3.1290 \(\int \frac{x^{14}}{3+b x^5} \, dx\)

Optimal. Leaf size=36 \[ -\frac{3 x^5}{5 b^2}+\frac{9 \log \left (b x^5+3\right )}{5 b^3}+\frac{x^{10}}{10 b} \]

[Out]

(-3*x^5)/(5*b^2) + x^10/(10*b) + (9*Log[3 + b*x^5])/(5*b^3)

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Rubi [A]  time = 0.0245118, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{3 x^5}{5 b^2}+\frac{9 \log \left (b x^5+3\right )}{5 b^3}+\frac{x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

Int[x^14/(3 + b*x^5),x]

[Out]

(-3*x^5)/(5*b^2) + x^10/(10*b) + (9*Log[3 + b*x^5])/(5*b^3)

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^{14}}{3+b x^5} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^2}{3+b x} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (-\frac{3}{b^2}+\frac{x}{b}+\frac{9}{b^2 (3+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{3 x^5}{5 b^2}+\frac{x^{10}}{10 b}+\frac{9 \log \left (3+b x^5\right )}{5 b^3}\\ \end{align*}

Mathematica [A]  time = 0.005524, size = 36, normalized size = 1. \[ -\frac{3 x^5}{5 b^2}+\frac{9 \log \left (b x^5+3\right )}{5 b^3}+\frac{x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/(3 + b*x^5),x]

[Out]

(-3*x^5)/(5*b^2) + x^10/(10*b) + (9*Log[3 + b*x^5])/(5*b^3)

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Maple [A]  time = 0.002, size = 31, normalized size = 0.9 \begin{align*} -{\frac{3\,{x}^{5}}{5\,{b}^{2}}}+{\frac{{x}^{10}}{10\,b}}+{\frac{9\,\ln \left ( b{x}^{5}+3 \right ) }{5\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(b*x^5+3),x)

[Out]

-3/5*x^5/b^2+1/10*x^10/b+9/5*ln(b*x^5+3)/b^3

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Maxima [A]  time = 1.00468, size = 41, normalized size = 1.14 \begin{align*} \frac{b x^{10} - 6 \, x^{5}}{10 \, b^{2}} + \frac{9 \, \log \left (b x^{5} + 3\right )}{5 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^5+3),x, algorithm="maxima")

[Out]

1/10*(b*x^10 - 6*x^5)/b^2 + 9/5*log(b*x^5 + 3)/b^3

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Fricas [A]  time = 1.72883, size = 69, normalized size = 1.92 \begin{align*} \frac{b^{2} x^{10} - 6 \, b x^{5} + 18 \, \log \left (b x^{5} + 3\right )}{10 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^5+3),x, algorithm="fricas")

[Out]

1/10*(b^2*x^10 - 6*b*x^5 + 18*log(b*x^5 + 3))/b^3

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Sympy [A]  time = 0.347588, size = 31, normalized size = 0.86 \begin{align*} \frac{x^{10}}{10 b} - \frac{3 x^{5}}{5 b^{2}} + \frac{9 \log{\left (b x^{5} + 3 \right )}}{5 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(b*x**5+3),x)

[Out]

x**10/(10*b) - 3*x**5/(5*b**2) + 9*log(b*x**5 + 3)/(5*b**3)

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Giac [A]  time = 1.37874, size = 42, normalized size = 1.17 \begin{align*} \frac{b x^{10} - 6 \, x^{5}}{10 \, b^{2}} + \frac{9 \, \log \left ({\left | b x^{5} + 3 \right |}\right )}{5 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^5+3),x, algorithm="giac")

[Out]

1/10*(b*x^10 - 6*x^5)/b^2 + 9/5*log(abs(b*x^5 + 3))/b^3