Optimal. Leaf size=118 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d) (5 b c-a d)}{2 c d^3}+\frac{x^3 (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^3}{3 d^2} \]
[Out]
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Rubi [A] time = 0.292757, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d) (5 b c-a d)}{2 c d^3}+\frac{x^3 (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^3}{3 d^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x^2)^2)/(c + d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 44.0846, size = 100, normalized size = 0.85 \[ \frac{b^{2} x^{3}}{3 d^{2}} + \frac{x^{3} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} - \frac{x \left (a d - 5 b c\right ) \left (a d - b c\right )}{2 c d^{3}} + \frac{\left (a d - 5 b c\right ) \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{c} d^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.120253, size = 105, normalized size = 0.89 \[ \frac{\left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{2 b x (b c-a d)}{d^3}+\frac{b^2 x^3}{3 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2)^2,x]
[Out]
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Maple [A] time = 0.013, size = 156, normalized size = 1.3 \[{\frac{{b}^{2}{x}^{3}}{3\,{d}^{2}}}+2\,{\frac{abx}{{d}^{2}}}-2\,{\frac{x{b}^{2}c}{{d}^{3}}}-{\frac{{a}^{2}x}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{xabc}{{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}{c}^{2}x}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}}{2\,d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{abc}{{d}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{5\,{b}^{2}{c}^{2}}{2\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x^2+a)^2/(d*x^2+c)^2,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((b*x^2 + a)^2*x^2/(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248417, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2} +{\left (5 \, b^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (2 \, b^{2} d^{2} x^{5} - 2 \,{\left (5 \, b^{2} c d - 6 \, a b d^{2}\right )} x^{3} - 3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x\right )} \sqrt{-c d}}{12 \,{\left (d^{4} x^{2} + c d^{3}\right )} \sqrt{-c d}}, \frac{3 \,{\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2} +{\left (5 \, b^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (2 \, b^{2} d^{2} x^{5} - 2 \,{\left (5 \, b^{2} c d - 6 \, a b d^{2}\right )} x^{3} - 3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x\right )} \sqrt{c d}}{6 \,{\left (d^{4} x^{2} + c d^{3}\right )} \sqrt{c d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^2/(d*x^2 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.94537, size = 245, normalized size = 2.08 \[ \frac{b^{2} x^{3}}{3 d^{2}} - \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c d^{3} + 2 d^{4} x^{2}} - \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (- \frac{c d^{3} \sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (\frac{c d^{3} \sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{x \left (2 a b d - 2 b^{2} c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.227917, size = 154, normalized size = 1.31 \[ \frac{{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} d^{3}} - \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} d^{3}} + \frac{b^{2} d^{4} x^{3} - 6 \, b^{2} c d^{3} x + 6 \, a b d^{4} x}{3 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^2/(d*x^2 + c)^2,x, algorithm="giac")
[Out]