3.1452 \(\int \frac {x^7}{a+b x^8} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log \left (a+b x^8\right )}{8 b} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {260} \[ \frac {\log \left (a+b x^8\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^8),x]

[Out]

Log[a + b*x^8]/(8*b)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (a+b x^8\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^8),x]

[Out]

Log[a + b*x^8]/(8*b)

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fricas [A]  time = 0.55, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{8} + a\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(b*x^8 + a)/b

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giac [A]  time = 0.18, size = 14, normalized size = 0.93 \[ \frac {\log \left ({\left | b x^{8} + a \right |}\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(abs(b*x^8 + a))/b

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {\ln \left (b \,x^{8}+a \right )}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{8} + a\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(b*x^8 + a)/b

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mupad [B]  time = 0.04, size = 13, normalized size = 0.87 \[ \frac {\ln \left (b\,x^8+a\right )}{8\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^8),x)

[Out]

log(a + b*x^8)/(8*b)

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sympy [A]  time = 0.40, size = 10, normalized size = 0.67 \[ \frac {\log {\left (a + b x^{8} \right )}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x**8)/(8*b)

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