3.90.2 \(\int \frac {20 \log ^2(2)}{640 x+448 x^2+(480 x+336 x^2) \log (\frac {-10-7 x}{2 x})+(120 x+84 x^2) \log ^2(\frac {-10-7 x}{2 x})+(10 x+7 x^2) \log ^3(\frac {-10-7 x}{2 x})} \, dx\)

Optimal. Leaf size=29 \[ \frac {x+\frac {x \log ^2(2)}{\left (4+\log \left (\frac {3}{2}-\frac {5 (1+x)}{x}\right )\right )^2}}{x} \]

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Rubi [F]  time = 0.15, 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 {20 \log ^2(2)}{640 x+448 x^2+\left (480 x+336 x^2\right ) \log \left (\frac {-10-7 x}{2 x}\right )+\left (120 x+84 x^2\right ) \log ^2\left (\frac {-10-7 x}{2 x}\right )+\left (10 x+7 x^2\right ) \log ^3\left (\frac {-10-7 x}{2 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*Log[2]^2)/(640*x + 448*x^2 + (480*x + 336*x^2)*Log[(-10 - 7*x)/(2*x)] + (120*x + 84*x^2)*Log[(-10 - 7*
x)/(2*x)]^2 + (10*x + 7*x^2)*Log[(-10 - 7*x)/(2*x)]^3),x]

[Out]

2*Log[2]^2*Defer[Int][1/(x*(4 + Log[-7/2 - 5/x])^3), x] - 14*Log[2]^2*Defer[Int][1/((10 + 7*x)*(4 + Log[-7/2 -
5/x])^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (20 \log ^2(2)\right ) \int \frac {1}{640 x+448 x^2+\left (480 x+336 x^2\right ) \log \left (\frac {-10-7 x}{2 x}\right )+\left (120 x+84 x^2\right ) \log ^2\left (\frac {-10-7 x}{2 x}\right )+\left (10 x+7 x^2\right ) \log ^3\left (\frac {-10-7 x}{2 x}\right )} \, dx\\ &=\left (20 \log ^2(2)\right ) \int \frac {1}{x (10+7 x) \left (4+\log \left (-\frac {7}{2}-\frac {5}{x}\right )\right )^3} \, dx\\ &=\left (20 \log ^2(2)\right ) \int \left (\frac {1}{10 x \left (4+\log \left (-\frac {7}{2}-\frac {5}{x}\right )\right )^3}-\frac {7}{10 (10+7 x) \left (4+\log \left (-\frac {7}{2}-\frac {5}{x}\right )\right )^3}\right ) \, dx\\ &=\left (2 \log ^2(2)\right ) \int \frac {1}{x \left (4+\log \left (-\frac {7}{2}-\frac {5}{x}\right )\right )^3} \, dx-\left (14 \log ^2(2)\right ) \int \frac {1}{(10+7 x) \left (4+\log \left (-\frac {7}{2}-\frac {5}{x}\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.66 \begin {gather*} \frac {\log ^2(2)}{\left (4+\log \left (-\frac {7}{2}-\frac {5}{x}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*Log[2]^2)/(640*x + 448*x^2 + (480*x + 336*x^2)*Log[(-10 - 7*x)/(2*x)] + (120*x + 84*x^2)*Log[(-1
0 - 7*x)/(2*x)]^2 + (10*x + 7*x^2)*Log[(-10 - 7*x)/(2*x)]^3),x]

[Out]

Log[2]^2/(4 + Log[-7/2 - 5/x])^2

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fricas [A]  time = 0.44, size = 35, normalized size = 1.21 \begin {gather*} \frac {\log \relax (2)^{2}}{\log \left (-\frac {7 \, x + 10}{2 \, x}\right )^{2} + 8 \, \log \left (-\frac {7 \, x + 10}{2 \, x}\right ) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*log(2)^2/((7*x^2+10*x)*log(1/2*(-7*x-10)/x)^3+(84*x^2+120*x)*log(1/2*(-7*x-10)/x)^2+(336*x^2+480*
x)*log(1/2*(-7*x-10)/x)+448*x^2+640*x),x, algorithm="fricas")

[Out]

log(2)^2/(log(-1/2*(7*x + 10)/x)^2 + 8*log(-1/2*(7*x + 10)/x) + 16)

