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

Optimal. Leaf size=19 \[ \frac {x^6}{6 a \left (a+b x^3\right )^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1584, 264} \begin {gather*} \frac {x^6}{6 a \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11/(a*x^2 + b*x^5)^3,x]

[Out]

x^6/(6*a*(a + b*x^3)^2)

Rule 264

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 1584

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*} \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {a+2 b x^3}{6 b^2 \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{11}}{\left (a x^2+b x^5\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^11/(a*x^2 + b*x^5)^3,x]

[Out]

IntegrateAlgebraic[x^11/(a*x^2 + b*x^5)^3, x]

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fricas [B]  time = 0.38, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.18, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.06, size = 31, normalized size = 1.63 \begin {gather*} \frac {a}{6 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {1}{3 \left (b \,x^{3}+a \right ) b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.29, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 5.11, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {a}{6\,b^2}+\frac {x^3}{3\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.40, size = 36, normalized size = 1.89 \begin {gather*} \frac {- a - 2 b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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