3.15.52 \(\int \frac {x^7}{a+b x^8} \, dx\) [1452]

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 = {266} \begin {gather*} \frac {\log \left (a+b x^8\right )}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^8),x]

[Out]

Log[a + b*x^8]/(8*b)

Rule 266

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 \begin {gather*} \frac {\log \left (a+b x^8\right )}{8 b} \end {gather*}

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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Maple [A]
time = 0.16, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\ln \left (b \,x^{8}+a \right )}{8 b}\)	\(14\)
default	\(\frac {\ln \left (b \,x^{8}+a \right )}{8 b}\)	\(14\)
norman	\(\frac {\ln \left (b \,x^{8}+a \right )}{8 b}\)	\(14\)
risch	\(\frac {\ln \left (b \,x^{8}+a \right )}{8 b}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (b x^{8} + a\right )}{8 \, b} \end {gather*}

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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Fricas [A]
time = 0.34, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (b x^{8} + a\right )}{8 \, b} \end {gather*}

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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Sympy [A]
time = 0.10, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (a + b x^{8} \right )}}{8 b} \end {gather*}

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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Giac [A]
time = 1.43, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | b x^{8} + a \right |}\right )}{8 \, b} \end {gather*}

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

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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