Optimal. Leaf size=50 \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]
[Out]
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Rubi [A] time = 0.0715088, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)*(c + d*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b d^{2} x^{7}}{7} + c^{2} \int a\, dx + \frac{c x^{3} \left (2 a d + b c\right )}{3} + \frac{d x^{5} \left (a d + 2 b c\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.017418, size = 50, normalized size = 1. \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)*(c + d*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 49, normalized size = 1. \[{\frac{b{d}^{2}{x}^{7}}{7}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,acd+{c}^{2}b \right ){x}^{3}}{3}}+a{c}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.39718, size = 65, normalized size = 1.3 \[ \frac{1}{7} \, b d^{2} x^{7} + \frac{1}{5} \,{\left (2 \, b c d + a d^{2}\right )} x^{5} + a c^{2} x + \frac{1}{3} \,{\left (b c^{2} + 2 \, a c d\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.177117, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} d^{2} b + \frac{2}{5} x^{5} d c b + \frac{1}{5} x^{5} d^{2} a + \frac{1}{3} x^{3} c^{2} b + \frac{2}{3} x^{3} d c a + x c^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.110744, size = 53, normalized size = 1.06 \[ a c^{2} x + \frac{b d^{2} x^{7}}{7} + x^{5} \left (\frac{a d^{2}}{5} + \frac{2 b c d}{5}\right ) + x^{3} \left (\frac{2 a c d}{3} + \frac{b c^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.227377, size = 68, normalized size = 1.36 \[ \frac{1}{7} \, b d^{2} x^{7} + \frac{2}{5} \, b c d x^{5} + \frac{1}{5} \, a d^{2} x^{5} + \frac{1}{3} \, b c^{2} x^{3} + \frac{2}{3} \, a c d x^{3} + a c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^2,x, algorithm="giac")
[Out]