3.1286 \(\int \frac{x^9}{2 b+b x^5} \, dx\)

Optimal. Leaf size=24 \[ \frac{x^5}{5 b}-\frac{2 \log \left (x^5+2\right )}{5 b} \]

[Out]

x^5/(5*b) - (2*Log[2 + x^5])/(5*b)

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Rubi [A]  time = 0.0138791, antiderivative size = 24, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{x^5}{5 b}-\frac{2 \log \left (x^5+2\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(2*b + b*x^5),x]

[Out]

x^5/(5*b) - (2*Log[2 + x^5])/(5*b)

Rule 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{x^9}{2 b+b x^5} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x}{2 b+b x} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{2}{b (2+x)}\right ) \, dx,x,x^5\right )\\ &=\frac{x^5}{5 b}-\frac{2 \log \left (2+x^5\right )}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0032358, size = 24, normalized size = 1. \[ \frac{x^5}{5 b}-\frac{2 \log \left (x^5+2\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(2*b + b*x^5),x]

[Out]

x^5/(5*b) - (2*Log[2 + x^5])/(5*b)

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Maple [A]  time = 0., size = 21, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,b}}-{\frac{2\,\ln \left ({x}^{5}+2 \right ) }{5\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b*x^5+2*b),x)

[Out]

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

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Maxima [A]  time = 1.02867, size = 27, normalized size = 1.12 \begin{align*} \frac{x^{5}}{5 \, b} - \frac{2 \, \log \left (x^{5} + 2\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^5+2*b),x, algorithm="maxima")

[Out]

1/5*x^5/b - 2/5*log(x^5 + 2)/b

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Fricas [A]  time = 1.92924, size = 41, normalized size = 1.71 \begin{align*} \frac{x^{5} - 2 \, \log \left (x^{5} + 2\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^5+2*b),x, algorithm="fricas")

[Out]

1/5*(x^5 - 2*log(x^5 + 2))/b

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Sympy [A]  time = 0.238175, size = 17, normalized size = 0.71 \begin{align*} \frac{x^{5}}{5 b} - \frac{2 \log{\left (x^{5} + 2 \right )}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x**5+2*b),x)

[Out]

x**5/(5*b) - 2*log(x**5 + 2)/(5*b)

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Giac [A]  time = 1.20889, size = 28, normalized size = 1.17 \begin{align*} \frac{x^{5}}{5 \, b} - \frac{2 \, \log \left ({\left | x^{5} + 2 \right |}\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^5+2*b),x, algorithm="giac")

[Out]

1/5*x^5/b - 2/5*log(abs(x^5 + 2))/b