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giac [A]  time = 0.18, size = 20, normalized size = 0.69 \begin {gather*} \frac {\log \relax (2)^{2}}{{\left (\log \left (-\frac {7 \, x + 10}{2 \, x}\right ) + 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*log(2)^2/((7*x^2+10*x)*log(1/2*(-7*x-10)/x)^3+(84*x^2+120*x)*log(1/2*(-7*x-10)/x)^2+(336*x^2+480*
x)*log(1/2*(-7*x-10)/x)+448*x^2+640*x),x, algorithm="giac")

[Out]

log(2)^2/(log(-1/2*(7*x + 10)/x) + 4)^2

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maple [A]  time = 0.14, size = 18, normalized size = 0.62

method	result	size
derivativedivides	\(\frac {\ln \relax (2)^{2}}{\left (\ln \left (-\frac {7}{2}-\frac {5}{x}\right )+4\right )^{2}}\)	\(18\)
default	\(\frac {\ln \relax (2)^{2}}{\left (\ln \left (-\frac {7}{2}-\frac {5}{x}\right )+4\right )^{2}}\)	\(18\)
norman	\(\frac {\ln \relax (2)^{2}}{\left (\ln \left (\frac {-7 x -10}{2 x}\right )+4\right )^{2}}\)	\(21\)
risch	\(\frac {\ln \relax (2)^{2}}{\left (\ln \left (\frac {-7 x -10}{2 x}\right )+4\right )^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(20*ln(2)^2/((7*x^2+10*x)*ln(1/2*(-7*x-10)/x)^3+(84*x^2+120*x)*ln(1/2*(-7*x-10)/x)^2+(336*x^2+480*x)*ln(1/2
*(-7*x-10)/x)+448*x^2+640*x),x,method=_RETURNVERBOSE)

[Out]

ln(2)^2/(ln(-7/2-5/x)+4)^2

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maxima [A]  time = 0.46, size = 51, normalized size = 1.76 \begin {gather*} \frac {\log \relax (2)^{2}}{\log \relax (2)^{2} + 2 \, {\left (\log \relax (2) - 4\right )} \log \relax (x) + \log \relax (x)^{2} - 2 \, {\left (\log \relax (2) + \log \relax (x) - 4\right )} \log \left (-7 \, x - 10\right ) + \log \left (-7 \, x - 10\right )^{2} - 8 \, \log \relax (2) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*log(2)^2/((7*x^2+10*x)*log(1/2*(-7*x-10)/x)^3+(84*x^2+120*x)*log(1/2*(-7*x-10)/x)^2+(336*x^2+480*
x)*log(1/2*(-7*x-10)/x)+448*x^2+640*x),x, algorithm="maxima")

[Out]

log(2)^2/(log(2)^2 + 2*(log(2) - 4)*log(x) + log(x)^2 - 2*(log(2) + log(x) - 4)*log(-7*x - 10) + log(-7*x - 10
)^2 - 8*log(2) + 16)

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mupad [B]  time = 5.76, size = 20, normalized size = 0.69 \begin {gather*} \frac {{\ln \relax (2)}^2}{{\left (\ln \left (-\frac {7\,x+10}{2\,x}\right )+4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*log(2)^2)/(640*x + log(-((7*x)/2 + 5)/x)^3*(10*x + 7*x^2) + log(-((7*x)/2 + 5)/x)^2*(120*x + 84*x^2) +
log(-((7*x)/2 + 5)/x)*(480*x + 336*x^2) + 448*x^2),x)

[Out]

log(2)^2/(log(-(7*x + 10)/(2*x)) + 4)^2

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sympy [A]  time = 0.15, size = 32, normalized size = 1.10 \begin {gather*} \frac {\log {\relax (2 )}^{2}}{\log {\left (\frac {- \frac {7 x}{2} - 5}{x} \right )}^{2} + 8 \log {\left (\frac {- \frac {7 x}{2} - 5}{x} \right )} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*ln(2)**2/((7*x**2+10*x)*ln(1/2*(-7*x-10)/x)**3+(84*x**2+120*x)*ln(1/2*(-7*x-10)/x)**2+(336*x**2+4
80*x)*ln(1/2*(-7*x-10)/x)+448*x**2+640*x),x)

[Out]

log(2)**2/(log((-7*x/2 - 5)/x)**2 + 8*log((-7*x/2 - 5)/x) + 16)

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