\(\int \frac {x^{14}}{3+b x^5} \, dx\) [1290]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int \frac {x^{14}}{3+b x^5} \, dx=-\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b}+\frac {9 \log \left (3+b x^5\right )}{5 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {9 \log \left (b x^5+3\right )}{5 b^3}-\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b} \]

[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 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*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {x^2}{3+b x} \, dx,x,x^5\right ) \\ & = \frac {1}{5} \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {x^{14}}{3+b x^5} \, dx=-\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b}+\frac {9 \log \left (3+b x^5\right )}{5 b^3} \]

[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)

Maple [A] (verified)

Time = 4.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83

method	result	size
meijerg	\(\frac {-\frac {b \,x^{5} \left (-b \,x^{5}+6\right )}{10}+\frac {9 \ln \left (1+\frac {b \,x^{5}}{3}\right )}{5}}{b^{3}}\)	\(30\)
parallelrisch	\(\frac {b^{2} x^{10}-6 b \,x^{5}+18 \ln \left (b \,x^{5}+3\right )}{10 b^{3}}\)	\(30\)
norman	\(-\frac {3 x^{5}}{5 b^{2}}+\frac {x^{10}}{10 b}+\frac {9 \ln \left (b \,x^{5}+3\right )}{5 b^{3}}\)	\(31\)
default	\(\frac {\frac {1}{2} b \,x^{10}-3 x^{5}}{5 b^{2}}+\frac {9 \ln \left (b \,x^{5}+3\right )}{5 b^{3}}\)	\(32\)
risch	\(\frac {x^{10}}{10 b}-\frac {3 x^{5}}{5 b^{2}}+\frac {9}{10 b^{3}}+\frac {9 \ln \left (b \,x^{5}+3\right )}{5 b^{3}}\)	\(36\)

[In]

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

[Out]

9/5/b^3*(-1/18*b*x^5*(-b*x^5+6)+ln(1+1/3*b*x^5))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {b^{2} x^{10} - 6 \, b x^{5} + 18 \, \log \left (b x^{5} + 3\right )}{10 \, b^{3}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {x^{10}}{10 b} - \frac {3 x^{5}}{5 b^{2}} + \frac {9 \log {\left (b x^{5} + 3 \right )}}{5 b^{3}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {b x^{10} - 6 \, x^{5}}{10 \, b^{2}} + \frac {9 \, \log \left (b x^{5} + 3\right )}{5 \, b^{3}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {b x^{10} - 6 \, x^{5}}{10 \, b^{2}} + \frac {9 \, \log \left ({\left | b x^{5} + 3 \right |}\right )}{5 \, b^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {18\,\ln \left (b\,x^5+3\right )-6\,b\,x^5+b^2\,x^{10}}{10\,b^3} \]

[In]

int(x^14/(b*x^5 + 3),x)

[Out]

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