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

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log \left (a+b x^8\right )}{8 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {266} \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log \left (a+b x^8\right )}{8 b} \]

[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*} \text {integral}& = \frac {\log \left (a+b x^8\right )}{8 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log \left (a+b x^8\right )}{8 b} \]

[In]

Integrate[x^7/(a + b*x^8),x]

[Out]

Log[a + b*x^8]/(8*b)

Maple [A] (verified)

Time = 3.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 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	\(\frac {\ln \left (b \,x^{8}+a \right )}{8 b}\)	\(14\)
parallelrisch	\(\frac {\ln \left (b \,x^{8}+a \right )}{8 b}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log \left (b x^{8} + a\right )}{8 \, b} \]

[In]

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

[Out]

1/8*log(b*x^8 + a)/b

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log {\left (a + b x^{8} \right )}}{8 b} \]

[In]

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

[Out]

log(a + b*x**8)/(8*b)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log \left (b x^{8} + a\right )}{8 \, b} \]

[In]

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

[Out]

1/8*log(b*x^8 + a)/b

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\log \left ({\left | b x^{8} + a \right |}\right )}{8 \, b} \]

[In]

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

[Out]

1/8*log(abs(b*x^8 + a))/b

Mupad [B] (verification not implemented)

Time = 5.73 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{a+b x^8} \, dx=\frac {\ln \left (b\,x^8+a\right )}{8\,b} \]

[In]

int(x^7/(a + b*x^8),x)

[Out]

log(a + b*x^8)/(8*b)