Optimal. Leaf size=104 \[ \frac{1}{3} a^2 B x^3+\frac{1}{4} a^2 C x^4+\frac{A \left (a+b x^2\right )^3}{6 b}+\frac{1}{7} b x^7 (2 a D+b B)+\frac{1}{5} a x^5 (a D+2 b B)+\frac{1}{3} a b C x^6+\frac{1}{8} b^2 C x^8+\frac{1}{9} b^2 D x^9 \]
[Out]
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Rubi [A] time = 0.316991, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{3} a^2 B x^3+\frac{1}{4} a^2 C x^4+\frac{A \left (a+b x^2\right )^3}{6 b}+\frac{1}{7} b x^7 (2 a D+b B)+\frac{1}{5} a x^5 (a D+2 b B)+\frac{1}{3} a b C x^6+\frac{1}{8} b^2 C x^8+\frac{1}{9} b^2 D x^9 \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]
[Out]
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Rubi in Sympy [A] time = 40.1159, size = 94, normalized size = 0.9 \[ \frac{A \left (a + b x^{2}\right )^{3}}{6 b} + \frac{B a^{2} x^{3}}{3} + \frac{C a^{2} x^{4}}{4} + \frac{C a b x^{6}}{3} + \frac{C b^{2} x^{8}}{8} + \frac{D b^{2} x^{9}}{9} + \frac{a x^{5} \left (2 B b + D a\right )}{5} + \frac{b x^{7} \left (B b + 2 D a\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**2+a)**2*(D*x**3+C*x**2+B*x+A),x)
[Out]
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Mathematica [A] time = 0.0961104, size = 92, normalized size = 0.88 \[ \frac{42 a^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+12 a b x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))+5 b^2 x^6 (84 A+x (72 B+7 x (9 C+8 D x)))}{2520} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]
[Out]
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Maple [A] time = 0.003, size = 102, normalized size = 1. \[{\frac{{b}^{2}D{x}^{9}}{9}}+{\frac{{b}^{2}C{x}^{8}}{8}}+{\frac{ \left ({b}^{2}B+2\,abD \right ){x}^{7}}{7}}+{\frac{ \left ({b}^{2}A+2\,abC \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,abB+{a}^{2}D \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,abA+{a}^{2}C \right ){x}^{4}}{4}}+{\frac{{a}^{2}B{x}^{3}}{3}}+{\frac{{a}^{2}A{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x)
[Out]
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Maxima [A] time = 1.34595, size = 136, normalized size = 1.31 \[ \frac{1}{9} \, D b^{2} x^{9} + \frac{1}{8} \, C b^{2} x^{8} + \frac{1}{7} \,{\left (2 \, D a b + B b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (2 \, C a b + A b^{2}\right )} x^{6} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{5} \,{\left (D a^{2} + 2 \, B a b\right )} x^{5} + \frac{1}{2} \, A a^{2} x^{2} + \frac{1}{4} \,{\left (C a^{2} + 2 \, A a b\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^2*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204869, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} b^{2} D + \frac{1}{8} x^{8} b^{2} C + \frac{2}{7} x^{7} b a D + \frac{1}{7} x^{7} b^{2} B + \frac{1}{3} x^{6} b a C + \frac{1}{6} x^{6} b^{2} A + \frac{1}{5} x^{5} a^{2} D + \frac{2}{5} x^{5} b a B + \frac{1}{4} x^{4} a^{2} C + \frac{1}{2} x^{4} b a A + \frac{1}{3} x^{3} a^{2} B + \frac{1}{2} x^{2} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^2*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.073285, size = 110, normalized size = 1.06 \[ \frac{A a^{2} x^{2}}{2} + \frac{B a^{2} x^{3}}{3} + \frac{C b^{2} x^{8}}{8} + \frac{D b^{2} x^{9}}{9} + x^{7} \left (\frac{B b^{2}}{7} + \frac{2 D a b}{7}\right ) + x^{6} \left (\frac{A b^{2}}{6} + \frac{C a b}{3}\right ) + x^{5} \left (\frac{2 B a b}{5} + \frac{D a^{2}}{5}\right ) + x^{4} \left (\frac{A a b}{2} + \frac{C a^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**2+a)**2*(D*x**3+C*x**2+B*x+A),x)
[Out]
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GIAC/XCAS [A] time = 0.220532, size = 142, normalized size = 1.37 \[ \frac{1}{9} \, D b^{2} x^{9} + \frac{1}{8} \, C b^{2} x^{8} + \frac{2}{7} \, D a b x^{7} + \frac{1}{7} \, B b^{2} x^{7} + \frac{1}{3} \, C a b x^{6} + \frac{1}{6} \, A b^{2} x^{6} + \frac{1}{5} \, D a^{2} x^{5} + \frac{2}{5} \, B a b x^{5} + \frac{1}{4} \, C a^{2} x^{4} + \frac{1}{2} \, A a b x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{2} \, A a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^2*x,x, algorithm="giac")
[Out]