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

Optimal. Leaf size=23 \[ -\frac {a^2}{3 x^3}-\frac {2 a b}{x}+b^2 x \]

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Rubi [A]  time = 0.01, antiderivative size = 23, 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}{3 x^3}-\frac {2 a b}{x}+b^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {a^2}{3 x^3}-\frac {2 a b}{x}+b^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a^2/x^3 - (2*a*b)/x + b^2*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^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.81, size = 26, normalized size = 1.13 \begin {gather*} \frac {3 \, b^{2} x^{4} - 6 \, a b x^{2} - a^{2}}{3 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/3*(3*b^2*x^4 - 6*a*b*x^2 - a^2)/x^3

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giac [A]  time = 0.20, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x - \frac {6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3

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maple [A]  time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x -\frac {2 a b}{x}-\frac {a^{2}}{3 x^{3}} \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^4,x)

[Out]

-1/3*a^2/x^3-2*a*b/x+b^2*x

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maxima [A]  time = 1.33, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x - \frac {6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3

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mupad [B]  time = 4.11, size = 24, normalized size = 1.04 \begin {gather*} b^2\,x-\frac {\frac {a^2}{3}+2\,b\,a\,x^2}{x^3} \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^4,x)

[Out]

b^2*x - (a^2/3 + 2*a*b*x^2)/x^3

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sympy [A]  time = 0.14, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x + \frac {- a^{2} - 6 a b x^{2}}{3 x^{3}} \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**4,x)

[Out]

b**2*x + (-a**2 - 6*a*b*x**2)/(3*x**3)

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