3.53.98 \(\int \frac {-7-3 x-90 x^2-\log (x)}{125 x^2+75 x^3+1365 x^4+541 x^5+4914 x^6+972 x^7+5832 x^8+(75 x^2+30 x^3+543 x^4+108 x^5+972 x^6) \log (x)+(15 x^2+3 x^3+54 x^4) \log ^2(x)+x^2 \log ^3(x)} \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{x \left (5+x+18 x^2+\log (x)\right )^2} \]

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Rubi [F]  time = 0.72, 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 {-7-3 x-90 x^2-\log (x)}{125 x^2+75 x^3+1365 x^4+541 x^5+4914 x^6+972 x^7+5832 x^8+\left (75 x^2+30 x^3+543 x^4+108 x^5+972 x^6\right ) \log (x)+\left (15 x^2+3 x^3+54 x^4\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7 - 3*x - 90*x^2 - Log[x])/(125*x^2 + 75*x^3 + 1365*x^4 + 541*x^5 + 4914*x^6 + 972*x^7 + 5832*x^8 + (75*
x^2 + 30*x^3 + 543*x^4 + 108*x^5 + 972*x^6)*Log[x] + (15*x^2 + 3*x^3 + 54*x^4)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

-72*Defer[Int][(5 + x + 18*x^2 + Log[x])^(-3), x] - 2*Defer[Int][1/(x^2*(5 + x + 18*x^2 + Log[x])^3), x] - 2*D
efer[Int][1/(x*(5 + x + 18*x^2 + Log[x])^3), x] - Defer[Int][1/(x^2*(5 + x + 18*x^2 + Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-7-3 x-90 x^2-\log (x)}{x^2 \left (5+x+18 x^2+\log (x)\right )^3} \, dx\\ &=\int \left (-\frac {2 \left (1+x+36 x^2\right )}{x^2 \left (5+x+18 x^2+\log (x)\right )^3}-\frac {1}{x^2 \left (5+x+18 x^2+\log (x)\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1+x+36 x^2}{x^2 \left (5+x+18 x^2+\log (x)\right )^3} \, dx\right )-\int \frac {1}{x^2 \left (5+x+18 x^2+\log (x)\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {36}{\left (5+x+18 x^2+\log (x)\right )^3}+\frac {1}{x^2 \left (5+x+18 x^2+\log (x)\right )^3}+\frac {1}{x \left (5+x+18 x^2+\log (x)\right )^3}\right ) \, dx\right )-\int \frac {1}{x^2 \left (5+x+18 x^2+\log (x)\right )^2} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \left (5+x+18 x^2+\log (x)\right )^3} \, dx\right )-2 \int \frac {1}{x \left (5+x+18 x^2+\log (x)\right )^3} \, dx-72 \int \frac {1}{\left (5+x+18 x^2+\log (x)\right )^3} \, dx-\int \frac {1}{x^2 \left (5+x+18 x^2+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.58, size = 16, normalized size = 1.00 \begin {gather*} \frac {1}{x \left (5+x+18 x^2+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7 - 3*x - 90*x^2 - Log[x])/(125*x^2 + 75*x^3 + 1365*x^4 + 541*x^5 + 4914*x^6 + 972*x^7 + 5832*x^8
+ (75*x^2 + 30*x^3 + 543*x^4 + 108*x^5 + 972*x^6)*Log[x] + (15*x^2 + 3*x^3 + 54*x^4)*Log[x]^2 + x^2*Log[x]^3),
x]

[Out]

1/(x*(5 + x + 18*x^2 + Log[x])^2)

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fricas [B]  time = 0.52, size = 48, normalized size = 3.00 \begin {gather*} \frac {1}{324 \, x^{5} + 36 \, x^{4} + 181 \, x^{3} + x \log \relax (x)^{2} + 10 \, x^{2} + 2 \, {\left (18 \, x^{3} + x^{2} + 5 \, x\right )} \log \relax (x) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-90*x^2-3*x-7)/(x^2*log(x)^3+(54*x^4+3*x^3+15*x^2)*log(x)^2+(972*x^6+108*x^5+543*x^4+30*x^3+
75*x^2)*log(x)+5832*x^8+972*x^7+4914*x^6+541*x^5+1365*x^4+75*x^3+125*x^2),x, algorithm="fricas")

[Out]

1/(324*x^5 + 36*x^4 + 181*x^3 + x*log(x)^2 + 10*x^2 + 2*(18*x^3 + x^2 + 5*x)*log(x) + 25*x)

