3.121 \(\int \frac{x^2 \left (a c+b c x^2\right )}{a+b x^2} \, dx\)

Optimal. Leaf size=8 \[ \frac{c x^3}{3} \]

[Out]

(c*x^3)/3

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Rubi [A]  time = 0.00635038, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a*c + b*c*x^2))/(a + b*x^2),x]

[Out]

(c*x^3)/3

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Rubi in Sympy [A]  time = 3.58967, size = 5, normalized size = 0.62 \[ \frac{c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*c*x**2+a*c)/(b*x**2+a),x)

[Out]

c*x**3/3

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Mathematica [A]  time = 0.000604448, size = 8, normalized size = 1. \[ \frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a*c + b*c*x^2))/(a + b*x^2),x]

[Out]

(c*x^3)/3

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Maple [A]  time = 0.001, size = 7, normalized size = 0.9 \[{\frac{c{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*c*x^2+a*c)/(b*x^2+a),x)

[Out]

1/3*c*x^3

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Maxima [A]  time = 1.33462, size = 8, normalized size = 1. \[ \frac{1}{3} \, c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3

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Fricas [A]  time = 0.213022, size = 8, normalized size = 1. \[ \frac{1}{3} \, c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3

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Sympy [A]  time = 0.104626, size = 5, normalized size = 0.62 \[ \frac{c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*c*x**2+a*c)/(b*x**2+a),x)

[Out]

c*x**3/3

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GIAC/XCAS [A]  time = 0.222676, size = 8, normalized size = 1. \[ \frac{1}{3} \, c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3