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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ a \log (x)+\frac {b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x,x]

[Out]

(b*x^2)/2 + a*Log[x]

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fricas [A]  time = 0.80, size = 11, normalized size = 0.85 \[ \frac {1}{2} \, b x^{2} + a \log \relax (x) \]

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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giac [A]  time = 1.02, size = 14, normalized size = 1.08 \[ \frac {1}{2} \, b x^{2} + \frac {1}{2} \, a \log \left (x^{2}\right ) \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \frac {b \,x^{2}}{2}+a \ln \relax (x ) \]

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 = 1.35, size = 14, normalized size = 1.08 \[ \frac {1}{2} \, b x^{2} + \frac {1}{2} \, a \log \left (x^{2}\right ) \]

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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mupad [B]  time = 0.02, size = 11, normalized size = 0.85 \[ \frac {b\,x^2}{2}+a\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x,x)

[Out]

(b*x^2)/2 + a*log(x)

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sympy [A]  time = 0.08, size = 10, normalized size = 0.77 \[ a \log {\relax (x )} + \frac {b x^{2}}{2} \]

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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