3.7.18 \(\int \frac {a+b x^4}{x^9} \, dx\) [618]

Optimal. Leaf size=17 \[ -\frac {a}{8 x^8}-\frac {b}{4 x^4} \]

[Out]

-1/8*a/x^8-1/4*b/x^4

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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}{8 x^8}-\frac {b}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^9,x]

[Out]

-1/8*a/x^8 - b/(4*x^4)

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+b x^4}{x^9} \, dx &=\int \left (\frac {a}{x^9}+\frac {b}{x^5}\right ) \, dx\\ &=-\frac {a}{8 x^8}-\frac {b}{4 x^4}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {a}{8 x^8}-\frac {b}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^9,x]

[Out]

-1/8*a/x^8 - b/(4*x^4)

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

method	result	size
gosper	\(-\frac {2 b \,x^{4}+a}{8 x^{8}}\)	\(14\)
default	\(-\frac {a}{8 x^{8}}-\frac {b}{4 x^{4}}\)	\(14\)
norman	\(\frac {-\frac {b \,x^{4}}{4}-\frac {a}{8}}{x^{8}}\)	\(15\)
risch	\(\frac {-\frac {b \,x^{4}}{4}-\frac {a}{8}}{x^{8}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*a/x^8-1/4*b/x^4

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Maxima [A]
time = 0.31, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2 \, b x^{4} + a}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^9,x, algorithm="maxima")

[Out]

-1/8*(2*b*x^4 + a)/x^8

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Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2 \, b x^{4} + a}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^9,x, algorithm="fricas")

[Out]

-1/8*(2*b*x^4 + a)/x^8

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Sympy [A]
time = 0.05, size = 14, normalized size = 0.82 \begin {gather*} \frac {- a - 2 b x^{4}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**9,x)

[Out]

(-a - 2*b*x**4)/(8*x**8)

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Giac [A]
time = 1.15, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2 \, b x^{4} + a}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^9,x, algorithm="giac")

[Out]

-1/8*(2*b*x^4 + a)/x^8

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2\,b\,x^4+a}{8\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)/x^9,x)

[Out]

-(a + 2*b*x^4)/(8*x^8)

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