3.413 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{x} \, dx\)

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 = 0.0072947, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {14} \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x,x]

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a^2+2 a b x^2+b^2 x^4}{x} \, dx &=\int \left (\frac{a^2}{x}+2 a b x+b^2 x^3\right ) \, dx\\ &=a b x^2+\frac{b^2 x^4}{4}+a^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0009761, 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^2 + 2*a*b*x^2 + b^2*x^4)/x,x]

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

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Maple [A]  time = 0.043, size = 22, normalized size = 1. \begin{align*} ab{x}^{2}+{\frac{{b}^{2}{x}^{4}}{4}}+{a}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*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 = 0.981399, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*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 = 1.44811, size = 49, normalized size = 2.13 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*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 = 0.247208, size = 20, normalized size = 0.87 \begin{align*} a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*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 [A]  time = 1.15573, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x,x, algorithm="giac")

[Out]

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