Optimal. Leaf size=94 \[ \frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9 \]
[Out]
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Rubi [A] time = 0.250629, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9 \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b d^{2} f x^{9}}{9} + c^{2} e \int a\, dx + \frac{c x^{3} \left (a c f + 2 a d e + b c e\right )}{3} + \frac{d x^{7} \left (a d f + 2 b c f + b d e\right )}{7} + x^{5} \left (\frac{2 a c d f}{5} + \frac{a d^{2} e}{5} + \frac{b c^{2} f}{5} + \frac{2 b c d e}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(d*x**2+c)**2*(f*x**2+e),x)
[Out]
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Mathematica [A] time = 0.0581505, size = 96, normalized size = 1.02 \[ \frac{1}{5} x^5 \left (2 a c d f+a d^2 e+b c^2 f+2 b c d e\right )+\frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]
[Out]
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Maple [A] time = 0.001, size = 101, normalized size = 1.1 \[{\frac{b{d}^{2}f{x}^{9}}{9}}+{\frac{ \left ( \left ( a{d}^{2}+2\,bcd \right ) f+b{d}^{2}e \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,acd+b{c}^{2} \right ) f+ \left ( a{d}^{2}+2\,bcd \right ) e \right ){x}^{5}}{5}}+{\frac{ \left ( a{c}^{2}f+ \left ( 2\,acd+b{c}^{2} \right ) e \right ){x}^{3}}{3}}+a{c}^{2}ex \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e),x)
[Out]
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Maxima [A] time = 1.35665, size = 135, normalized size = 1.44 \[ \frac{1}{9} \, b d^{2} f x^{9} + \frac{1}{7} \,{\left (b d^{2} e +{\left (2 \, b c d + a d^{2}\right )} f\right )} x^{7} + \frac{1}{5} \,{\left ({\left (2 \, b c d + a d^{2}\right )} e +{\left (b c^{2} + 2 \, a c d\right )} f\right )} x^{5} + a c^{2} e x + \frac{1}{3} \,{\left (a c^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^2*(f*x^2 + e),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.182081, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} f d^{2} b + \frac{1}{7} x^{7} e d^{2} b + \frac{2}{7} x^{7} f d c b + \frac{1}{7} x^{7} f d^{2} a + \frac{2}{5} x^{5} e d c b + \frac{1}{5} x^{5} f c^{2} b + \frac{1}{5} x^{5} e d^{2} a + \frac{2}{5} x^{5} f d c a + \frac{1}{3} x^{3} e c^{2} b + \frac{2}{3} x^{3} e d c a + \frac{1}{3} x^{3} f c^{2} a + x e c^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^2*(f*x^2 + e),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.074659, size = 121, normalized size = 1.29 \[ a c^{2} e x + \frac{b d^{2} f x^{9}}{9} + x^{7} \left (\frac{a d^{2} f}{7} + \frac{2 b c d f}{7} + \frac{b d^{2} e}{7}\right ) + x^{5} \left (\frac{2 a c d f}{5} + \frac{a d^{2} e}{5} + \frac{b c^{2} f}{5} + \frac{2 b c d e}{5}\right ) + x^{3} \left (\frac{a c^{2} f}{3} + \frac{2 a c d e}{3} + \frac{b c^{2} e}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(d*x**2+c)**2*(f*x**2+e),x)
[Out]
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GIAC/XCAS [A] time = 0.227169, size = 162, normalized size = 1.72 \[ \frac{1}{9} \, b d^{2} f x^{9} + \frac{2}{7} \, b c d f x^{7} + \frac{1}{7} \, a d^{2} f x^{7} + \frac{1}{7} \, b d^{2} x^{7} e + \frac{1}{5} \, b c^{2} f x^{5} + \frac{2}{5} \, a c d f x^{5} + \frac{2}{5} \, b c d x^{5} e + \frac{1}{5} \, a d^{2} x^{5} e + \frac{1}{3} \, a c^{2} f x^{3} + \frac{1}{3} \, b c^{2} x^{3} e + \frac{2}{3} \, a c d x^{3} e + a c^{2} x e \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^2*(f*x^2 + e),x, algorithm="giac")
[Out]