3.12.71 \(\int \frac {-45-105 x-24 x^2}{36 x^2+105 x^3+88 x^4+16 x^5+(72 x+114 x^2+24 x^3) \log (4+x)+(36+9 x) \log ^2(4+x)} \, dx\)

Optimal. Leaf size=15 \[ \frac {1}{x+\frac {4 x^2}{3}+\log (4+x)} \]

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Rubi [A]  time = 0.21, antiderivative size = 18, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6688, 12, 6686} \begin {gather*} \frac {3}{x (4 x+3)+3 \log (x+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-45 - 105*x - 24*x^2)/(36*x^2 + 105*x^3 + 88*x^4 + 16*x^5 + (72*x + 114*x^2 + 24*x^3)*Log[4 + x] + (36 +
9*x)*Log[4 + x]^2),x]

[Out]

3/(x*(3 + 4*x) + 3*Log[4 + x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-15-35 x-8 x^2\right )}{(4+x) (x (3+4 x)+3 \log (4+x))^2} \, dx\\ &=3 \int \frac {-15-35 x-8 x^2}{(4+x) (x (3+4 x)+3 \log (4+x))^2} \, dx\\ &=\frac {3}{x (3+4 x)+3 \log (4+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.20 \begin {gather*} \frac {3}{x (3+4 x)+3 \log (4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-45 - 105*x - 24*x^2)/(36*x^2 + 105*x^3 + 88*x^4 + 16*x^5 + (72*x + 114*x^2 + 24*x^3)*Log[4 + x] +
(36 + 9*x)*Log[4 + x]^2),x]

[Out]

3/(x*(3 + 4*x) + 3*Log[4 + x])

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fricas [A]  time = 0.59, size = 19, normalized size = 1.27 \begin {gather*} \frac {3}{4 \, x^{2} + 3 \, x + 3 \, \log \left (x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2-105*x-45)/((9*x+36)*log(4+x)^2+(24*x^3+114*x^2+72*x)*log(4+x)+16*x^5+88*x^4+105*x^3+36*x^2)
,x, algorithm="fricas")

[Out]

3/(4*x^2 + 3*x + 3*log(x + 4))

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giac [A]  time = 0.38, size = 19, normalized size = 1.27 \begin {gather*} \frac {3}{4 \, x^{2} + 3 \, x + 3 \, \log \left (x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2-105*x-45)/((9*x+36)*log(4+x)^2+(24*x^3+114*x^2+72*x)*log(4+x)+16*x^5+88*x^4+105*x^3+36*x^2)
,x, algorithm="giac")

[Out]

3/(4*x^2 + 3*x + 3*log(x + 4))

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maple [A]  time = 0.05, size = 20, normalized size = 1.33

method	result	size
norman	\(\frac {3}{4 x^{2}+3 \ln \left (4+x \right )+3 x}\)	\(20\)
risch	\(\frac {3}{4 x^{2}+3 \ln \left (4+x \right )+3 x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-24*x^2-105*x-45)/((9*x+36)*ln(4+x)^2+(24*x^3+114*x^2+72*x)*ln(4+x)+16*x^5+88*x^4+105*x^3+36*x^2),x,metho
d=_RETURNVERBOSE)

[Out]

3/(4*x^2+3*ln(4+x)+3*x)

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maxima [A]  time = 0.44, size = 19, normalized size = 1.27 \begin {gather*} \frac {3}{4 \, x^{2} + 3 \, x + 3 \, \log \left (x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2-105*x-45)/((9*x+36)*log(4+x)^2+(24*x^3+114*x^2+72*x)*log(4+x)+16*x^5+88*x^4+105*x^3+36*x^2)
,x, algorithm="maxima")

[Out]

3/(4*x^2 + 3*x + 3*log(x + 4))

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mupad [B]  time = 1.06, size = 19, normalized size = 1.27 \begin {gather*} \frac {3}{3\,x+3\,\ln \left (x+4\right )+4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(105*x + 24*x^2 + 45)/(log(x + 4)^2*(9*x + 36) + log(x + 4)*(72*x + 114*x^2 + 24*x^3) + 36*x^2 + 105*x^3
+ 88*x^4 + 16*x^5),x)

[Out]

3/(3*x + 3*log(x + 4) + 4*x^2)

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sympy [A]  time = 0.13, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{4 x^{2} + 3 x + 3 \log {\left (x + 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x**2-105*x-45)/((9*x+36)*ln(4+x)**2+(24*x**3+114*x**2+72*x)*ln(4+x)+16*x**5+88*x**4+105*x**3+36
*x**2),x)

[Out]

3/(4*x**2 + 3*x + 3*log(x + 4))

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