3.2.44 \(\int \frac {b x^2+c x^4}{x^8} \, dx\) [144]

Optimal. Leaf size=17 \[ -\frac {b}{5 x^5}-\frac {c}{3 x^3} \]

[Out]

-1/5*b/x^5-1/3*c/x^3

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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} -\frac {b}{5 x^5}-\frac {c}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^8,x]

[Out]

-1/5*b/x^5 - c/(3*x^3)

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 {b x^2+c x^4}{x^8} \, dx &=\int \left (\frac {b}{x^6}+\frac {c}{x^4}\right ) \, dx\\ &=-\frac {b}{5 x^5}-\frac {c}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {b}{5 x^5}-\frac {c}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^8,x]

[Out]

-1/5*b/x^5 - c/(3*x^3)

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
default	\(-\frac {b}{5 x^{5}}-\frac {c}{3 x^{3}}\)	\(14\)
risch	\(\frac {-\frac {c \,x^{2}}{3}-\frac {b}{5}}{x^{5}}\)	\(15\)
gosper	\(-\frac {5 c \,x^{2}+3 b}{15 x^{5}}\)	\(16\)
norman	\(\frac {-\frac {1}{5} b \,x^{2}-\frac {1}{3} c \,x^{4}}{x^{7}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/5*b/x^5-1/3*c/x^3

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Maxima [A]
time = 0.31, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5 \, c x^{2} + 3 \, b}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^8,x, algorithm="maxima")

[Out]

-1/15*(5*c*x^2 + 3*b)/x^5

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Fricas [A]
time = 0.32, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5 \, c x^{2} + 3 \, b}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^8,x, algorithm="fricas")

[Out]

-1/15*(5*c*x^2 + 3*b)/x^5

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Sympy [A]
time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} \frac {- 3 b - 5 c x^{2}}{15 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)/x**8,x)

[Out]

(-3*b - 5*c*x**2)/(15*x**5)

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Giac [A]
time = 5.93, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5 \, c x^{2} + 3 \, b}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^8,x, algorithm="giac")

[Out]

-1/15*(5*c*x^2 + 3*b)/x^5

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Mupad [B]
time = 0.03, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5\,c\,x^2+3\,b}{15\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^8,x)

[Out]

-(3*b + 5*c*x^2)/(15*x^5)

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