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

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

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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} \begin {gather*} b \log (x)-\frac {a}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^3,x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^3,x]

[Out]

-1/2*a/x^2 + b*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x^2}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)/x^3,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)/x^3, x]

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fricas [A]  time = 1.28, size = 17, normalized size = 1.31 \begin {gather*} \frac {2 \, b x^{2} \log \relax (x) - a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*x^2*log(x) - a)/x^2

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giac [A]  time = 1.06, size = 20, normalized size = 1.54 \begin {gather*} \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {b x^{2} + a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*log(x^2) - 1/2*(b*x^2 + a)/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x^3,x)

[Out]

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

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maxima [A]  time = 1.36, size = 14, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*log(x^2) - 1/2*a/x^2

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mupad [B]  time = 4.93, size = 11, normalized size = 0.85 \begin {gather*} b\,\ln \relax (x)-\frac {a}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^3,x)

[Out]

b*log(x) - a/(2*x^2)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.77 \begin {gather*} - \frac {a}{2 x^{2}} + b \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/x**3,x)

[Out]

-a/(2*x**2) + b*log(x)

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