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giac [B]  time = 0.18, size = 101, normalized size = 6.31 \begin {gather*} \frac {36 \, x^{2} + x + 1}{11664 \, x^{7} + 1620 \, x^{6} + 1296 \, x^{5} \log \relax (x) + 6876 \, x^{5} + 108 \, x^{4} \log \relax (x) + 36 \, x^{3} \log \relax (x)^{2} + 577 \, x^{4} + 398 \, x^{3} \log \relax (x) + x^{2} \log \relax (x)^{2} + 1091 \, x^{3} + 12 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + 35 \, x^{2} + 10 \, x \log \relax (x) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-90*x^2-3*x-7)/(x^2*log(x)^3+(54*x^4+3*x^3+15*x^2)*log(x)^2+(972*x^6+108*x^5+543*x^4+30*x^3+
75*x^2)*log(x)+5832*x^8+972*x^7+4914*x^6+541*x^5+1365*x^4+75*x^3+125*x^2),x, algorithm="giac")

[Out]

(36*x^2 + x + 1)/(11664*x^7 + 1620*x^6 + 1296*x^5*log(x) + 6876*x^5 + 108*x^4*log(x) + 36*x^3*log(x)^2 + 577*x
^4 + 398*x^3*log(x) + x^2*log(x)^2 + 1091*x^3 + 12*x^2*log(x) + x*log(x)^2 + 35*x^2 + 10*x*log(x) + 25*x)

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maple [A]  time = 0.02, size = 17, normalized size = 1.06

method	result	size
risch	\(\frac {1}{x \left (5+18 x^{2}+x +\ln \relax (x )\right )^{2}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x)-90*x^2-3*x-7)/(x^2*ln(x)^3+(54*x^4+3*x^3+15*x^2)*ln(x)^2+(972*x^6+108*x^5+543*x^4+30*x^3+75*x^2)*l
n(x)+5832*x^8+972*x^7+4914*x^6+541*x^5+1365*x^4+75*x^3+125*x^2),x,method=_RETURNVERBOSE)

[Out]

1/x/(5+18*x^2+x+ln(x))^2

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maxima [B]  time = 0.42, size = 48, normalized size = 3.00 \begin {gather*} \frac {1}{324 \, x^{5} + 36 \, x^{4} + 181 \, x^{3} + x \log \relax (x)^{2} + 10 \, x^{2} + 2 \, {\left (18 \, x^{3} + x^{2} + 5 \, x\right )} \log \relax (x) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-90*x^2-3*x-7)/(x^2*log(x)^3+(54*x^4+3*x^3+15*x^2)*log(x)^2+(972*x^6+108*x^5+543*x^4+30*x^3+
75*x^2)*log(x)+5832*x^8+972*x^7+4914*x^6+541*x^5+1365*x^4+75*x^3+125*x^2),x, algorithm="maxima")

[Out]

1/(324*x^5 + 36*x^4 + 181*x^3 + x*log(x)^2 + 10*x^2 + 2*(18*x^3 + x^2 + 5*x)*log(x) + 25*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {3\,x+\ln \relax (x)+90\,x^2+7}{x^2\,{\ln \relax (x)}^3+{\ln \relax (x)}^2\,\left (54\,x^4+3\,x^3+15\,x^2\right )+\ln \relax (x)\,\left (972\,x^6+108\,x^5+543\,x^4+30\,x^3+75\,x^2\right )+125\,x^2+75\,x^3+1365\,x^4+541\,x^5+4914\,x^6+972\,x^7+5832\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + log(x) + 90*x^2 + 7)/(x^2*log(x)^3 + log(x)^2*(15*x^2 + 3*x^3 + 54*x^4) + log(x)*(75*x^2 + 30*x^3
+ 543*x^4 + 108*x^5 + 972*x^6) + 125*x^2 + 75*x^3 + 1365*x^4 + 541*x^5 + 4914*x^6 + 972*x^7 + 5832*x^8),x)

[Out]

int(-(3*x + log(x) + 90*x^2 + 7)/(x^2*log(x)^3 + log(x)^2*(15*x^2 + 3*x^3 + 54*x^4) + log(x)*(75*x^2 + 30*x^3
+ 543*x^4 + 108*x^5 + 972*x^6) + 125*x^2 + 75*x^3 + 1365*x^4 + 541*x^5 + 4914*x^6 + 972*x^7 + 5832*x^8), x)

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sympy [B]  time = 0.17, size = 48, normalized size = 3.00 \begin {gather*} \frac {1}{324 x^{5} + 36 x^{4} + 181 x^{3} + 10 x^{2} + x \log {\relax (x )}^{2} + 25 x + \left (36 x^{3} + 2 x^{2} + 10 x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)-90*x**2-3*x-7)/(x**2*ln(x)**3+(54*x**4+3*x**3+15*x**2)*ln(x)**2+(972*x**6+108*x**5+543*x**4+
30*x**3+75*x**2)*ln(x)+5832*x**8+972*x**7+4914*x**6+541*x**5+1365*x**4+75*x**3+125*x**2),x)

[Out]

1/(324*x**5 + 36*x**4 + 181*x**3 + 10*x**2 + x*log(x)**2 + 25*x + (36*x**3 + 2*x**2 + 10*x)*log(x))

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