3.15.72 \(\int \frac {48-336 x-16 x^2+80 x^3-64 x^4+(288-64 x^2+128 x^3) \log (5 x)-64 x^2 \log ^2(5 x)}{225 x^9-450 x^8 \log (5 x)+225 x^7 \log ^2(5 x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {16 \left (x^2+\frac {-3+x^2}{-x+\log (5 x)}\right )}{225 x^6} \]

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Rubi [F]  time = 0.99, 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 {48-336 x-16 x^2+80 x^3-64 x^4+\left (288-64 x^2+128 x^3\right ) \log (5 x)-64 x^2 \log ^2(5 x)}{225 x^9-450 x^8 \log (5 x)+225 x^7 \log ^2(5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(48 - 336*x - 16*x^2 + 80*x^3 - 64*x^4 + (288 - 64*x^2 + 128*x^3)*Log[5*x] - 64*x^2*Log[5*x]^2)/(225*x^9 -
450*x^8*Log[5*x] + 225*x^7*Log[5*x]^2),x]

[Out]

16/(225*x^4) + (16*Defer[Int][1/(x^7*(x - Log[5*x])^2), x])/75 - (16*Defer[Int][1/(x^6*(x - Log[5*x])^2), x])/
75 - (16*Defer[Int][1/(x^5*(x - Log[5*x])^2), x])/225 + (16*Defer[Int][1/(x^4*(x - Log[5*x])^2), x])/225 - (32
*Defer[Int][1/(x^7*(x - Log[5*x])), x])/25 + (64*Defer[Int][1/(x^5*(x - Log[5*x])), x])/225

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48-336 x-16 x^2+80 x^3-64 x^4+\left (288-64 x^2+128 x^3\right ) \log (5 x)-64 x^2 \log ^2(5 x)}{225 x^7 (x-\log (5 x))^2} \, dx\\ &=\frac {1}{225} \int \frac {48-336 x-16 x^2+80 x^3-64 x^4+\left (288-64 x^2+128 x^3\right ) \log (5 x)-64 x^2 \log ^2(5 x)}{x^7 (x-\log (5 x))^2} \, dx\\ &=\frac {1}{225} \int \left (-\frac {64}{x^5}+\frac {16 \left (3-3 x-x^2+x^3\right )}{x^7 (x-\log (5 x))^2}+\frac {32 \left (-9+2 x^2\right )}{x^7 (x-\log (5 x))}\right ) \, dx\\ &=\frac {16}{225 x^4}+\frac {16}{225} \int \frac {3-3 x-x^2+x^3}{x^7 (x-\log (5 x))^2} \, dx+\frac {32}{225} \int \frac {-9+2 x^2}{x^7 (x-\log (5 x))} \, dx\\ &=\frac {16}{225 x^4}+\frac {16}{225} \int \left (\frac {3}{x^7 (x-\log (5 x))^2}-\frac {3}{x^6 (x-\log (5 x))^2}-\frac {1}{x^5 (x-\log (5 x))^2}+\frac {1}{x^4 (x-\log (5 x))^2}\right ) \, dx+\frac {32}{225} \int \left (-\frac {9}{x^7 (x-\log (5 x))}+\frac {2}{x^5 (x-\log (5 x))}\right ) \, dx\\ &=\frac {16}{225 x^4}-\frac {16}{225} \int \frac {1}{x^5 (x-\log (5 x))^2} \, dx+\frac {16}{225} \int \frac {1}{x^4 (x-\log (5 x))^2} \, dx+\frac {16}{75} \int \frac {1}{x^7 (x-\log (5 x))^2} \, dx-\frac {16}{75} \int \frac {1}{x^6 (x-\log (5 x))^2} \, dx+\frac {64}{225} \int \frac {1}{x^5 (x-\log (5 x))} \, dx-\frac {32}{25} \int \frac {1}{x^7 (x-\log (5 x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.81, size = 31, normalized size = 1.15 \begin {gather*} -\frac {16}{225} \left (-\frac {1}{x^4}+\frac {3-x^2}{x^6 (-x+\log (5 x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48 - 336*x - 16*x^2 + 80*x^3 - 64*x^4 + (288 - 64*x^2 + 128*x^3)*Log[5*x] - 64*x^2*Log[5*x]^2)/(225
*x^9 - 450*x^8*Log[5*x] + 225*x^7*Log[5*x]^2),x]

