3.49 \(\int \left (a+c x^2\right )^4 \, dx\)

Optimal. Leaf size=51 \[ a^4 x+\frac{4}{3} a^3 c x^3+\frac{6}{5} a^2 c^2 x^5+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9} \]

[Out]

a^4*x + (4*a^3*c*x^3)/3 + (6*a^2*c^2*x^5)/5 + (4*a*c^3*x^7)/7 + (c^4*x^9)/9

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Rubi [A]  time = 0.0438834, antiderivative size = 51, 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 \[ a^4 x+\frac{4}{3} a^3 c x^3+\frac{6}{5} a^2 c^2 x^5+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^4,x]

[Out]

a^4*x + (4*a^3*c*x^3)/3 + (6*a^2*c^2*x^5)/5 + (4*a*c^3*x^7)/7 + (c^4*x^9)/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{3} c x^{3}}{3} + \frac{6 a^{2} c^{2} x^{5}}{5} + \frac{4 a c^{3} x^{7}}{7} + \frac{c^{4} x^{9}}{9} + \int a^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**4,x)

[Out]

4*a**3*c*x**3/3 + 6*a**2*c**2*x**5/5 + 4*a*c**3*x**7/7 + c**4*x**9/9 + Integral(
a**4, x)

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Mathematica [A]  time = 0.00265362, size = 51, normalized size = 1. \[ a^4 x+\frac{4}{3} a^3 c x^3+\frac{6}{5} a^2 c^2 x^5+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^4,x]

[Out]

a^4*x + (4*a^3*c*x^3)/3 + (6*a^2*c^2*x^5)/5 + (4*a*c^3*x^7)/7 + (c^4*x^9)/9

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Maple [A]  time = 0.001, size = 44, normalized size = 0.9 \[{a}^{4}x+{\frac{4\,{a}^{3}c{x}^{3}}{3}}+{\frac{6\,{a}^{2}{c}^{2}{x}^{5}}{5}}+{\frac{4\,a{c}^{3}{x}^{7}}{7}}+{\frac{{c}^{4}{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^4,x)

[Out]

a^4*x+4/3*a^3*c*x^3+6/5*a^2*c^2*x^5+4/7*a*c^3*x^7+1/9*c^4*x^9

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Maxima [A]  time = 0.701094, size = 58, normalized size = 1.14 \[ \frac{1}{9} \, c^{4} x^{9} + \frac{4}{7} \, a c^{3} x^{7} + \frac{6}{5} \, a^{2} c^{2} x^{5} + \frac{4}{3} \, a^{3} c x^{3} + a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^4,x, algorithm="maxima")

[Out]

1/9*c^4*x^9 + 4/7*a*c^3*x^7 + 6/5*a^2*c^2*x^5 + 4/3*a^3*c*x^3 + a^4*x

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Fricas [A]  time = 0.185464, size = 1, normalized size = 0.02 \[ \frac{1}{9} x^{9} c^{4} + \frac{4}{7} x^{7} c^{3} a + \frac{6}{5} x^{5} c^{2} a^{2} + \frac{4}{3} x^{3} c a^{3} + x a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^4,x, algorithm="fricas")

[Out]

1/9*x^9*c^4 + 4/7*x^7*c^3*a + 6/5*x^5*c^2*a^2 + 4/3*x^3*c*a^3 + x*a^4

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Sympy [A]  time = 0.106809, size = 49, normalized size = 0.96 \[ a^{4} x + \frac{4 a^{3} c x^{3}}{3} + \frac{6 a^{2} c^{2} x^{5}}{5} + \frac{4 a c^{3} x^{7}}{7} + \frac{c^{4} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**4,x)

[Out]

a**4*x + 4*a**3*c*x**3/3 + 6*a**2*c**2*x**5/5 + 4*a*c**3*x**7/7 + c**4*x**9/9

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GIAC/XCAS [A]  time = 0.208722, size = 58, normalized size = 1.14 \[ \frac{1}{9} \, c^{4} x^{9} + \frac{4}{7} \, a c^{3} x^{7} + \frac{6}{5} \, a^{2} c^{2} x^{5} + \frac{4}{3} \, a^{3} c x^{3} + a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^4,x, algorithm="giac")

[Out]

1/9*c^4*x^9 + 4/7*a*c^3*x^7 + 6/5*a^2*c^2*x^5 + 4/3*a^3*c*x^3 + a^4*x