3.1.25 \(\int \frac {(a+b x^2)^2}{x^7} \, dx\)

Optimal. Leaf size=19 \[ -\frac {\left (a+b x^2\right )^3}{6 a x^6} \]

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} -\frac {\left (a+b x^2\right )^3}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^7,x]

[Out]

-(a + b*x^2)^3/(6*a*x^6)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^7} \, dx &=-\frac {\left (a+b x^2\right )^3}{6 a x^6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.58 \begin {gather*} -\frac {a^2}{6 x^6}-\frac {a b}{2 x^4}-\frac {b^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^7,x]

[Out]

-1/6*a^2/x^6 - (a*b)/(2*x^4) - b^2/(2*x^2)

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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^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^2/x^7,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^2/x^7, x]

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fricas [A]  time = 1.19, size = 24, normalized size = 1.26 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7,x, algorithm="fricas")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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giac [A]  time = 1.11, size = 24, normalized size = 1.26 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7,x, algorithm="giac")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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maple [A]  time = 0.01, size = 25, normalized size = 1.32 \begin {gather*} -\frac {b^{2}}{2 x^{2}}-\frac {a b}{2 x^{4}}-\frac {a^{2}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^7,x)

[Out]

-1/6*a^2/x^6-1/2*b^2/x^2-1/2*a*b/x^4

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maxima [A]  time = 1.38, size = 24, normalized size = 1.26 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7,x, algorithm="maxima")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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mupad [B]  time = 0.03, size = 26, normalized size = 1.37 \begin {gather*} -\frac {\frac {a^2}{6}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^4}{2}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^7,x)

[Out]

-(a^2/6 + (b^2*x^4)/2 + (a*b*x^2)/2)/x^6

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sympy [A]  time = 0.19, size = 26, normalized size = 1.37 \begin {gather*} \frac {- a^{2} - 3 a b x^{2} - 3 b^{2} x^{4}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**7,x)

[Out]

(-a**2 - 3*a*b*x**2 - 3*b**2*x**4)/(6*x**6)

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