3.66.5 \(\int \frac {-x+30 x^3-6 x^4+(10-2 x-30 x^3+6 x^4) \log (5-x)+(15 x^3-3 x^4) \log ^2(5-x)}{(-15 x^3+3 x^4) \log ^2(5-x)} \, dx\)

Optimal. Leaf size=24 \[ -x+\frac {2 \left (-5+\frac {1}{6 x^2}+x\right )}{\log (5-x)} \]

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Rubi [F]  time = 0.64, 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 {-x+30 x^3-6 x^4+\left (10-2 x-30 x^3+6 x^4\right ) \log (5-x)+\left (15 x^3-3 x^4\right ) \log ^2(5-x)}{\left (-15 x^3+3 x^4\right ) \log ^2(5-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x + 30*x^3 - 6*x^4 + (10 - 2*x - 30*x^3 + 6*x^4)*Log[5 - x] + (15*x^3 - 3*x^4)*Log[5 - x]^2)/((-15*x^3 +
3*x^4)*Log[5 - x]^2),x]

[Out]

-x + 1/(75*Log[5 - x]) - (2*(5 - x))/Log[5 - x] + Defer[Int][1/(x^2*Log[5 - x]^2), x]/15 + Defer[Int][1/(x*Log
[5 - x]^2), x]/75 - (2*Defer[Int][1/(x^3*Log[5 - x]), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+30 x^3-6 x^4+\left (10-2 x-30 x^3+6 x^4\right ) \log (5-x)+\left (15 x^3-3 x^4\right ) \log ^2(5-x)}{x^3 (-15+3 x) \log ^2(5-x)} \, dx\\ &=\int \left (-1+\frac {-1+30 x^2-6 x^3}{3 (-5+x) x^2 \log ^2(5-x)}+\frac {2 \left (-1+3 x^3\right )}{3 x^3 \log (5-x)}\right ) \, dx\\ &=-x+\frac {1}{3} \int \frac {-1+30 x^2-6 x^3}{(-5+x) x^2 \log ^2(5-x)} \, dx+\frac {2}{3} \int \frac {-1+3 x^3}{x^3 \log (5-x)} \, dx\\ &=-x+\frac {1}{3} \int \left (-\frac {6}{\log ^2(5-x)}-\frac {1}{25 (-5+x) \log ^2(5-x)}+\frac {1}{5 x^2 \log ^2(5-x)}+\frac {1}{25 x \log ^2(5-x)}\right ) \, dx+\frac {2}{3} \int \left (\frac {3}{\log (5-x)}-\frac {1}{x^3 \log (5-x)}\right ) \, dx\\ &=-x-\frac {1}{75} \int \frac {1}{(-5+x) \log ^2(5-x)} \, dx+\frac {1}{75} \int \frac {1}{x \log ^2(5-x)} \, dx+\frac {1}{15} \int \frac {1}{x^2 \log ^2(5-x)} \, dx-\frac {2}{3} \int \frac {1}{x^3 \log (5-x)} \, dx-2 \int \frac {1}{\log ^2(5-x)} \, dx+2 \int \frac {1}{\log (5-x)} \, dx\\ &=-x+\frac {1}{75} \int \frac {1}{x \log ^2(5-x)} \, dx-\frac {1}{75} \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,5-x\right )+\frac {1}{15} \int \frac {1}{x^2 \log ^2(5-x)} \, dx-\frac {2}{3} \int \frac {1}{x^3 \log (5-x)} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,5-x\right )-2 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )\\ &=-x-\frac {2 (5-x)}{\log (5-x)}-2 \text {li}(5-x)+\frac {1}{75} \int \frac {1}{x \log ^2(5-x)} \, dx-\frac {1}{75} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (5-x)\right )+\frac {1}{15} \int \frac {1}{x^2 \log ^2(5-x)} \, dx-\frac {2}{3} \int \frac {1}{x^3 \log (5-x)} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )\\ &=-x+\frac {1}{75 \log (5-x)}-\frac {2 (5-x)}{\log (5-x)}+\frac {1}{75} \int \frac {1}{x \log ^2(5-x)} \, dx+\frac {1}{15} \int \frac {1}{x^2 \log ^2(5-x)} \, dx-\frac {2}{3} \int \frac {1}{x^3 \log (5-x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{3} \left (-3 x+\frac {-30+\frac {1}{x^2}+6 x}{\log (5-x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + 30*x^3 - 6*x^4 + (10 - 2*x - 30*x^3 + 6*x^4)*Log[5 - x] + (15*x^3 - 3*x^4)*Log[5 - x]^2)/((-15
*x^3 + 3*x^4)*Log[5 - x]^2),x]

