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

Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{b^2 x^6}{6} \]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

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Rubi [A]  time = 0.0397211, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3)^2/x,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

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Rubi in Sympy [A]  time = 6.93237, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 2*a*b*x**5/5 + b**2*x**6/6

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Mathematica [A]  time = 0.00241619, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{{b}^{2}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^2/x,x)

[Out]

1/4*a^2*x^4+2/5*a*b*x^5+1/6*b^2*x^6

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Maxima [A]  time = 1.40459, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4

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Fricas [A]  time = 0.198338, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4

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Sympy [A]  time = 0.086975, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 2*a*b*x**5/5 + b**2*x**6/6

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GIAC/XCAS [A]  time = 0.217166, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4