Optimal. Leaf size=54 \[ -\frac{a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^4}{4 b} \]
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Rubi [A] time = 0.131333, antiderivative size = 54, 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 \[ -\frac{a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^4}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^2))/(a + b*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} - \left (\frac{a d}{2} - \frac{b c}{2}\right ) \int ^{x^{2}} \frac{1}{b^{2}}\, dx + \frac{d \int ^{x^{2}} x\, dx}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**2+c)/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0319273, size = 47, normalized size = 0.87 \[ \frac{b x^2 \left (-2 a d+2 b c+b d x^2\right )+2 a (a d-b c) \log \left (a+b x^2\right )}{4 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^2))/(a + b*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 62, normalized size = 1.2 \[{\frac{d{x}^{4}}{4\,b}}-{\frac{ad{x}^{2}}{2\,{b}^{2}}}+{\frac{c{x}^{2}}{2\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\,{b}^{3}}}-{\frac{ac\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^2+c)/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.34607, size = 68, normalized size = 1.26 \[ \frac{b d x^{4} + 2 \,{\left (b c - a d\right )} x^{2}}{4 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x^3/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232231, size = 69, normalized size = 1.28 \[ \frac{b^{2} d x^{4} + 2 \,{\left (b^{2} c - a b d\right )} x^{2} - 2 \,{\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x^3/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.65353, size = 44, normalized size = 0.81 \[ \frac{a \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{d x^{4}}{4 b} - \frac{x^{2} \left (a d - b c\right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(d*x**2+c)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.233297, size = 70, normalized size = 1.3 \[ \frac{b d x^{4} + 2 \, b c x^{2} - 2 \, a d x^{2}}{4 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x^3/(b*x^2 + a),x, algorithm="giac")
[Out]