3.1.24 \(\int \frac {(a+b x^2)^2}{x^6} \, dx\)

Optimal. Leaf size=28 \[ -\frac {a^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {b^2}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} -\frac {a^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {b^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^6,x]

[Out]

-a^2/(5*x^5) - (2*a*b)/(3*x^3) - b^2/x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^6} \, dx &=\int \left (\frac {a^2}{x^6}+\frac {2 a b}{x^4}+\frac {b^2}{x^2}\right ) \, dx\\ &=-\frac {a^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {b^2}{x}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {a^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {b^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^6,x]

[Out]

-1/5*a^2/x^5 - (2*a*b)/(3*x^3) - b^2/x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^2}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^2/x^6,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^2/x^6, x]

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fricas [A]  time = 1.03, size = 26, normalized size = 0.93 \begin {gather*} -\frac {15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^6,x, algorithm="fricas")

[Out]

-1/15*(15*b^2*x^4 + 10*a*b*x^2 + 3*a^2)/x^5

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giac [A]  time = 1.05, size = 26, normalized size = 0.93 \begin {gather*} -\frac {15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^6,x, algorithm="giac")

[Out]

-1/15*(15*b^2*x^4 + 10*a*b*x^2 + 3*a^2)/x^5

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maple [A]  time = 0.00, size = 25, normalized size = 0.89 \begin {gather*} -\frac {b^{2}}{x}-\frac {2 a b}{3 x^{3}}-\frac {a^{2}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^6,x)

[Out]

-1/5*a^2/x^5-2/3*a*b/x^3-b^2/x

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maxima [A]  time = 1.30, size = 26, normalized size = 0.93 \begin {gather*} -\frac {15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^6,x, algorithm="maxima")

[Out]

-1/15*(15*b^2*x^4 + 10*a*b*x^2 + 3*a^2)/x^5

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mupad [B]  time = 0.03, size = 25, normalized size = 0.89 \begin {gather*} -\frac {\frac {a^2}{5}+\frac {2\,a\,b\,x^2}{3}+b^2\,x^4}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^6,x)

[Out]

-(a^2/5 + b^2*x^4 + (2*a*b*x^2)/3)/x^5

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sympy [A]  time = 0.17, size = 27, normalized size = 0.96 \begin {gather*} \frac {- 3 a^{2} - 10 a b x^{2} - 15 b^{2} x^{4}}{15 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**6,x)

[Out]

(-3*a**2 - 10*a*b*x**2 - 15*b**2*x**4)/(15*x**5)

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