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

Optimal. Leaf size=26 \[ a^2 \log (x)+\frac{2}{3} a b x^3+\frac{b^2 x^6}{6} \]

[Out]

(2*a*b*x^3)/3 + (b^2*x^6)/6 + a^2*Log[x]

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Rubi [A]  time = 0.0338788, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ a^2 \log (x)+\frac{2}{3} a b x^3+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2/x,x]

[Out]

(2*a*b*x^3)/3 + (b^2*x^6)/6 + a^2*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00127193, size = 26, normalized size = 1. \[ a^2 \log (x)+\frac{2}{3} a b x^3+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2/x,x]

[Out]

(2*a*b*x^3)/3 + (b^2*x^6)/6 + a^2*Log[x]

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Maple [A]  time = 0.002, size = 23, normalized size = 0.9 \[{\frac{2\,ab{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{6}}{6}}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/x,x)

[Out]

2/3*a*b*x^3+1/6*b^2*x^6+a^2*ln(x)

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Maxima [A]  time = 1.42346, size = 34, normalized size = 1.31 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{3} \, a b x^{3} + \frac{1}{3} \, a^{2} \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/3*a*b*x^3 + 1/3*a^2*log(x^3)

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Fricas [A]  time = 0.243299, size = 30, normalized size = 1.15 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{3} \, a b x^{3} + a^{2} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/3*a*b*x^3 + a^2*log(x)

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Sympy [A]  time = 1.05256, size = 24, normalized size = 0.92 \[ a^{2} \log{\left (x \right )} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2/x,x)

[Out]

a**2*log(x) + 2*a*b*x**3/3 + b**2*x**6/6

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GIAC/XCAS [A]  time = 0.224094, size = 31, normalized size = 1.19 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{2}{3} \, a b x^{3} + a^{2}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^2*x^6 + 2/3*a*b*x^3 + a^2*ln(abs(x))