3.98.46 \(\int \frac {-192 x^2-208 x^3-56 x^4+(-384 x-576 x^2-304 x^3-56 x^4) \log (\frac {3}{2+x})}{(1728+3024 x+1764 x^2+343 x^3) \log ^3(\frac {3}{2+x})} \, dx\)

Optimal. Leaf size=30 \[ 4 \left (5-\frac {x^2}{\left (6+\frac {x}{2+x}\right )^2 \log ^2\left (\frac {3}{2+x}\right )}\right ) \]

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Rubi [F]  time = 1.29, 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 {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-192*x^2 - 208*x^3 - 56*x^4 + (-384*x - 576*x^2 - 304*x^3 - 56*x^4)*Log[3/(2 + x)])/((1728 + 3024*x + 176
4*x^2 + 343*x^3)*Log[3/(2 + x)]^3),x]

[Out]

(40*(2 + x))/(343*Log[3/(2 + x)]^2) - (4*x*(2 + x))/(49*Log[3/(2 + x)]^2) - (2304*Defer[Int][1/((12 + 7*x)^2*L
og[3/(2 + x)]^3), x])/343 - (768*Defer[Int][1/((12 + 7*x)*Log[3/(2 + x)]^3), x])/343 + (4608*Defer[Int][1/((12
+ 7*x)^3*Log[3/(2 + x)]^2), x])/343 + (1920*Defer[Int][1/((12 + 7*x)^2*Log[3/(2 + x)]^2), x])/343

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x (2+x) \left (-x (12+7 x)-\left (24+24 x+7 x^2\right ) \log \left (\frac {3}{2+x}\right )\right )}{(12+7 x)^3 \log ^3\left (\frac {3}{2+x}\right )} \, dx\\ &=8 \int \frac {x (2+x) \left (-x (12+7 x)-\left (24+24 x+7 x^2\right ) \log \left (\frac {3}{2+x}\right )\right )}{(12+7 x)^3 \log ^3\left (\frac {3}{2+x}\right )} \, dx\\ &=8 \int \left (-\frac {x^2 (2+x)}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )}-\frac {x (2+x) \left (24+24 x+7 x^2\right )}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )}\right ) \, dx\\ &=-\left (8 \int \frac {x^2 (2+x)}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx\right )-8 \int \frac {x (2+x) \left (24+24 x+7 x^2\right )}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=-\left (8 \int \left (-\frac {10}{343 \log ^3\left (\frac {3}{2+x}\right )}+\frac {x}{49 \log ^3\left (\frac {3}{2+x}\right )}+\frac {288}{343 (12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )}+\frac {96}{343 (12+7 x) \log ^3\left (\frac {3}{2+x}\right )}\right ) \, dx\right )-8 \int \left (\frac {2}{343 \log ^2\left (\frac {3}{2+x}\right )}+\frac {x}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {576}{343 (12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )}-\frac {240}{343 (12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )}\right ) \, dx\\ &=-\left (\frac {16}{343} \int \frac {1}{\log ^2\left (\frac {3}{2+x}\right )} \, dx\right )-\frac {8}{49} \int \frac {x}{\log ^3\left (\frac {3}{2+x}\right )} \, dx-\frac {8}{49} \int \frac {x}{\log ^2\left (\frac {3}{2+x}\right )} \, dx+\frac {80}{343} \int \frac {1}{\log ^3\left (\frac {3}{2+x}\right )} \, dx-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {8 x (2+x)}{49 \log \left (\frac {3}{2+x}\right )}-\frac {16}{343} \operatorname {Subst}\left (\int \frac {1}{\log ^2\left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {8}{49} \int \frac {1}{\log ^2\left (\frac {3}{2+x}\right )} \, dx+\frac {8}{49} \int \frac {x}{\log ^2\left (\frac {3}{2+x}\right )} \, dx+\frac {80}{343} \operatorname {Subst}\left (\int \frac {1}{\log ^3\left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {16}{49} \int \frac {1}{\log \left (\frac {3}{2+x}\right )} \, dx+\frac {16}{49} \int \frac {x}{\log \left (\frac {3}{2+x}\right )} \, dx-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=\frac {40 (2+x)}{343 \log ^2\left (\frac {3}{2+x}\right )}-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {16 (2+x)}{343 \log \left (\frac {3}{2+x}\right )}+\frac {16}{343} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )-\frac {40}{343} \operatorname {Subst}\left (\int \frac {1}{\log ^2\left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {8}{49} \operatorname {Subst}\left (\int \frac {1}{\log ^2\left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {16}{49} \int \left (-\frac {2}{\log \left (\frac {3}{2+x}\right )}+\frac {2+x}{\log \left (\frac {3}{2+x}\right )}\right ) \, dx-\frac {16}{49} \int \frac {1}{\log \left (\frac {3}{2+x}\right )} \, dx-\frac {16}{49} \int \frac {x}{\log \left (\frac {3}{2+x}\right )} \, dx+\frac {16}{49} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=\frac {40 (2+x)}{343 \log ^2\left (\frac {3}{2+x}\right )}-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}+\frac {40}{343} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )-\frac {48}{343} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )-\frac {8}{49} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )-\frac {16}{49} \int \left (-\frac {2}{\log \left (\frac {3}{2+x}\right )}+\frac {2+x}{\log \left (\frac {3}{2+x}\right )}\right ) \, dx+\frac {16}{49} \int \frac {2+x}{\log \left (\frac {3}{2+x}\right )} \, dx-\frac {16}{49} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )-\frac {32}{49} \int \frac {1}{\log \left (\frac {3}{2+x}\right )} \, dx-\frac {48}{49} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=-\frac {384}{343} \text {Ei}\left (-\log \left (\frac {3}{2+x}\right )\right )+\frac {40 (2+x)}{343 \log ^2\left (\frac {3}{2+x}\right )}-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {16}{49} \int \frac {2+x}{\log \left (\frac {3}{2+x}\right )} \, dx+\frac {16}{49} \operatorname {Subst}\left (\int \frac {x}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )-\frac {120}{343} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )+\frac {24}{49} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )+\frac {32}{49} \int \frac {1}{\log \left (\frac {3}{2+x}\right )} \, dx-\frac {32}{49} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {48}{49} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=\frac {40 (2+x)}{343 \log ^2\left (\frac {3}{2+x}\right )}-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {16}{49} \operatorname {Subst}\left (\int \frac {x}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {32}{49} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {3}{x}\right )} \, dx,x,2+x\right )+\frac {96}{49} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx-\frac {144}{49} \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=-\frac {144}{49} \text {Ei}\left (-2 \log \left (\frac {3}{2+x}\right )\right )+\frac {96}{49} \text {Ei}\left (-\log \left (\frac {3}{2+x}\right )\right )+\frac {40 (2+x)}{343 \log ^2\left (\frac {3}{2+x}\right )}-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {96}{49} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {144}{49} \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log \left (\frac {3}{2+x}\right )\right )+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ &=\frac {40 (2+x)}{343 \log ^2\left (\frac {3}{2+x}\right )}-\frac {4 x (2+x)}{49 \log ^2\left (\frac {3}{2+x}\right )}-\frac {768}{343} \int \frac {1}{(12+7 x) \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {1920}{343} \int \frac {1}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \, dx-\frac {2304}{343} \int \frac {1}{(12+7 x)^2 \log ^3\left (\frac {3}{2+x}\right )} \, dx+\frac {4608}{343} \int \frac {1}{(12+7 x)^3 \log ^2\left (\frac {3}{2+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 27, normalized size = 0.90 \begin {gather*} -\frac {4 x^2 (2+x)^2}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-192*x^2 - 208*x^3 - 56*x^4 + (-384*x - 576*x^2 - 304*x^3 - 56*x^4)*Log[3/(2 + x)])/((1728 + 3024*x
+ 1764*x^2 + 343*x^3)*Log[3/(2 + x)]^3),x]

