3/2} (A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{a^2 (b B-a D) \log \left (a+b x^2\right )}{2 b^4}-\frac{a x (A b-a C)}{b^3}+\frac{x^3 (A b-a C)}{3 b^2}-\frac{a x^2 (b B-a D)}{2 b^3}+\frac{x^4 (b B-a D)}{4 b^2}+\frac{C x^5}{5 b}+\frac{D x^6}{6 b} \]
[Out]
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Rubi [A] time = 0.324638, antiderivative size = 151, 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 \[ \frac{a^{3/2} (A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{a^2 (b B-a D) \log \left (a+b x^2\right )}{2 b^4}-\frac{a x (A b-a C)}{b^3}+\frac{x^3 (A b-a C)}{3 b^2}-\frac{a x^2 (b B-a D)}{2 b^3}+\frac{x^4 (b B-a D)}{4 b^2}+\frac{C x^5}{5 b}+\frac{D x^6}{6 b} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{C x^{5}}{5 b} + \frac{D x^{6}}{6 b} + \frac{a^{\frac{3}{2}} \left (A b - C a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{a^{2} \left (B b - D a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{a \left (B b - D a\right ) \int x\, dx}{b^{3}} + \frac{x^{4} \left (B b - D a\right )}{4 b^{2}} + \frac{x^{3} \left (A b - C a\right )}{3 b^{2}} - \frac{\left (A b - C a\right ) \int a\, dx}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(D*x**3+C*x**2+B*x+A)/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.156339, size = 130, normalized size = 0.86 \[ \frac{-60 a^{3/2} \sqrt{b} (a C-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+b x \left (30 a^2 (2 C+D x)-5 a b (12 A+x (6 B+x (4 C+3 D x)))+b^2 x^2 (20 A+x (15 B+2 x (6 C+5 D x)))\right )-30 a^2 (a D-b B) \log \left (a+b x^2\right )}{60 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
[Out]
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Maple [A] time = 0.008, size = 176, normalized size = 1.2 \[{\frac{D{x}^{6}}{6\,b}}+{\frac{C{x}^{5}}{5\,b}}+{\frac{B{x}^{4}}{4\,b}}-{\frac{D{x}^{4}a}{4\,{b}^{2}}}+{\frac{A{x}^{3}}{3\,b}}-{\frac{C{x}^{3}a}{3\,{b}^{2}}}-{\frac{Ba{x}^{2}}{2\,{b}^{2}}}+{\frac{D{x}^{2}{a}^{2}}{2\,{b}^{3}}}-{\frac{aAx}{{b}^{2}}}+{\frac{Cx{a}^{2}}{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{3}}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) D}{2\,{b}^{4}}}+{\frac{A{a}^{2}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{a}^{3}C}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x)
[Out]
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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)*x^4/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.262755, size = 1, normalized size = 0.01 \[ \left [\frac{10 \, D b^{3} x^{6} + 12 \, C b^{3} x^{5} - 15 \,{\left (D a b^{2} - B b^{3}\right )} x^{4} - 20 \,{\left (C a b^{2} - A b^{3}\right )} x^{3} + 30 \,{\left (D a^{2} b - B a b^{2}\right )} x^{2} - 30 \,{\left (C a^{2} b - A a b^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 60 \,{\left (C a^{2} b - A a b^{2}\right )} x - 30 \,{\left (D a^{3} - B a^{2} b\right )} \log \left (b x^{2} + a\right )}{60 \, b^{4}}, \frac{10 \, D b^{3} x^{6} + 12 \, C b^{3} x^{5} - 15 \,{\left (D a b^{2} - B b^{3}\right )} x^{4} - 20 \,{\left (C a b^{2} - A b^{3}\right )} x^{3} + 30 \,{\left (D a^{2} b - B a b^{2}\right )} x^{2} - 60 \,{\left (C a^{2} b - A a b^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 60 \,{\left (C a^{2} b - A a b^{2}\right )} x - 30 \,{\left (D a^{3} - B a^{2} b\right )} \log \left (b x^{2} + a\right )}{60 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*x^4/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.07887, size = 308, normalized size = 2.04 \[ \frac{C x^{5}}{5 b} + \frac{D x^{6}}{6 b} + \left (- \frac{a^{2} \left (- B b + D a\right )}{2 b^{4}} - \frac{\sqrt{- a^{3} b^{9}} \left (- A b + C a\right )}{2 b^{8}}\right ) \log{\left (x + \frac{B a^{2} b - D a^{3} - 2 b^{4} \left (- \frac{a^{2} \left (- B b + D a\right )}{2 b^{4}} - \frac{\sqrt{- a^{3} b^{9}} \left (- A b + C a\right )}{2 b^{8}}\right )}{- A a b^{2} + C a^{2} b} \right )} + \left (- \frac{a^{2} \left (- B b + D a\right )}{2 b^{4}} + \frac{\sqrt{- a^{3} b^{9}} \left (- A b + C a\right )}{2 b^{8}}\right ) \log{\left (x + \frac{B a^{2} b - D a^{3} - 2 b^{4} \left (- \frac{a^{2} \left (- B b + D a\right )}{2 b^{4}} + \frac{\sqrt{- a^{3} b^{9}} \left (- A b + C a\right )}{2 b^{8}}\right )}{- A a b^{2} + C a^{2} b} \right )} - \frac{x^{4} \left (- B b + D a\right )}{4 b^{2}} - \frac{x^{3} \left (- A b + C a\right )}{3 b^{2}} + \frac{x^{2} \left (- B a b + D a^{2}\right )}{2 b^{3}} + \frac{x \left (- A a b + C a^{2}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(D*x**3+C*x**2+B*x+A)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.221778, size = 217, normalized size = 1.44 \[ -\frac{{\left (C a^{3} - A a^{2} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{{\left (D a^{3} - B a^{2} b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{10 \, D b^{5} x^{6} + 12 \, C b^{5} x^{5} - 15 \, D a b^{4} x^{4} + 15 \, B b^{5} x^{4} - 20 \, C a b^{4} x^{3} + 20 \, A b^{5} x^{3} + 30 \, D a^{2} b^{3} x^{2} - 30 \, B a b^{4} x^{2} + 60 \, C a^{2} b^{3} x - 60 \, A a b^{4} x}{60 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*x^4/(b*x^2 + a),x, algorithm="giac")
[Out]