Optimal. Leaf size=45 \[ a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} c^2 d x^5 \]
[Out]
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Rubi [A] time = 0.052609, 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 \[ a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} c^2 d x^5 \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 a c d x^{3}}{3} + \frac{c^{2} d x^{5}}{5} + d \int a^{2}\, dx + \frac{e \left (a + c x^{2}\right )^{3}}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.00396939, size = 60, normalized size = 1.33 \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a c d x^3+\frac{1}{2} a c e x^4+\frac{1}{5} c^2 d x^5+\frac{1}{6} c^2 e x^6 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 51, normalized size = 1.1 \[{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{{c}^{2}d{x}^{5}}{5}}+{\frac{ace{x}^{4}}{2}}+{\frac{2\,acd{x}^{3}}{3}}+{\frac{e{a}^{2}{x}^{2}}{2}}+{a}^{2}dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.694721, size = 68, normalized size = 1.51 \[ \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{5} \, c^{2} d x^{5} + \frac{1}{2} \, a c e x^{4} + \frac{2}{3} \, a c d x^{3} + \frac{1}{2} \, a^{2} e x^{2} + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.18655, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{1}{2} x^{4} e c a + \frac{2}{3} x^{3} d c a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.110155, size = 58, normalized size = 1.29 \[ a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{2 a c d x^{3}}{3} + \frac{a c e x^{4}}{2} + \frac{c^{2} d x^{5}}{5} + \frac{c^{2} e x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210782, size = 72, normalized size = 1.6 \[ \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{1}{2} \, a c x^{4} e + \frac{2}{3} \, a c d x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d),x, algorithm="giac")
[Out]