Optimal. Leaf size=71 \[ \frac{d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac{\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3} \]
[Out]
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Rubi [A] time = 0.27408, antiderivative size = 71, 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 \[ \frac{d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac{\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^2)^2*(c + d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 29.2518, size = 60, normalized size = 0.85 \[ \frac{d^{2} \left (a + b x^{2}\right )^{5}}{10 b^{3}} - \frac{d \left (a + b x^{2}\right )^{4} \left (a d - b c\right )}{4 b^{3}} + \frac{\left (a + b x^{2}\right )^{3} \left (a d - b c\right )^{2}}{6 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**2+a)**2*(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.0432828, size = 81, normalized size = 1.14 \[ \frac{1}{60} x^2 \left (10 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+30 a^2 c^2+15 b d x^6 (a d+b c)+30 a c x^2 (a d+b c)+6 b^2 d^2 x^8\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^2)^2*(c + d*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 90, normalized size = 1.3 \[{\frac{{b}^{2}{d}^{2}{x}^{10}}{10}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{4}}{4}}+{\frac{{a}^{2}{c}^{2}{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^2+a)^2*(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.35656, size = 115, normalized size = 1.62 \[ \frac{1}{10} \, b^{2} d^{2} x^{10} + \frac{1}{4} \,{\left (b^{2} c d + a b d^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + \frac{1}{2} \, a^{2} c^{2} x^{2} + \frac{1}{2} \,{\left (a b c^{2} + a^{2} c d\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20764, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} d^{2} b^{2} + \frac{1}{4} x^{8} d c b^{2} + \frac{1}{4} x^{8} d^{2} b a + \frac{1}{6} x^{6} c^{2} b^{2} + \frac{2}{3} x^{6} d c b a + \frac{1}{6} x^{6} d^{2} a^{2} + \frac{1}{2} x^{4} c^{2} b a + \frac{1}{2} x^{4} d c a^{2} + \frac{1}{2} x^{2} c^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.152933, size = 94, normalized size = 1.32 \[ \frac{a^{2} c^{2} x^{2}}{2} + \frac{b^{2} d^{2} x^{10}}{10} + x^{8} \left (\frac{a b d^{2}}{4} + \frac{b^{2} c d}{4}\right ) + x^{6} \left (\frac{a^{2} d^{2}}{6} + \frac{2 a b c d}{3} + \frac{b^{2} c^{2}}{6}\right ) + x^{4} \left (\frac{a^{2} c d}{2} + \frac{a b c^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**2+a)**2*(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.222479, size = 127, normalized size = 1.79 \[ \frac{1}{10} \, b^{2} d^{2} x^{10} + \frac{1}{4} \, b^{2} c d x^{8} + \frac{1}{4} \, a b d^{2} x^{8} + \frac{1}{6} \, b^{2} c^{2} x^{6} + \frac{2}{3} \, a b c d x^{6} + \frac{1}{6} \, a^{2} d^{2} x^{6} + \frac{1}{2} \, a b c^{2} x^{4} + \frac{1}{2} \, a^{2} c d x^{4} + \frac{1}{2} \, a^{2} c^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x,x, algorithm="giac")
[Out]