3.3.43 \(\int \frac {a^2+2 a b x^2+b^2 x^4}{x^3} \, dx\)

Optimal. Leaf size=27 \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3,x]

[Out]

-1/2*a^2/x^2 + (b^2*x^2)/2 + 2*a*b*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^2+2 a b x^2+b^2 x^4}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3, x]

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fricas [A]  time = 0.68, size = 27, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{4} + 4 \, a b x^{2} \log \relax (x) - a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} \frac {b^{2} x^{2}}{2}+2 a b \ln \relax (x )-\frac {a^{2}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)/x^3,x)

[Out]

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

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maxima [A]  time = 1.32, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 23, normalized size = 0.85 \begin {gather*} \frac {b^2\,x^2}{2}-\frac {a^2}{2\,x^2}+2\,a\,b\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)/x^3,x)

[Out]

(b^2*x^2)/2 - a^2/(2*x^2) + 2*a*b*log(x)

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sympy [A]  time = 0.14, size = 24, normalized size = 0.89 \begin {gather*} - \frac {a^{2}}{2 x^{2}} + 2 a b \log {\relax (x )} + \frac {b^{2} x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)/x**3,x)

[Out]

-a**2/(2*x**2) + 2*a*b*log(x) + b**2*x**2/2

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