Optimal. Leaf size=79 \[ -\frac{c (b c-a d)^2 \log \left (c+d x^2\right )}{2 d^4}+\frac{x^2 (b c-a d)^2}{2 d^3}-\frac{b x^4 (b c-2 a d)}{4 d^2}+\frac{b^2 x^6}{6 d} \]
[Out]
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Rubi [A] time = 0.202514, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{c (b c-a d)^2 \log \left (c+d x^2\right )}{2 d^4}+\frac{x^2 (b c-a d)^2}{2 d^3}-\frac{b x^4 (b c-2 a d)}{4 d^2}+\frac{b^2 x^6}{6 d} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(a + b*x^2)^2)/(c + d*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{6}}{6 d} + \frac{b \left (2 a d - b c\right ) \int ^{x^{2}} x\, dx}{2 d^{2}} - \frac{c \left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 d^{4}} + \frac{\left (a d - b c\right )^{2} \int ^{x^{2}} \frac{1}{d^{3}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0688523, size = 82, normalized size = 1.04 \[ \frac{d x^2 \left (6 a^2 d^2+6 a b d \left (d x^2-2 c\right )+b^2 \left (6 c^2-3 c d x^2+2 d^2 x^4\right )\right )-6 c (b c-a d)^2 \log \left (c+d x^2\right )}{12 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2),x]
[Out]
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Maple [A] time = 0.006, size = 124, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{6}}{6\,d}}+{\frac{{x}^{4}ab}{2\,d}}-{\frac{{b}^{2}c{x}^{4}}{4\,{d}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2\,d}}-{\frac{abc{x}^{2}}{{d}^{2}}}+{\frac{{x}^{2}{b}^{2}{c}^{2}}{2\,{d}^{3}}}-{\frac{c\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{d}^{2}}}+{\frac{{c}^{2}\ln \left ( d{x}^{2}+c \right ) ab}{{d}^{3}}}-{\frac{{c}^{3}\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,{d}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^2+a)^2/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.34784, size = 135, normalized size = 1.71 \[ \frac{2 \, b^{2} d^{2} x^{6} - 3 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{4} + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{12 \, d^{3}} - \frac{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^3/(d*x^2 + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225583, size = 136, normalized size = 1.72 \[ \frac{2 \, b^{2} d^{3} x^{6} - 3 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 6 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2} - 6 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (d x^{2} + c\right )}{12 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^3/(d*x^2 + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.19881, size = 83, normalized size = 1.05 \[ \frac{b^{2} x^{6}}{6 d} - \frac{c \left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 d^{4}} + \frac{x^{4} \left (2 a b d - b^{2} c\right )}{4 d^{2}} + \frac{x^{2} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.221766, size = 144, normalized size = 1.82 \[ \frac{2 \, b^{2} d^{2} x^{6} - 3 \, b^{2} c d x^{4} + 6 \, a b d^{2} x^{4} + 6 \, b^{2} c^{2} x^{2} - 12 \, a b c d x^{2} + 6 \, a^{2} d^{2} x^{2}}{12 \, d^{3}} - \frac{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^3/(d*x^2 + c),x, algorithm="giac")
[Out]