\(\int \frac {x^{11}}{(a x^2+b x^5)^3} \, dx\) [283]

Optimal result

Integrand size = 17, antiderivative size = 19 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=\frac {x^6}{6 a \left (a+b x^3\right )^2} \]

[Out]

1/6*x^6/a/(b*x^3+a)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 270} \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=\frac {x^6}{6 a \left (a+b x^3\right )^2} \]

[In]

Int[x^11/(a*x^2 + b*x^5)^3,x]

[Out]

x^6/(6*a*(a + b*x^3)^2)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5}{\left (a+b x^3\right )^3} \, dx \\ & = \frac {x^6}{6 a \left (a+b x^3\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=-\frac {a+2 b x^3}{6 b^2 \left (a+b x^3\right )^2} \]

[In]

Integrate[x^11/(a*x^2 + b*x^5)^3,x]

[Out]

-1/6*(a + 2*b*x^3)/(b^2*(a + b*x^3)^2)

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21

method	result	size
gosper	\(-\frac {2 b \,x^{3}+a}{6 \left (b \,x^{3}+a \right )^{2} b^{2}}\)	\(23\)
parallelrisch	\(\frac {-2 b \,x^{3}-a}{6 b^{2} \left (b \,x^{3}+a \right )^{2}}\)	\(25\)
risch	\(\frac {-\frac {x^{3}}{3 b}-\frac {a}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}\)	\(26\)
default	\(\frac {a}{6 b^{2} \left (b \,x^{3}+a \right )^{2}}-\frac {1}{3 b^{2} \left (b \,x^{3}+a \right )}\)	\(31\)
norman	\(\frac {-\frac {x^{8}}{3 b}-\frac {a \,x^{5}}{6 b^{2}}}{x^{5} \left (b \,x^{3}+a \right )^{2}}\)	\(32\)

[In]

int(x^11/(b*x^5+a*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/6*(2*b*x^3+a)/(b*x^3+a)^2/b^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=-\frac {2 \, b x^{3} + a}{6 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]

[In]

integrate(x^11/(b*x^5+a*x^2)^3,x, algorithm="fricas")

[Out]

-1/6*(2*b*x^3 + a)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=\frac {- a - 2 b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \]

[In]

integrate(x**11/(b*x**5+a*x**2)**3,x)

[Out]

(-a - 2*b*x**3)/(6*a**2*b**2 + 12*a*b**3*x**3 + 6*b**4*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=-\frac {2 \, b x^{3} + a}{6 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]

[In]

integrate(x^11/(b*x^5+a*x^2)^3,x, algorithm="maxima")

[Out]

-1/6*(2*b*x^3 + a)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=-\frac {2 \, b x^{3} + a}{6 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \]

[In]

integrate(x^11/(b*x^5+a*x^2)^3,x, algorithm="giac")

[Out]

-1/6*(2*b*x^3 + a)/((b*x^3 + a)^2*b^2)

Mupad [B] (verification not implemented)

Time = 8.94 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx=-\frac {\frac {a}{6\,b^2}+\frac {x^3}{3\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \]

[In]

int(x^11/(a*x^2 + b*x^5)^3,x)

[Out]

-(a/(6*b^2) + x^3/(3*b))/(a^2 + b^2*x^6 + 2*a*b*x^3)