3.1437 \(\int \frac{a+b x^7}{x^8} \, dx\)

Optimal. Leaf size=13 \[ b \log (x)-\frac{a}{7 x^7} \]

[Out]

-a/(7*x^7) + b*Log[x]

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Rubi [A]  time = 0.0045949, antiderivative size = 13, 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} \[ b \log (x)-\frac{a}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^7)/x^8,x]

[Out]

-a/(7*x^7) + b*Log[x]

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

Mathematica [A]  time = 0.0022285, size = 13, normalized size = 1. \[ b \log (x)-\frac{a}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^7)/x^8,x]

[Out]

-a/(7*x^7) + b*Log[x]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.9 \begin{align*} -{\frac{a}{7\,{x}^{7}}}+b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^7+a)/x^8,x)

[Out]

-1/7*a/x^7+b*ln(x)

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Maxima [A]  time = 1.05293, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{7} \, b \log \left (x^{7}\right ) - \frac{a}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^7+a)/x^8,x, algorithm="maxima")

[Out]

1/7*b*log(x^7) - 1/7*a/x^7

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Fricas [A]  time = 1.72718, size = 41, normalized size = 3.15 \begin{align*} \frac{7 \, b x^{7} \log \left (x\right ) - a}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^7+a)/x^8,x, algorithm="fricas")

[Out]

1/7*(7*b*x^7*log(x) - a)/x^7

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Sympy [A]  time = 0.280122, size = 10, normalized size = 0.77 \begin{align*} - \frac{a}{7 x^{7}} + b \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**7+a)/x**8,x)

[Out]

-a/(7*x**7) + b*log(x)

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Giac [A]  time = 1.1302, size = 24, normalized size = 1.85 \begin{align*} b \log \left ({\left | x \right |}\right ) - \frac{b x^{7} + a}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^7+a)/x^8,x, algorithm="giac")

[Out]

b*log(abs(x)) - 1/7*(b*x^7 + a)/x^7