3.85.15 \(\int \frac {-2 x^4+3 x^5+4 x^6+(5 x^4+12 x^5) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx\)

Optimal. Leaf size=14 \[ x^4 \left (1+\frac {x}{(x+\log (x))^2}\right ) \]

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Rubi [F]  time = 0.46, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^4 + 3*x^5 + 4*x^6 + (5*x^4 + 12*x^5)*Log[x] + 12*x^4*Log[x]^2 + 4*x^3*Log[x]^3)/(x^3 + 3*x^2*Log[x]
+ 3*x*Log[x]^2 + Log[x]^3),x]

[Out]

x^4 - 2*Defer[Int][x^4/(x + Log[x])^3, x] - 2*Defer[Int][x^5/(x + Log[x])^3, x] + 5*Defer[Int][x^4/(x + Log[x]
)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (x \left (-2+3 x+4 x^2\right )+x (5+12 x) \log (x)+12 x \log ^2(x)+4 \log ^3(x)\right )}{(x+\log (x))^3} \, dx\\ &=\int \left (4 x^3-\frac {2 x^4 (1+x)}{(x+\log (x))^3}+\frac {5 x^4}{(x+\log (x))^2}\right ) \, dx\\ &=x^4-2 \int \frac {x^4 (1+x)}{(x+\log (x))^3} \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx\\ &=x^4-2 \int \left (\frac {x^4}{(x+\log (x))^3}+\frac {x^5}{(x+\log (x))^3}\right ) \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx\\ &=x^4-2 \int \frac {x^4}{(x+\log (x))^3} \, dx-2 \int \frac {x^5}{(x+\log (x))^3} \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 14, normalized size = 1.00 \begin {gather*} x^4+\frac {x^5}{(x+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^4 + 3*x^5 + 4*x^6 + (5*x^4 + 12*x^5)*Log[x] + 12*x^4*Log[x]^2 + 4*x^3*Log[x]^3)/(x^3 + 3*x^2*L
og[x] + 3*x*Log[x]^2 + Log[x]^3),x]

[Out]

x^4 + x^5/(x + Log[x])^2

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fricas [B]  time = 0.91, size = 38, normalized size = 2.71 \begin {gather*} \frac {x^{6} + 2 \, x^{5} \log \relax (x) + x^{4} \log \relax (x)^{2} + x^{5}}{x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)^3+12*x^4*log(x)^2+(12*x^5+5*x^4)*log(x)+4*x^6+3*x^5-2*x^4)/(log(x)^3+3*x*log(x)^2+3*x^
2*log(x)+x^3),x, algorithm="fricas")

[Out]

(x^6 + 2*x^5*log(x) + x^4*log(x)^2 + x^5)/(x^2 + 2*x*log(x) + log(x)^2)

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giac [B]  time = 0.21, size = 43, normalized size = 3.07 \begin {gather*} x^{4} + \frac {x^{6} + x^{5}}{x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)^3+12*x^4*log(x)^2+(12*x^5+5*x^4)*log(x)+4*x^6+3*x^5-2*x^4)/(log(x)^3+3*x*log(x)^2+3*x^
2*log(x)+x^3),x, algorithm="giac")

[Out]

x^4 + (x^6 + x^5)/(x^3 + 2*x^2*log(x) + x*log(x)^2 + x^2 + 2*x*log(x) + log(x)^2)

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maple [A]  time = 0.03, size = 15, normalized size = 1.07

method	result	size
risch	\(x^{4}+\frac {x^{5}}{\left (x +\ln \relax (x )\right )^{2}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*ln(x)^3+12*x^4*ln(x)^2+(12*x^5+5*x^4)*ln(x)+4*x^6+3*x^5-2*x^4)/(ln(x)^3+3*x*ln(x)^2+3*x^2*ln(x)+x^3
),x,method=_RETURNVERBOSE)

[Out]

x^4+x^5/(x+ln(x))^2

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maxima [B]  time = 0.39, size = 38, normalized size = 2.71 \begin {gather*} \frac {x^{6} + 2 \, x^{5} \log \relax (x) + x^{4} \log \relax (x)^{2} + x^{5}}{x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)^3+12*x^4*log(x)^2+(12*x^5+5*x^4)*log(x)+4*x^6+3*x^5-2*x^4)/(log(x)^3+3*x*log(x)^2+3*x^
2*log(x)+x^3),x, algorithm="maxima")

[Out]

(x^6 + 2*x^5*log(x) + x^4*log(x)^2 + x^5)/(x^2 + 2*x*log(x) + log(x)^2)

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mupad [B]  time = 5.32, size = 26, normalized size = 1.86 \begin {gather*} x^4-\frac {x^6-x^4\,\left (x^2+x\right )}{{\left (x+\ln \relax (x)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(5*x^4 + 12*x^5) + 4*x^3*log(x)^3 + 12*x^4*log(x)^2 - 2*x^4 + 3*x^5 + 4*x^6)/(3*x*log(x)^2 + 3*x^2
*log(x) + log(x)^3 + x^3),x)

[Out]

x^4 - (x^6 - x^4*(x + x^2))/(x + log(x))^2

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sympy [A]  time = 0.11, size = 20, normalized size = 1.43 \begin {gather*} \frac {x^{5}}{x^{2} + 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3*ln(x)**3+12*x**4*ln(x)**2+(12*x**5+5*x**4)*ln(x)+4*x**6+3*x**5-2*x**4)/(ln(x)**3+3*x*ln(x)**
2+3*x**2*ln(x)+x**3),x)

[Out]

x**5/(x**2 + 2*x*log(x) + log(x)**2) + x**4

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