3.13.92 \(\int \frac {x^4}{3+b x^5} \, dx\) [1292]

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

[Out]

1/5*ln(b*x^5+3)/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 (b x^5+3\right )}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(3 + b*x^5),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (3+b x^5\right )}{5 b} \end {gather*}

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 = 0.16, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\ln \left (b \,x^{5}+3\right )}{5 b}\)	\(14\)
default	\(\frac {\ln \left (b \,x^{5}+3\right )}{5 b}\)	\(14\)
norman	\(\frac {\ln \left (b \,x^{5}+3\right )}{5 b}\)	\(14\)
risch	\(\frac {\ln \left (b \,x^{5}+3\right )}{5 b}\)	\(14\)
meijerg	\(\frac {\ln \left (1+\frac {b \,x^{5}}{3}\right )}{5 b}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (b x^{5} + 3\right )}{5 \, b} \end {gather*}

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 = 0.36, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (b x^{5} + 3\right )}{5 \, b} \end {gather*}

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 = 0.06, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (b x^{5} + 3 \right )}}{5 b} \end {gather*}

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 = 1.94, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | b x^{5} + 3 \right |}\right )}{5 \, b} \end {gather*}

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

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Mupad [B]
time = 1.02, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (b\,x^5+3\right )}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^5 + 3),x)

[Out]

log(b*x^5 + 3)/(5*b)

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