Optimal. Leaf size=75 \[ \frac{a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{x^4 (A b-a B)}{4 b^2}+\frac{B x^6}{6 b} \]
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Rubi [A] time = 0.194155, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{x^4 (A b-a B)}{4 b^2}+\frac{B x^6}{6 b} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^2))/(a + b*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B x^{6}}{6 b} + \frac{a^{2} \left (A b - B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{\left (A b - B a\right ) \int ^{x^{2}} x\, dx}{2 b^{2}} - \frac{\left (A b - B a\right ) \int ^{x^{2}} a\, dx}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**2+A)/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0537712, size = 71, normalized size = 0.95 \[ \frac{b x^2 \left (6 a^2 B-3 a b \left (2 A+B x^2\right )+b^2 x^2 \left (3 A+2 B x^2\right )\right )+6 a^2 (A b-a B) \log \left (a+b x^2\right )}{12 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^2))/(a + b*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 86, normalized size = 1.2 \[{\frac{B{x}^{6}}{6\,b}}+{\frac{A{x}^{4}}{4\,b}}-{\frac{B{x}^{4}a}{4\,{b}^{2}}}-{\frac{aA{x}^{2}}{2\,{b}^{2}}}+{\frac{B{x}^{2}{a}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{3}}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^2+A)/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.35327, size = 100, normalized size = 1.33 \[ \frac{2 \, B b^{2} x^{6} - 3 \,{\left (B a b - A b^{2}\right )} x^{4} + 6 \,{\left (B a^{2} - A a b\right )} x^{2}}{12 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227649, size = 101, normalized size = 1.35 \[ \frac{2 \, B b^{3} x^{6} - 3 \,{\left (B a b^{2} - A b^{3}\right )} x^{4} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 6 \,{\left (B a^{3} - A a^{2} b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.81479, size = 65, normalized size = 0.87 \[ \frac{B x^{6}}{6 b} - \frac{a^{2} \left (- A b + B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x^{2} \left (- A a b + B a^{2}\right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**2+A)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.244918, size = 104, normalized size = 1.39 \[ \frac{2 \, B b^{2} x^{6} - 3 \, B a b x^{4} + 3 \, A b^{2} x^{4} + 6 \, B a^{2} x^{2} - 6 \, A a b x^{2}}{12 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a),x, algorithm="giac")
[Out]