3.19.16 \(\int \frac {a+\frac {b}{x^2}}{x^6} \, dx\) [1816]

Optimal. Leaf size=17 \[ -\frac {b}{7 x^7}-\frac {a}{5 x^5} \]

[Out]

-1/7*b/x^7-1/5*a/x^5

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Rubi [A]
time = 0.00, antiderivative size = 17, 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*} -\frac {a}{5 x^5}-\frac {b}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x^6,x]

[Out]

-1/7*b/x^7 - a/(5*x^5)

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+\frac {b}{x^2}}{x^6} \, dx &=\int \left (\frac {b}{x^8}+\frac {a}{x^6}\right ) \, dx\\ &=-\frac {b}{7 x^7}-\frac {a}{5 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {b}{7 x^7}-\frac {a}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^6,x]

[Out]

-1/7*b/x^7 - a/(5*x^5)

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Maple [A]
time = 0.02, size = 14, normalized size = 0.82

method	result	size
default	\(-\frac {b}{7 x^{7}}-\frac {a}{5 x^{5}}\)	\(14\)
norman	\(\frac {-\frac {a \,x^{2}}{5}-\frac {b}{7}}{x^{7}}\)	\(15\)
risch	\(\frac {-\frac {a \,x^{2}}{5}-\frac {b}{7}}{x^{7}}\)	\(15\)
gosper	\(-\frac {7 a \,x^{2}+5 b}{35 x^{7}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/x^2+a)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/7*b/x^7-1/5*a/x^5

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Maxima [A]
time = 0.29, size = 15, normalized size = 0.88 \begin {gather*} -\frac {7 \, a x^{2} + 5 \, b}{35 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(7*a*x^2 + 5*b)/x^7

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Fricas [A]
time = 0.35, size = 15, normalized size = 0.88 \begin {gather*} -\frac {7 \, a x^{2} + 5 \, b}{35 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(7*a*x^2 + 5*b)/x^7

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Sympy [A]
time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} \frac {- 7 a x^{2} - 5 b}{35 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-7*a*x**2 - 5*b)/(35*x**7)

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Giac [A]
time = 0.58, size = 15, normalized size = 0.88 \begin {gather*} -\frac {7 \, a x^{2} + 5 \, b}{35 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(7*a*x^2 + 5*b)/x^7

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Mupad [B]
time = 0.03, size = 15, normalized size = 0.88 \begin {gather*} -\frac {7\,a\,x^2+5\,b}{35\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/x^6,x)

[Out]

-(5*b + 7*a*x^2)/(35*x^7)

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