[Out]

(-4*x^2*(2 + x)^2)/((12 + 7*x)^2*Log[3/(2 + x)]^2)

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fricas [A]  time = 0.74, size = 38, normalized size = 1.27 \begin {gather*} -\frac {4 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )}}{{\left (49 \, x^{2} + 168 \, x + 144\right )} \log \left (\frac {3}{x + 2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-56*x^4-304*x^3-576*x^2-384*x)*log(3/(2+x))-56*x^4-208*x^3-192*x^2)/(343*x^3+1764*x^2+3024*x+1728)
/log(3/(2+x))^3,x, algorithm="fricas")

[Out]

-4*(x^4 + 4*x^3 + 4*x^2)/((49*x^2 + 168*x + 144)*log(3/(x + 2))^2)

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giac [B]  time = 0.14, size = 72, normalized size = 2.40 \begin {gather*} \frac {4 \, {\left (\frac {4}{x + 2} - \frac {4}{{\left (x + 2\right )}^{2}} - 1\right )}}{\frac {49 \, \log \left (\frac {3}{x + 2}\right )^{2}}{{\left (x + 2\right )}^{2}} - \frac {28 \, \log \left (\frac {3}{x + 2}\right )^{2}}{{\left (x + 2\right )}^{3}} + \frac {4 \, \log \left (\frac {3}{x + 2}\right )^{2}}{{\left (x + 2\right )}^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-56*x^4-304*x^3-576*x^2-384*x)*log(3/(2+x))-56*x^4-208*x^3-192*x^2)/(343*x^3+1764*x^2+3024*x+1728)
/log(3/(2+x))^3,x, algorithm="giac")

