3.15.60 \(\int \frac {1}{x^9 (a+b x^8)} \, dx\) [1460]

Optimal. Leaf size=35 \[ -\frac {1}{8 a x^8}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^8\right )}{8 a^2} \]

[Out]

-1/8/a/x^8-b*ln(x)/a^2+1/8*b*ln(b*x^8+a)/a^2

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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46} \begin {gather*} \frac {b \log \left (a+b x^8\right )}{8 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{8 a x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^8)),x]

[Out]

-1/8*1/(a*x^8) - (b*Log[x])/a^2 + (b*Log[a + b*x^8])/(8*a^2)

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x^9 \left (a+b x^8\right )} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,x^8\right )\\ &=\frac {1}{8} \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,x^8\right )\\ &=-\frac {1}{8 a x^8}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^8\right )}{8 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{8 a x^8}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^8\right )}{8 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^8)),x]

[Out]

-1/8*1/(a*x^8) - (b*Log[x])/a^2 + (b*Log[a + b*x^8])/(8*a^2)

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Maple [A]
time = 0.19, size = 32, normalized size = 0.91

method	result	size
default	\(-\frac {1}{8 a \,x^{8}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{8}+a \right )}{8 a^{2}}\)	\(32\)
norman	\(-\frac {1}{8 a \,x^{8}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{8}+a \right )}{8 a^{2}}\)	\(32\)
risch	\(-\frac {1}{8 a \,x^{8}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (-b \,x^{8}-a \right )}{8 a^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^8+a),x,method=_RETURNVERBOSE)

[Out]

-1/8/a/x^8-b*ln(x)/a^2+1/8*b*ln(b*x^8+a)/a^2

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Maxima [A]
time = 0.29, size = 33, normalized size = 0.94 \begin {gather*} \frac {b \log \left (b x^{8} + a\right )}{8 \, a^{2}} - \frac {b \log \left (x^{8}\right )}{8 \, a^{2}} - \frac {1}{8 \, a x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a),x, algorithm="maxima")

[Out]

1/8*b*log(b*x^8 + a)/a^2 - 1/8*b*log(x^8)/a^2 - 1/8/(a*x^8)

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Fricas [A]
time = 0.37, size = 33, normalized size = 0.94 \begin {gather*} \frac {b x^{8} \log \left (b x^{8} + a\right ) - 8 \, b x^{8} \log \left (x\right ) - a}{8 \, a^{2} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a),x, algorithm="fricas")

[Out]

1/8*(b*x^8*log(b*x^8 + a) - 8*b*x^8*log(x) - a)/(a^2*x^8)

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Sympy [A]
time = 0.29, size = 31, normalized size = 0.89 \begin {gather*} - \frac {1}{8 a x^{8}} - \frac {b \log {\left (x \right )}}{a^{2}} + \frac {b \log {\left (\frac {a}{b} + x^{8} \right )}}{8 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**8+a),x)

[Out]

-1/(8*a*x**8) - b*log(x)/a**2 + b*log(a/b + x**8)/(8*a**2)

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Giac [A]
time = 1.27, size = 43, normalized size = 1.23 \begin {gather*} -\frac {b \log \left (x^{8}\right )}{8 \, a^{2}} + \frac {b \log \left ({\left | b x^{8} + a \right |}\right )}{8 \, a^{2}} + \frac {b x^{8} - a}{8 \, a^{2} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a),x, algorithm="giac")

[Out]

-1/8*b*log(x^8)/a^2 + 1/8*b*log(abs(b*x^8 + a))/a^2 + 1/8*(b*x^8 - a)/(a^2*x^8)

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Mupad [B]
time = 0.08, size = 31, normalized size = 0.89 \begin {gather*} \frac {b\,\ln \left (b\,x^8+a\right )}{8\,a^2}-\frac {1}{8\,a\,x^8}-\frac {b\,\ln \left (x\right )}{a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^8)),x)

[Out]

(b*log(a + b*x^8))/(8*a^2) - 1/(8*a*x^8) - (b*log(x))/a^2

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