3.13.85 \(\int \frac {x^{14}}{2 b+b x^5} \, dx\) [1285]

Optimal. Leaf size=34 \[ -\frac {2 x^5}{5 b}+\frac {x^{10}}{10 b}+\frac {4 \log \left (2+x^5\right )}{5 b} \]

[Out]

-2/5*x^5/b+1/10*x^10/b+4/5*ln(x^5+2)/b

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} \frac {x^{10}}{10 b}-\frac {2 x^5}{5 b}+\frac {4 \log \left (x^5+2\right )}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^14/(2*b + b*x^5),x]

[Out]

(-2*x^5)/(5*b) + x^10/(10*b) + (4*Log[2 + x^5])/(5*b)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{14}}{2 b+b x^5} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {x^2}{2 b+b x} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \left (-\frac {2}{b}+\frac {x}{b}+\frac {4}{b (2+x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {2 x^5}{5 b}+\frac {x^{10}}{10 b}+\frac {4 \log \left (2+x^5\right )}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.74 \begin {gather*} \frac {-12-4 x^5+x^{10}+8 \log \left (2+x^5\right )}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^14/(2*b + b*x^5),x]

[Out]

(-12 - 4*x^5 + x^10 + 8*Log[2 + x^5])/(10*b)

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Maple [A]
time = 0.18, size = 24, normalized size = 0.71

method	result	size
default	\(\frac {\frac {x^{10}}{10}-\frac {2 x^{5}}{5}+\frac {4 \ln \left (x^{5}+2\right )}{5}}{b}\)	\(24\)
meijerg	\(\frac {-\frac {x^{5} \left (-\frac {3 x^{5}}{2}+6\right )}{15}+\frac {4 \ln \left (1+\frac {x^{5}}{2}\right )}{5}}{b}\)	\(27\)
norman	\(-\frac {2 x^{5}}{5 b}+\frac {x^{10}}{10 b}+\frac {4 \ln \left (x^{5}+2\right )}{5 b}\)	\(29\)
risch	\(\frac {x^{10}}{10 b}-\frac {2 x^{5}}{5 b}+\frac {2}{5 b}+\frac {4 \ln \left (x^{5}+2\right )}{5 b}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(b*x^5+2*b),x,method=_RETURNVERBOSE)

[Out]

1/b*(1/10*x^10-2/5*x^5+4/5*ln(x^5+2))

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Maxima [A]
time = 0.29, size = 26, normalized size = 0.76 \begin {gather*} \frac {x^{10} - 4 \, x^{5}}{10 \, b} + \frac {4 \, \log \left (x^{5} + 2\right )}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^5+2*b),x, algorithm="maxima")

[Out]

1/10*(x^10 - 4*x^5)/b + 4/5*log(x^5 + 2)/b

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Fricas [A]
time = 0.37, size = 22, normalized size = 0.65 \begin {gather*} \frac {x^{10} - 4 \, x^{5} + 8 \, \log \left (x^{5} + 2\right )}{10 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^5+2*b),x, algorithm="fricas")

[Out]

1/10*(x^10 - 4*x^5 + 8*log(x^5 + 2))/b

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Sympy [A]
time = 0.09, size = 26, normalized size = 0.76 \begin {gather*} \frac {x^{10}}{10 b} - \frac {2 x^{5}}{5 b} + \frac {4 \log {\left (x^{5} + 2 \right )}}{5 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(b*x**5+2*b),x)

[Out]

x**10/(10*b) - 2*x**5/(5*b) + 4*log(x**5 + 2)/(5*b)

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Giac [A]
time = 2.08, size = 30, normalized size = 0.88 \begin {gather*} \frac {4 \, \log \left ({\left | x^{5} + 2 \right |}\right )}{5 \, b} + \frac {b x^{10} - 4 \, b x^{5}}{10 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^5+2*b),x, algorithm="giac")

[Out]

4/5*log(abs(x^5 + 2))/b + 1/10*(b*x^10 - 4*b*x^5)/b^2

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Mupad [B]
time = 0.06, size = 22, normalized size = 0.65 \begin {gather*} \frac {8\,\ln \left (x^5+2\right )-4\,x^5+x^{10}}{10\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(2*b + b*x^5),x)

[Out]

(8*log(x^5 + 2) - 4*x^5 + x^10)/(10*b)

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