Optimal. Leaf size=55 \[ \frac{1}{4} a^2 c x^4+\frac{1}{8} b x^8 (2 a d+b c)+\frac{1}{6} a x^6 (a d+2 b c)+\frac{1}{10} b^2 d x^{10} \]
[Out]
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Rubi [A] time = 0.182199, antiderivative size = 55, 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 \[ \frac{1}{4} a^2 c x^4+\frac{1}{8} b x^8 (2 a d+b c)+\frac{1}{6} a x^6 (a d+2 b c)+\frac{1}{10} b^2 d x^{10} \]
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. \[ \frac{a^{2} c \int ^{x^{2}} x\, dx}{2} + \frac{a x^{6} \left (a d + 2 b c\right )}{6} + \frac{b^{2} d x^{10}}{10} + \frac{b x^{8} \left (2 a d + b c\right )}{8} \]
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 = 0.0131721, size = 55, normalized size = 1. \[ \frac{1}{4} a^2 c x^4+\frac{1}{8} b x^8 (2 a d+b c)+\frac{1}{6} a x^6 (a d+2 b c)+\frac{1}{10} b^2 d x^{10} \]
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 = 0.001, size = 52, normalized size = 1. \[{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{6}}{6}}+{\frac{{a}^{2}c{x}^{4}}{4}} \]
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 = 1.32444, size = 69, normalized size = 1.25 \[ \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{8} \,{\left (b^{2} c + 2 \, a b d\right )} x^{8} + \frac{1}{4} \, a^{2} c x^{4} + \frac{1}{6} \,{\left (2 \, a b c + a^{2} d\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.199242, size = 1, normalized size = 0.02 \[ \frac{1}{10} x^{10} d b^{2} + \frac{1}{8} x^{8} c b^{2} + \frac{1}{4} x^{8} d b a + \frac{1}{3} x^{6} c b a + \frac{1}{6} x^{6} d a^{2} + \frac{1}{4} x^{4} c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.115505, size = 53, normalized size = 0.96 \[ \frac{a^{2} c x^{4}}{4} + \frac{b^{2} d x^{10}}{10} + x^{8} \left (\frac{a b d}{4} + \frac{b^{2} c}{8}\right ) + x^{6} \left (\frac{a^{2} d}{6} + \frac{a b c}{3}\right ) \]
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 = 0.222238, size = 72, normalized size = 1.31 \[ \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{8} \, b^{2} c x^{8} + \frac{1}{4} \, a b d x^{8} + \frac{1}{3} \, a b c x^{6} + \frac{1}{6} \, a^{2} d x^{6} + \frac{1}{4} \, a^{2} c x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^3,x, algorithm="giac")
[Out]