Optimal. Leaf size=130 \[ \frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11} \]
[Out]
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Rubi [A] time = 0.380114, antiderivative size = 130, 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 \[ \frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b d^{3} f x^{11}}{11} + c^{3} e \int a\, dx + \frac{c^{2} x^{3} \left (a c f + 3 a d e + b c e\right )}{3} + \frac{c x^{5} \left (3 a c d f + 3 a d^{2} e + b c^{2} f + 3 b c d e\right )}{5} + \frac{d^{2} x^{9} \left (a d f + 3 b c f + b d e\right )}{9} + \frac{d x^{7} \left (3 a c d f + a d^{2} e + 3 b c^{2} f + 3 b c d e\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(d*x**2+c)**3*(f*x**2+e),x)
[Out]
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Mathematica [A] time = 0.0966672, size = 130, normalized size = 1. \[ \frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]
[Out]
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Maple [A] time = 0., size = 149, normalized size = 1.2 \[{\frac{b{d}^{3}f{x}^{11}}{11}}+{\frac{ \left ( \left ( a{d}^{3}+3\,bc{d}^{2} \right ) f+b{d}^{3}e \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ) f+ \left ( a{d}^{3}+3\,bc{d}^{2} \right ) e \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,a{c}^{2}d+b{c}^{3} \right ) f+ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ) e \right ){x}^{5}}{5}}+{\frac{ \left ( a{c}^{3}f+ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ) e \right ){x}^{3}}{3}}+a{c}^{3}ex \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(d*x^2+c)^3*(f*x^2+e),x)
[Out]
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Maxima [A] time = 1.36694, size = 197, normalized size = 1.52 \[ \frac{1}{11} \, b d^{3} f x^{11} + \frac{1}{9} \,{\left (b d^{3} e +{\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} x^{9} + \frac{1}{7} \,{\left ({\left (3 \, b c d^{2} + a d^{3}\right )} e + 3 \,{\left (b c^{2} d + a c d^{2}\right )} f\right )} x^{7} + a c^{3} e x + \frac{1}{5} \,{\left (3 \,{\left (b c^{2} d + a c d^{2}\right )} e +{\left (b c^{3} + 3 \, a c^{2} d\right )} f\right )} x^{5} + \frac{1}{3} \,{\left (a c^{3} f +{\left (b c^{3} + 3 \, a c^{2} d\right )} e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^3*(f*x^2 + e),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.182995, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} f d^{3} b + \frac{1}{9} x^{9} e d^{3} b + \frac{1}{3} x^{9} f d^{2} c b + \frac{1}{9} x^{9} f d^{3} a + \frac{3}{7} x^{7} e d^{2} c b + \frac{3}{7} x^{7} f d c^{2} b + \frac{1}{7} x^{7} e d^{3} a + \frac{3}{7} x^{7} f d^{2} c a + \frac{3}{5} x^{5} e d c^{2} b + \frac{1}{5} x^{5} f c^{3} b + \frac{3}{5} x^{5} e d^{2} c a + \frac{3}{5} x^{5} f d c^{2} a + \frac{1}{3} x^{3} e c^{3} b + x^{3} e d c^{2} a + \frac{1}{3} x^{3} f c^{3} a + x e c^{3} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^3*(f*x^2 + e),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.087388, size = 173, normalized size = 1.33 \[ a c^{3} e x + \frac{b d^{3} f x^{11}}{11} + x^{9} \left (\frac{a d^{3} f}{9} + \frac{b c d^{2} f}{3} + \frac{b d^{3} e}{9}\right ) + x^{7} \left (\frac{3 a c d^{2} f}{7} + \frac{a d^{3} e}{7} + \frac{3 b c^{2} d f}{7} + \frac{3 b c d^{2} e}{7}\right ) + x^{5} \left (\frac{3 a c^{2} d f}{5} + \frac{3 a c d^{2} e}{5} + \frac{b c^{3} f}{5} + \frac{3 b c^{2} d e}{5}\right ) + x^{3} \left (\frac{a c^{3} f}{3} + a c^{2} d e + \frac{b c^{3} e}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(d*x**2+c)**3*(f*x**2+e),x)
[Out]
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GIAC/XCAS [A] time = 0.225356, size = 234, normalized size = 1.8 \[ \frac{1}{11} \, b d^{3} f x^{11} + \frac{1}{3} \, b c d^{2} f x^{9} + \frac{1}{9} \, a d^{3} f x^{9} + \frac{1}{9} \, b d^{3} x^{9} e + \frac{3}{7} \, b c^{2} d f x^{7} + \frac{3}{7} \, a c d^{2} f x^{7} + \frac{3}{7} \, b c d^{2} x^{7} e + \frac{1}{7} \, a d^{3} x^{7} e + \frac{1}{5} \, b c^{3} f x^{5} + \frac{3}{5} \, a c^{2} d f x^{5} + \frac{3}{5} \, b c^{2} d x^{5} e + \frac{3}{5} \, a c d^{2} x^{5} e + \frac{1}{3} \, a c^{3} f x^{3} + \frac{1}{3} \, b c^{3} x^{3} e + a c^{2} d x^{3} e + a c^{3} x e \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)^3*(f*x^2 + e),x, algorithm="giac")
[Out]