3.6.30 \(\int \frac {-32-68 x-66 x^2}{(16 x+17 x^2+11 x^3) \log ^2(-32 x-34 x^2-22 x^3)} \, dx\)

Optimal. Leaf size=33 \[ \frac {2}{\log \left (4 x \left ((4-x) x+\frac {1}{2} \left (-x-(4+3 x)^2\right )\right )\right )} \]

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Rubi [A]  time = 0.14, antiderivative size = 19, normalized size of antiderivative = 0.58, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1594, 6686} \begin {gather*} \frac {2}{\log \left (-22 x^3-34 x^2-32 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32 - 68*x - 66*x^2)/((16*x + 17*x^2 + 11*x^3)*Log[-32*x - 34*x^2 - 22*x^3]^2),x]

[Out]

2/Log[-32*x - 34*x^2 - 22*x^3]

Rule 1594

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32-68 x-66 x^2}{x \left (16+17 x+11 x^2\right ) \log ^2\left (-32 x-34 x^2-22 x^3\right )} \, dx\\ &=\frac {2}{\log \left (-32 x-34 x^2-22 x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.55 \begin {gather*} \frac {2}{\log \left (-2 x \left (16+17 x+11 x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 - 68*x - 66*x^2)/((16*x + 17*x^2 + 11*x^3)*Log[-32*x - 34*x^2 - 22*x^3]^2),x]

[Out]

2/Log[-2*x*(16 + 17*x + 11*x^2)]

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fricas [A]  time = 0.51, size = 19, normalized size = 0.58 \begin {gather*} \frac {2}{\log \left (-22 \, x^{3} - 34 \, x^{2} - 32 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-66*x^2-68*x-32)/(11*x^3+17*x^2+16*x)/log(-22*x^3-34*x^2-32*x)^2,x, algorithm="fricas")

[Out]

2/log(-22*x^3 - 34*x^2 - 32*x)

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giac [A]  time = 0.44, size = 19, normalized size = 0.58 \begin {gather*} \frac {2}{\log \left (-22 \, x^{3} - 34 \, x^{2} - 32 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-66*x^2-68*x-32)/(11*x^3+17*x^2+16*x)/log(-22*x^3-34*x^2-32*x)^2,x, algorithm="giac")

[Out]

2/log(-22*x^3 - 34*x^2 - 32*x)

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maple [A]  time = 0.04, size = 20, normalized size = 0.61

method	result	size
norman	\(\frac {2}{\ln \left (-22 x^{3}-34 x^{2}-32 x \right )}\)	\(20\)
risch	\(\frac {2}{\ln \left (-22 x^{3}-34 x^{2}-32 x \right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-66*x^2-68*x-32)/(11*x^3+17*x^2+16*x)/ln(-22*x^3-34*x^2-32*x)^2,x,method=_RETURNVERBOSE)

[Out]

2/ln(-22*x^3-34*x^2-32*x)

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maxima [C]  time = 0.70, size = 23, normalized size = 0.70 \begin {gather*} \frac {2}{i \, \pi + \log \relax (2) + \log \left (11 \, x^{2} + 17 \, x + 16\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-66*x^2-68*x-32)/(11*x^3+17*x^2+16*x)/log(-22*x^3-34*x^2-32*x)^2,x, algorithm="maxima")

[Out]

2/(I*pi + log(2) + log(11*x^2 + 17*x + 16) + log(x))

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mupad [B]  time = 0.76, size = 19, normalized size = 0.58 \begin {gather*} \frac {2}{\ln \left (-22\,x^3-34\,x^2-32\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(68*x + 66*x^2 + 32)/(log(- 32*x - 34*x^2 - 22*x^3)^2*(16*x + 17*x^2 + 11*x^3)),x)

[Out]

2/log(- 32*x - 34*x^2 - 22*x^3)

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sympy [A]  time = 0.21, size = 17, normalized size = 0.52 \begin {gather*} \frac {2}{\log {\left (- 22 x^{3} - 34 x^{2} - 32 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-66*x**2-68*x-32)/(11*x**3+17*x**2+16*x)/ln(-22*x**3-34*x**2-32*x)**2,x)

[Out]

2/log(-22*x**3 - 34*x**2 - 32*x)

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