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

Optimal. Leaf size=13 \[ a \log (x)+\frac{b x^2}{2} \]

[Out]

(b*x^2)/2 + a*Log[x]

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Rubi [A]  time = 0.0038555, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ a \log (x)+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x,x]

[Out]

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

Mathematica [A]  time = 0.0009411, size = 13, normalized size = 1. \[ a \log (x)+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x,x]

[Out]

(b*x^2)/2 + a*Log[x]

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \begin{align*}{\frac{b{x}^{2}}{2}}+a\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x,x)

[Out]

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

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Maxima [A]  time = 2.22765, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{2} \, b x^{2} + \frac{1}{2} \, a \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + 1/2*a*log(x^2)

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Fricas [A]  time = 1.46844, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{2} \, b x^{2} + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*log(x)

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Sympy [A]  time = 0.080684, size = 10, normalized size = 0.77 \begin{align*} a \log{\left (x \right )} + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/x,x)

[Out]

a*log(x) + b*x**2/2

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Giac [A]  time = 2.511, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{2} \, b x^{2} + \frac{1}{2} \, a \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + 1/2*a*log(x^2)