3.50.21 \(\int \frac {48 x+116 x^2+58 x^3-14 x^4+(-48 x-84 x^2-47 x^3-7 x^4) \log (x)}{(-192-720 x-756 x^2-15 x^3+189 x^4-45 x^5+3 x^6) \log ^3(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {2+\frac {7 x}{3}}{\left (-5-\frac {4}{x}+x\right )^2 \log ^2(x)} \]

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Rubi [F]  time = 0.49, 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 {48 x+116 x^2+58 x^3-14 x^4+\left (-48 x-84 x^2-47 x^3-7 x^4\right ) \log (x)}{\left (-192-720 x-756 x^2-15 x^3+189 x^4-45 x^5+3 x^6\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(48*x + 116*x^2 + 58*x^3 - 14*x^4 + (-48*x - 84*x^2 - 47*x^3 - 7*x^4)*Log[x])/((-192 - 720*x - 756*x^2 - 1
5*x^3 + 189*x^4 - 45*x^5 + 3*x^6)*Log[x]^3),x]

[Out]

(-2*Defer[Int][(x*(6 + 7*x))/((-4 - 5*x + x^2)^2*Log[x]^3), x])/3 - Defer[Int][(x*(48 + 84*x + 47*x^2 + 7*x^3)
)/((-4 - 5*x + x^2)^3*Log[x]^2), x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (2 \left (-24-58 x-29 x^2+7 x^3\right )+\left (48+84 x+47 x^2+7 x^3\right ) \log (x)\right )}{3 \left (4+5 x-x^2\right )^3 \log ^3(x)} \, dx\\ &=\frac {1}{3} \int \frac {x \left (2 \left (-24-58 x-29 x^2+7 x^3\right )+\left (48+84 x+47 x^2+7 x^3\right ) \log (x)\right )}{\left (4+5 x-x^2\right )^3 \log ^3(x)} \, dx\\ &=\frac {1}{3} \int \left (-\frac {2 x (6+7 x)}{\left (-4-5 x+x^2\right )^2 \log ^3(x)}-\frac {x \left (48+84 x+47 x^2+7 x^3\right )}{\left (-4-5 x+x^2\right )^3 \log ^2(x)}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x \left (48+84 x+47 x^2+7 x^3\right )}{\left (-4-5 x+x^2\right )^3 \log ^2(x)} \, dx\right )-\frac {2}{3} \int \frac {x (6+7 x)}{\left (-4-5 x+x^2\right )^2 \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^2 (6+7 x)}{3 \left (-4-5 x+x^2\right )^2 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48*x + 116*x^2 + 58*x^3 - 14*x^4 + (-48*x - 84*x^2 - 47*x^3 - 7*x^4)*Log[x])/((-192 - 720*x - 756*x
^2 - 15*x^3 + 189*x^4 - 45*x^5 + 3*x^6)*Log[x]^3),x]

[Out]

(x^2*(6 + 7*x))/(3*(-4 - 5*x + x^2)^2*Log[x]^2)

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fricas [A]  time = 0.62, size = 37, normalized size = 1.68 \begin {gather*} \frac {7 \, x^{3} + 6 \, x^{2}}{3 \, {\left (x^{4} - 10 \, x^{3} + 17 \, x^{2} + 40 \, x + 16\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^4-47*x^3-84*x^2-48*x)*log(x)-14*x^4+58*x^3+116*x^2+48*x)/(3*x^6-45*x^5+189*x^4-15*x^3-756*x^2
-720*x-192)/log(x)^3,x, algorithm="fricas")

[Out]

1/3*(7*x^3 + 6*x^2)/((x^4 - 10*x^3 + 17*x^2 + 40*x + 16)*log(x)^2)

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giac [B]  time = 0.21, size = 55, normalized size = 2.50 \begin {gather*} \frac {7 \, x^{3} + 6 \, x^{2}}{3 \, {\left (x^{4} \log \relax (x)^{2} - 10 \, x^{3} \log \relax (x)^{2} + 17 \, x^{2} \log \relax (x)^{2} + 40 \, x \log \relax (x)^{2} + 16 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^4-47*x^3-84*x^2-48*x)*log(x)-14*x^4+58*x^3+116*x^2+48*x)/(3*x^6-45*x^5+189*x^4-15*x^3-756*x^2
-720*x-192)/log(x)^3,x, algorithm="giac")

[Out]

1/3*(7*x^3 + 6*x^2)/(x^4*log(x)^2 - 10*x^3*log(x)^2 + 17*x^2*log(x)^2 + 40*x*log(x)^2 + 16*log(x)^2)

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maple [A]  time = 0.08, size = 35, normalized size = 1.59

method	result	size
risch	\(\frac {x^{2} \left (7 x +6\right )}{3 \left (x^{4}-10 x^{3}+17 x^{2}+40 x +16\right ) \ln \relax (x )^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7*x^4-47*x^3-84*x^2-48*x)*ln(x)-14*x^4+58*x^3+116*x^2+48*x)/(3*x^6-45*x^5+189*x^4-15*x^3-756*x^2-720*x-
192)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*x^2*(7*x+6)/(x^4-10*x^3+17*x^2+40*x+16)/ln(x)^2

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maxima [A]  time = 0.39, size = 37, normalized size = 1.68 \begin {gather*} \frac {7 \, x^{3} + 6 \, x^{2}}{3 \, {\left (x^{4} - 10 \, x^{3} + 17 \, x^{2} + 40 \, x + 16\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^4-47*x^3-84*x^2-48*x)*log(x)-14*x^4+58*x^3+116*x^2+48*x)/(3*x^6-45*x^5+189*x^4-15*x^3-756*x^2
-720*x-192)/log(x)^3,x, algorithm="maxima")

[Out]

1/3*(7*x^3 + 6*x^2)/((x^4 - 10*x^3 + 17*x^2 + 40*x + 16)*log(x)^2)

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mupad [B]  time = 4.85, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^2\,\left (7\,x+6\right )}{3\,{\ln \relax (x)}^2\,{\left (-x^2+5\,x+4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x - log(x)*(48*x + 84*x^2 + 47*x^3 + 7*x^4) + 116*x^2 + 58*x^3 - 14*x^4)/(log(x)^3*(720*x + 756*x^2 +
15*x^3 - 189*x^4 + 45*x^5 - 3*x^6 + 192)),x)

[Out]

(x^2*(7*x + 6))/(3*log(x)^2*(5*x - x^2 + 4)^2)

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sympy [A]  time = 0.22, size = 34, normalized size = 1.55 \begin {gather*} \frac {7 x^{3} + 6 x^{2}}{\left (3 x^{4} - 30 x^{3} + 51 x^{2} + 120 x + 48\right ) \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x**4-47*x**3-84*x**2-48*x)*ln(x)-14*x**4+58*x**3+116*x**2+48*x)/(3*x**6-45*x**5+189*x**4-15*x**
3-756*x**2-720*x-192)/ln(x)**3,x)

[Out]

(7*x**3 + 6*x**2)/((3*x**4 - 30*x**3 + 51*x**2 + 120*x + 48)*log(x)**2)

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