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

Optimal. Leaf size=23 \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

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Rubi [A]  time = 0.0382137, antiderivative size = 23, 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 \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/x,x]

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00161879, size = 23, normalized size = 1. \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/x,x]

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

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Maple [A]  time = 0.003, size = 22, normalized size = 1. \[ ab{x}^{2}+{\frac{{b}^{2}{x}^{4}}{4}}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x,x)

[Out]

a*b*x^2+1/4*b^2*x^4+a^2*ln(x)

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Maxima [A]  time = 1.32409, size = 32, normalized size = 1.39 \[ \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^2*x^4 + a*b*x^2 + 1/2*a^2*log(x^2)

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Fricas [A]  time = 0.208355, size = 28, normalized size = 1.22 \[ \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + a^{2} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^2*x^4 + a*b*x^2 + a^2*log(x)

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Sympy [A]  time = 1.02012, size = 20, normalized size = 0.87 \[ a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x,x)

[Out]

a**2*log(x) + a*b*x**2 + b**2*x**4/4

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GIAC/XCAS [A]  time = 0.222284, size = 32, normalized size = 1.39 \[ \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + \frac{1}{2} \, a^{2}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^2*x^4 + a*b*x^2 + 1/2*a^2*ln(x^2)