3.1292 \(\int \frac{x^4}{3+b x^5} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (b x^5+3\right )}{5 b} \]

[Out]

Log[3 + b*x^5]/(5*b)

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Rubi [A]  time = 0.0033289, antiderivative size = 15, normalized size of antiderivative = 1., 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} \[ \frac{\log \left (b x^5+3\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(3 + b*x^5),x]

[Out]

Log[3 + b*x^5]/(5*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^4}{3+b x^5} \, dx &=\frac{\log \left (3+b x^5\right )}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0024207, size = 15, normalized size = 1. \[ \frac{\log \left (b x^5+3\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(3 + b*x^5),x]

[Out]

Log[3 + b*x^5]/(5*b)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( b{x}^{5}+3 \right ) }{5\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^5+3),x)

[Out]

1/5*ln(b*x^5+3)/b

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Maxima [A]  time = 0.998187, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (b x^{5} + 3\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+3),x, algorithm="maxima")

[Out]

1/5*log(b*x^5 + 3)/b

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Fricas [A]  time = 1.65299, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (b x^{5} + 3\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+3),x, algorithm="fricas")

[Out]

1/5*log(b*x^5 + 3)/b

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Sympy [A]  time = 0.181443, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (b x^{5} + 3 \right )}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**5+3),x)

[Out]

log(b*x**5 + 3)/(5*b)

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Giac [A]  time = 1.37958, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | b x^{5} + 3 \right |}\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+3),x, algorithm="giac")

[Out]

1/5*log(abs(b*x^5 + 3))/b