Optimal. Leaf size=61 \[ \frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2 (b c-a d)}{2 b^2}+\frac{\left (c+d x^2\right )^2}{4 b} \]
[Out]
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Rubi [A] time = 0.114315, antiderivative size = 61, 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 \[ \frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2 (b c-a d)}{2 b^2}+\frac{\left (c+d x^2\right )^2}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^2)^2)/(a + b*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (c + d x^{2}\right )^{2}}{4 b} - \frac{\left (a d - b c\right ) \int ^{x^{2}} d\, dx}{2 b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**2+c)**2/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0385013, size = 49, normalized size = 0.8 \[ \frac{b d x^2 \left (-2 a d+4 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (a+b x^2\right )}{4 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^2)^2)/(a + b*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 85, normalized size = 1.4 \[{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{a{d}^{2}{x}^{2}}{2\,{b}^{2}}}+{\frac{d{x}^{2}c}{b}}+{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) cad}{{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^2+c)^2/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.34031, size = 89, normalized size = 1.46 \[ \frac{b d^{2} x^{4} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*x/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234495, size = 90, normalized size = 1.48 \[ \frac{b^{2} d^{2} x^{4} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*x/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.97639, size = 51, normalized size = 0.84 \[ \frac{d^{2} x^{4}}{4 b} - \frac{x^{2} \left (a d^{2} - 2 b c d\right )}{2 b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**2+c)**2/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.225176, size = 90, normalized size = 1.48 \[ \frac{b d^{2} x^{4} + 4 \, b c d x^{2} - 2 \, a d^{2} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*x/(b*x^2 + a),x, algorithm="giac")
[Out]