3.13.87 \(\int \frac {x^4}{2 b+b x^5} \, dx\) [1287]

Optimal. Leaf size=13 \[ \frac {\log \left (2+x^5\right )}{5 b} \]

[Out]

1/5*ln(x^5+2)/b

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {266} \begin {gather*} \frac {\log \left (x^5+2\right )}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(2*b + b*x^5),x]

[Out]

Log[2 + x^5]/(5*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^4}{2 b+b x^5} \, dx &=\frac {\log \left (2+x^5\right )}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.31 \begin {gather*} \frac {\log \left (2 b+b x^5\right )}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(2*b + b*x^5),x]

[Out]

Log[2*b + b*x^5]/(5*b)

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Maple [A]
time = 0.16, size = 12, normalized size = 0.92

method	result	size
default	\(\frac {\ln \left (x^{5}+2\right )}{5 b}\)	\(12\)
norman	\(\frac {\ln \left (x^{5}+2\right )}{5 b}\)	\(12\)
risch	\(\frac {\ln \left (x^{5}+2\right )}{5 b}\)	\(12\)
meijerg	\(\frac {\ln \left (1+\frac {x^{5}}{2}\right )}{5 b}\)	\(14\)
derivativedivides	\(\frac {\ln \left (b \,x^{5}+2 b \right )}{5 b}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^5+2*b),x,method=_RETURNVERBOSE)

[Out]

1/5*ln(x^5+2)/b

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Maxima [A]
time = 0.30, size = 15, normalized size = 1.15 \begin {gather*} \frac {\log \left (b x^{5} + 2 \, b\right )}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+2*b),x, algorithm="maxima")

[Out]

1/5*log(b*x^5 + 2*b)/b

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Fricas [A]
time = 0.35, size = 11, normalized size = 0.85 \begin {gather*} \frac {\log \left (x^{5} + 2\right )}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+2*b),x, algorithm="fricas")

[Out]

1/5*log(x^5 + 2)/b

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Sympy [A]
time = 0.07, size = 8, normalized size = 0.62 \begin {gather*} \frac {\log {\left (x^{5} + 2 \right )}}{5 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**5+2*b),x)

[Out]

log(x**5 + 2)/(5*b)

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Giac [A]
time = 2.39, size = 16, normalized size = 1.23 \begin {gather*} \frac {\log \left ({\left | b x^{5} + 2 \, b \right |}\right )}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+2*b),x, algorithm="giac")

[Out]

1/5*log(abs(b*x^5 + 2*b))/b

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Mupad [B]
time = 0.04, size = 11, normalized size = 0.85 \begin {gather*} \frac {\ln \left (x^5+2\right )}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(2*b + b*x^5),x)

[Out]

log(x^5 + 2)/(5*b)

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