[Out]

(-3*x + (-30 + x^(-2) + 6*x)/Log[5 - x])/3

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fricas [A]  time = 0.61, size = 36, normalized size = 1.50 \begin {gather*} -\frac {3 \, x^{3} \log \left (-x + 5\right ) - 6 \, x^{3} + 30 \, x^{2} - 1}{3 \, x^{2} \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^4+15*x^3)*log(5-x)^2+(6*x^4-30*x^3-2*x+10)*log(5-x)-6*x^4+30*x^3-x)/(3*x^4-15*x^3)/log(5-x)^2
,x, algorithm="fricas")

[Out]

-1/3*(3*x^3*log(-x + 5) - 6*x^3 + 30*x^2 - 1)/(x^2*log(-x + 5))

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giac [A]  time = 0.18, size = 29, normalized size = 1.21 \begin {gather*} -x + \frac {6 \, x^{3} - 30 \, x^{2} + 1}{3 \, x^{2} \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^4+15*x^3)*log(5-x)^2+(6*x^4-30*x^3-2*x+10)*log(5-x)-6*x^4+30*x^3-x)/(3*x^4-15*x^3)/log(5-x)^2
,x, algorithm="giac")

[Out]

-x + 1/3*(6*x^3 - 30*x^2 + 1)/(x^2*log(-x + 5))

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maple [A]  time = 0.41, size = 30, normalized size = 1.25

method	result	size
risch	\(-x +\frac {6 x^{3}-30 x^{2}+1}{3 x^{2} \ln \left (5-x \right )}\)	\(30\)
norman	\(\frac {\frac {1}{3}-10 x^{2}+2 x^{3}-x^{3} \ln \left (5-x \right )}{\ln \left (5-x \right ) x^{2}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^4+15*x^3)*ln(5-x)^2+(6*x^4-30*x^3-2*x+10)*ln(5-x)-6*x^4+30*x^3-x)/(3*x^4-15*x^3)/ln(5-x)^2,x,method
=_RETURNVERBOSE)

[Out]

-x+1/3*(6*x^3-30*x^2+1)/x^2/ln(5-x)

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maxima [A]  time = 0.41, size = 36, normalized size = 1.50 \begin {gather*} -\frac {3 \, x^{3} \log \left (-x + 5\right ) - 6 \, x^{3} + 30 \, x^{2} - 1}{3 \, x^{2} \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^4+15*x^3)*log(5-x)^2+(6*x^4-30*x^3-2*x+10)*log(5-x)-6*x^4+30*x^3-x)/(3*x^4-15*x^3)/log(5-x)^2
,x, algorithm="maxima")

[Out]

-1/3*(3*x^3*log(-x + 5) - 6*x^3 + 30*x^2 - 1)/(x^2*log(-x + 5))

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mupad [B]  time = 0.20, size = 28, normalized size = 1.17 \begin {gather*} \frac {2\,x^3-10\,x^2+\frac {1}{3}}{x^2\,\ln \left (5-x\right )}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - log(5 - x)^2*(15*x^3 - 3*x^4) + log(5 - x)*(2*x + 30*x^3 - 6*x^4 - 10) - 30*x^3 + 6*x^4)/(log(5 - x)^
2*(15*x^3 - 3*x^4)),x)

[Out]

(2*x^3 - 10*x^2 + 1/3)/(x^2*log(5 - x)) - x

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sympy [A]  time = 0.12, size = 22, normalized size = 0.92 \begin {gather*} - x + \frac {6 x^{3} - 30 x^{2} + 1}{3 x^{2} \log {\left (5 - x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**4+15*x**3)*ln(5-x)**2+(6*x**4-30*x**3-2*x+10)*ln(5-x)-6*x**4+30*x**3-x)/(3*x**4-15*x**3)/ln(
5-x)**2,x)

[Out]

-x + (6*x**3 - 30*x**2 + 1)/(3*x**2*log(5 - x))

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