3.823 \(\int \frac {a+b x^2+c x^4}{x^7} \, dx\)

Optimal. Leaf size=25 \[ -\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ -\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^7,x]

[Out]

-a/(6*x^6) - b/(4*x^4) - c/(2*x^2)

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^2+c x^4}{x^7} \, dx &=\int \left (\frac {a}{x^7}+\frac {b}{x^5}+\frac {c}{x^3}\right ) \, dx\\ &=-\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \[ -\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^7,x]

[Out]

-1/6*a/x^6 - b/(4*x^4) - c/(2*x^2)

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fricas [A]  time = 0.67, size = 21, normalized size = 0.84 \[ -\frac {6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^7,x, algorithm="fricas")

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

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giac [A]  time = 0.15, size = 21, normalized size = 0.84 \[ -\frac {6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^7,x, algorithm="giac")

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \[ -\frac {c}{2 x^{2}}-\frac {b}{4 x^{4}}-\frac {a}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^7,x)

[Out]

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

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maxima [A]  time = 1.30, size = 21, normalized size = 0.84 \[ -\frac {6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^7,x, algorithm="maxima")

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

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mupad [B]  time = 0.03, size = 21, normalized size = 0.84 \[ -\frac {\frac {c\,x^4}{2}+\frac {b\,x^2}{4}+\frac {a}{6}}{x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^7,x)

[Out]

-(a/6 + (b*x^2)/4 + (c*x^4)/2)/x^6

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sympy [A]  time = 0.34, size = 22, normalized size = 0.88 \[ \frac {- 2 a - 3 b x^{2} - 6 c x^{4}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**7,x)

[Out]

(-2*a - 3*b*x**2 - 6*c*x**4)/(12*x**6)

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