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

Optimal. Leaf size=8 \[ \frac{c x^2}{2} \]

[Out]

(c*x^2)/2

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Rubi [A]  time = 0.00598656, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{c x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ c \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*Integral(x, x)

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Mathematica [A]  time = 0.000429417, size = 8, normalized size = 1. \[ \frac{c x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2)/2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*x^2

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Maxima [A]  time = 1.3497, size = 8, normalized size = 1. \[ \frac{1}{2} \, c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*x^2

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Fricas [A]  time = 0.215096, size = 8, normalized size = 1. \[ \frac{1}{2} \, c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*x^2

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Sympy [A]  time = 0.098101, size = 5, normalized size = 0.62 \[ \frac{c x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**2/2

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GIAC/XCAS [A]  time = 0.228116, size = 8, normalized size = 1. \[ \frac{1}{2} \, c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*x^2