3.1.79 \(\int \frac {x^5}{(b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac {1}{2 c \left (b+c x^2\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, 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, 261} \begin {gather*} -\frac {1}{2 c \left (b+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(b*x^2 + c*x^4)^2,x]

[Out]

-1/(2*c*(b + c*x^2))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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^5}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {1}{2 c \left (b+c x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{2 c \left (b+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(b*x^2 + c*x^4)^2,x]

[Out]

-1/2*1/(c*(b + c*x^2))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{\left (b x^2+c x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5/(b*x^2 + c*x^4)^2,x]

[Out]

IntegrateAlgebraic[x^5/(b*x^2 + c*x^4)^2, x]

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fricas [A]  time = 0.62, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{2 \, {\left (c^{2} x^{2} + b c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

-1/2/(c^2*x^2 + b*c)

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giac [A]  time = 0.17, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2 \, {\left (c x^{2} + b\right )} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

-1/2/((c*x^2 + b)*c)

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{2 \left (c \,x^{2}+b \right ) c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(c*x^4+b*x^2)^2,x)

[Out]

-1/2/c/(c*x^2+b)

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maxima [A]  time = 1.32, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{2 \, {\left (c^{2} x^{2} + b c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

-1/2/(c^2*x^2 + b*c)

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mupad [B]  time = 0.02, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2\,c\,\left (c\,x^2+b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2 + c*x^4)^2,x)

[Out]

-1/(2*c*(b + c*x^2))

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sympy [A]  time = 0.18, size = 15, normalized size = 0.94 \begin {gather*} - \frac {1}{2 b c + 2 c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**4+b*x**2)**2,x)

[Out]

-1/(2*b*c + 2*c**2*x**2)

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