3.51 \(\int \left (a+c x^2\right )^2 \, dx\)

Optimal. Leaf size=25 \[ a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5} \]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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Rubi [A]  time = 0.0193471, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^2,x]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5} + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**2,x)

[Out]

2*a*c*x**3/3 + c**2*x**5/5 + Integral(a**2, x)

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Mathematica [A]  time = 0.00170871, size = 25, normalized size = 1. \[ a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^2,x]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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Maple [A]  time = 0.002, size = 22, normalized size = 0.9 \[{a}^{2}x+{\frac{2\,a{x}^{3}c}{3}}+{\frac{{c}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^2,x)

[Out]

a^2*x+2/3*a*x^3*c+1/5*c^2*x^5

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Maxima [A]  time = 0.703682, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, a c x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2,x, algorithm="maxima")

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x

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Fricas [A]  time = 0.191099, size = 1, normalized size = 0.04 \[ \frac{1}{5} x^{5} c^{2} + \frac{2}{3} x^{3} c a + x a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2,x, algorithm="fricas")

[Out]

1/5*x^5*c^2 + 2/3*x^3*c*a + x*a^2

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Sympy [A]  time = 0.087772, size = 22, normalized size = 0.88 \[ a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**2,x)

[Out]

a**2*x + 2*a*c*x**3/3 + c**2*x**5/5

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GIAC/XCAS [A]  time = 0.209627, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, a c x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2,x, algorithm="giac")

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x