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

Optimal. Leaf size=17 \[ -\frac{a}{6 x^6}-\frac{b}{4 x^4} \]

[Out]

-a/(6*x^6) - b/(4*x^4)

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Rubi [A]  time = 0.0047639, antiderivative size = 17, normalized size of antiderivative = 1., 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} \[ -\frac{a}{6 x^6}-\frac{b}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^7,x]

[Out]

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

Mathematica [A]  time = 0.002024, size = 17, normalized size = 1. \[ -\frac{a}{6 x^6}-\frac{b}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^7,x]

[Out]

-a/(6*x^6) - b/(4*x^4)

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Maple [A]  time = 0.004, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.60667, size = 20, normalized size = 1.18 \begin{align*} -\frac{3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b*x^2 + 2*a)/x^6

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Fricas [A]  time = 1.40074, size = 36, normalized size = 2.12 \begin{align*} -\frac{3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b*x^2 + 2*a)/x^6

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Sympy [A]  time = 0.271296, size = 15, normalized size = 0.88 \begin{align*} - \frac{2 a + 3 b x^{2}}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a + 3*b*x**2)/(12*x**6)

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Giac [A]  time = 2.63978, size = 20, normalized size = 1.18 \begin{align*} -\frac{3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b*x^2 + 2*a)/x^6