[Out]

(-16*(-x^(-4) + (3 - x^2)/(x^6*(-x + Log[5*x]))))/225

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fricas [A]  time = 1.04, size = 36, normalized size = 1.33 \begin {gather*} \frac {16 \, {\left (x^{3} - x^{2} \log \left (5 \, x\right ) - x^{2} + 3\right )}}{225 \, {\left (x^{7} - x^{6} \log \left (5 \, x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2*log(5*x)^2+(128*x^3-64*x^2+288)*log(5*x)-64*x^4+80*x^3-16*x^2-336*x+48)/(225*x^7*log(5*x)^2
-450*x^8*log(5*x)+225*x^9),x, algorithm="fricas")

[Out]

16/225*(x^3 - x^2*log(5*x) - x^2 + 3)/(x^7 - x^6*log(5*x))

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giac [A]  time = 0.29, size = 28, normalized size = 1.04 \begin {gather*} -\frac {16 \, {\left (x^{2} - 3\right )}}{225 \, {\left (x^{7} - x^{6} \log \left (5 \, x\right )\right )}} + \frac {16}{225 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2*log(5*x)^2+(128*x^3-64*x^2+288)*log(5*x)-64*x^4+80*x^3-16*x^2-336*x+48)/(225*x^7*log(5*x)^2
-450*x^8*log(5*x)+225*x^9),x, algorithm="giac")

[Out]

-16/225*(x^2 - 3)/(x^7 - x^6*log(5*x)) + 16/225/x^4

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maple [A]  time = 0.04, size = 27, normalized size = 1.00

method	result	size
risch	\(\frac {16}{225 x^{4}}-\frac {16 \left (x^{2}-3\right )}{225 x^{6} \left (x -\ln \left (5 x \right )\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-64*x^2*ln(5*x)^2+(128*x^3-64*x^2+288)*ln(5*x)-64*x^4+80*x^3-16*x^2-336*x+48)/(225*x^7*ln(5*x)^2-450*x^8*
ln(5*x)+225*x^9),x,method=_RETURNVERBOSE)

[Out]

16/225/x^4-16/225*(x^2-3)/x^6/(x-ln(5*x))

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maxima [A]  time = 0.96, size = 43, normalized size = 1.59 \begin {gather*} \frac {16 \, {\left (x^{3} - x^{2} {\left (\log \relax (5) + 1\right )} - x^{2} \log \relax (x) + 3\right )}}{225 \, {\left (x^{7} - x^{6} \log \relax (5) - x^{6} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2*log(5*x)^2+(128*x^3-64*x^2+288)*log(5*x)-64*x^4+80*x^3-16*x^2-336*x+48)/(225*x^7*log(5*x)^2
-450*x^8*log(5*x)+225*x^9),x, algorithm="maxima")

[Out]

16/225*(x^3 - x^2*(log(5) + 1) - x^2*log(x) + 3)/(x^7 - x^6*log(5) - x^6*log(x))

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mupad [B]  time = 1.14, size = 28, normalized size = 1.04 \begin {gather*} \frac {16}{225\,x^4}-\frac {\frac {16\,x^2}{225}-\frac {16}{75}}{x^6\,\left (x-\ln \left (5\,x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(336*x - log(5*x)*(128*x^3 - 64*x^2 + 288) + 16*x^2 - 80*x^3 + 64*x^4 + 64*x^2*log(5*x)^2 - 48)/(225*x^9
- 450*x^8*log(5*x) + 225*x^7*log(5*x)^2),x)

[Out]

16/(225*x^4) - ((16*x^2)/225 - 16/75)/(x^6*(x - log(5*x)))

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sympy [A]  time = 0.19, size = 27, normalized size = 1.00 \begin {gather*} \frac {16 x^{2} - 48}{- 225 x^{7} + 225 x^{6} \log {\left (5 x \right )}} + \frac {16}{225 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x**2*ln(5*x)**2+(128*x**3-64*x**2+288)*ln(5*x)-64*x**4+80*x**3-16*x**2-336*x+48)/(225*x**7*ln(5
*x)**2-450*x**8*ln(5*x)+225*x**9),x)

[Out]

(16*x**2 - 48)/(-225*x**7 + 225*x**6*log(5*x)) + 16/(225*x**4)

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