3.50 \(\int \left (a+c x^2\right )^3 \, dx\)

Optimal. Leaf size=35 \[ a^3 x+a^2 c x^3+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7} \]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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Rubi [A]  time = 0.0290113, antiderivative size = 35, 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 \[ a^3 x+a^2 c x^3+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^3,x]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} c x^{3} + \frac{3 a c^{2} x^{5}}{5} + \frac{c^{3} x^{7}}{7} + \int a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**3,x)

[Out]

a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7 + Integral(a**3, x)

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Mathematica [A]  time = 0.00196918, size = 35, normalized size = 1. \[ a^3 x+a^2 c x^3+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^3,x]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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Maple [A]  time = 0.002, size = 32, normalized size = 0.9 \[{a}^{3}x+{a}^{2}c{x}^{3}+{\frac{3\,a{c}^{2}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3,x)

[Out]

a^3*x+a^2*c*x^3+3/5*a*c^2*x^5+1/7*c^3*x^7

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Maxima [A]  time = 0.694496, size = 42, normalized size = 1.2 \[ \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3,x, algorithm="maxima")

[Out]

1/7*c^3*x^7 + 3/5*a*c^2*x^5 + a^2*c*x^3 + a^3*x

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Fricas [A]  time = 0.187323, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} c^{3} + \frac{3}{5} x^{5} c^{2} a + x^{3} c a^{2} + x a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/7*x^7*c^3 + 3/5*x^5*c^2*a + x^3*c*a^2 + x*a^3

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Sympy [A]  time = 0.098392, size = 32, normalized size = 0.91 \[ a^{3} x + a^{2} c x^{3} + \frac{3 a c^{2} x^{5}}{5} + \frac{c^{3} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**3,x)

[Out]

a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7

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GIAC/XCAS [A]  time = 0.207707, size = 42, normalized size = 1.2 \[ \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3,x, algorithm="giac")

[Out]

1/7*c^3*x^7 + 3/5*a*c^2*x^5 + a^2*c*x^3 + a^3*x