3.1271 \(\int \frac {x^4}{a+b x^5} \, dx\)

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

[Out]

1/5*ln(b*x^5+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} \[ \frac {\log \left (a+b x^5\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^5),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (a+b x^5\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^5),x]

[Out]

Log[a + b*x^5]/(5*b)

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fricas [A]  time = 0.71, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{5} + a\right )}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*log(b*x^5 + a)/b

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giac [A]  time = 0.17, size = 14, normalized size = 0.93 \[ \frac {\log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*log(abs(b*x^5 + a))/b

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {\ln \left (b \,x^{5}+a \right )}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.11, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{5} + a\right )}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*log(b*x^5 + a)/b

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mupad [B]  time = 0.03, size = 13, normalized size = 0.87 \[ \frac {\ln \left (b\,x^5+a\right )}{5\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x^5),x)

[Out]

log(a + b*x^5)/(5*b)

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sympy [A]  time = 0.29, size = 10, normalized size = 0.67 \[ \frac {\log {\left (a + b x^{5} \right )}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x**5)/(5*b)

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