Optimal. Leaf size=45 \[ a^2 A x+\frac{2}{3} a A c x^3+\frac{B \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} A c^2 x^5 \]
[Out]
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Rubi [A] time = 0.0448162, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^2 A x+\frac{2}{3} a A c x^3+\frac{B \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} A c^2 x^5 \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 A a c x^{3}}{3} + \frac{A c^{2} x^{5}}{5} + A \int a^{2}\, dx + \frac{B \left (a + c x^{2}\right )^{3}}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.00389291, size = 60, normalized size = 1.33 \[ a^2 A x+\frac{1}{2} a^2 B x^2+\frac{2}{3} a A c x^3+\frac{1}{2} a B c x^4+\frac{1}{5} A c^2 x^5+\frac{1}{6} B c^2 x^6 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0., size = 51, normalized size = 1.1 \[{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{A{c}^{2}{x}^{5}}{5}}+{\frac{aBc{x}^{4}}{2}}+{\frac{2\,aAc{x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{2}}{2}}+{a}^{2}Ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.697596, size = 68, normalized size = 1.51 \[ \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254742, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} c^{2} B + \frac{1}{5} x^{5} c^{2} A + \frac{1}{2} x^{4} c a B + \frac{2}{3} x^{3} c a A + \frac{1}{2} x^{2} a^{2} B + x a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.120201, size = 58, normalized size = 1.29 \[ A a^{2} x + \frac{2 A a c x^{3}}{3} + \frac{A c^{2} x^{5}}{5} + \frac{B a^{2} x^{2}}{2} + \frac{B a c x^{4}}{2} + \frac{B c^{2} x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.291391, size = 68, normalized size = 1.51 \[ \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A),x, algorithm="giac")
[Out]