3.10 \(\int \frac{\left (a x+b x^3\right )^2}{x^2} \, dx\)

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

[Out]

a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5

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Rubi [A]  time = 0.022954, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3)^2/x^2,x]

[Out]

a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x)**2/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.00167767, size = 25, normalized size = 1. \[ a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^3)^2/x^2,x]

[Out]

a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x)^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.37403, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, b^{2} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 2/3*a*b*x^3 + a^2*x

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Fricas [A]  time = 0.195497, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, b^{2} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 2/3*a*b*x^3 + a^2*x

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Sympy [A]  time = 0.047896, size = 22, normalized size = 0.88 \[ a^{2} x + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x)**2/x**2,x)

[Out]

a**2*x + 2*a*b*x**3/3 + b**2*x**5/5

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GIAC/XCAS [A]  time = 0.215578, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, b^{2} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 2/3*a*b*x^3 + a^2*x