[Out]

4*(4/(x + 2) - 4/(x + 2)^2 - 1)/(49*log(3/(x + 2))^2/(x + 2)^2 - 28*log(3/(x + 2))^2/(x + 2)^3 + 4*log(3/(x +
2))^2/(x + 2)^4)

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maple [A]  time = 0.14, size = 33, normalized size = 1.10

method	result	size
risch	\(-\frac {4 x^{2} \left (2+x \right )^{2}}{\left (49 x^{2}+168 x +144\right ) \ln \left (\frac {3}{2+x}\right )^{2}}\)	\(33\)
norman	\(\frac {-4 x^{4}-16 x^{3}-16 x^{2}}{\left (7 x +12\right )^{2} \ln \left (\frac {3}{2+x}\right )^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-56*x^4-304*x^3-576*x^2-384*x)*ln(3/(2+x))-56*x^4-208*x^3-192*x^2)/(343*x^3+1764*x^2+3024*x+1728)/ln(3/(
2+x))^3,x,method=_RETURNVERBOSE)

[Out]

-4*x^2*(2+x)^2/(49*x^2+168*x+144)/ln(3/(2+x))^2

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maxima [B]  time = 0.47, size = 81, normalized size = 2.70 \begin {gather*} -\frac {4 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )}}{49 \, x^{2} \log \relax (3)^{2} + 168 \, x \log \relax (3)^{2} + {\left (49 \, x^{2} + 168 \, x + 144\right )} \log \left (x + 2\right )^{2} + 144 \, \log \relax (3)^{2} - 2 \, {\left (49 \, x^{2} \log \relax (3) + 168 \, x \log \relax (3) + 144 \, \log \relax (3)\right )} \log \left (x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-56*x^4-304*x^3-576*x^2-384*x)*log(3/(2+x))-56*x^4-208*x^3-192*x^2)/(343*x^3+1764*x^2+3024*x+1728)
/log(3/(2+x))^3,x, algorithm="maxima")

[Out]

-4*(x^4 + 4*x^3 + 4*x^2)/(49*x^2*log(3)^2 + 168*x*log(3)^2 + (49*x^2 + 168*x + 144)*log(x + 2)^2 + 144*log(3)^
2 - 2*(49*x^2*log(3) + 168*x*log(3) + 144*log(3))*log(x + 2))

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mupad [B]  time = 6.08, size = 200, normalized size = 6.67 \begin {gather*} \frac {\frac {4\,\left (x+2\right )\,\left (7\,x^4+38\,x^3+72\,x^2+48\,x\right )}{{\left (7\,x+12\right )}^3}+\frac {8\,\ln \left (\frac {3}{x+2}\right )\,\left (x+2\right )\,\left (49\,x^5+392\,x^4+1248\,x^3+1992\,x^2+1632\,x+576\right )}{{\left (7\,x+12\right )}^4}}{\ln \left (\frac {3}{x+2}\right )}-\frac {\frac {4\,x^2\,{\left (x+2\right )}^2}{{\left (7\,x+12\right )}^2}+\frac {4\,x\,\ln \left (\frac {3}{x+2}\right )\,\left (x+2\right )\,\left (7\,x^3+38\,x^2+72\,x+48\right )}{{\left (7\,x+12\right )}^3}}{{\ln \left (\frac {3}{x+2}\right )}^2}-\frac {176\,x}{343}-\frac {8\,x^2}{49}-\frac {\frac {960\,x^3}{16807}+\frac {44928\,x^2}{117649}+\frac {694272\,x}{823543}+\frac {506880}{823543}}{x^4+\frac {48\,x^3}{7}+\frac {864\,x^2}{49}+\frac {6912\,x}{343}+\frac {20736}{2401}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(3/(x + 2))*(384*x + 576*x^2 + 304*x^3 + 56*x^4) + 192*x^2 + 208*x^3 + 56*x^4)/(log(3/(x + 2))^3*(302
4*x + 1764*x^2 + 343*x^3 + 1728)),x)

[Out]

((4*(x + 2)*(48*x + 72*x^2 + 38*x^3 + 7*x^4))/(7*x + 12)^3 + (8*log(3/(x + 2))*(x + 2)*(1632*x + 1992*x^2 + 12
48*x^3 + 392*x^4 + 49*x^5 + 576))/(7*x + 12)^4)/log(3/(x + 2)) - ((4*x^2*(x + 2)^2)/(7*x + 12)^2 + (4*x*log(3/
(x + 2))*(x + 2)*(72*x + 38*x^2 + 7*x^3 + 48))/(7*x + 12)^3)/log(3/(x + 2))^2 - (176*x)/343 - (8*x^2)/49 - ((6
94272*x)/823543 + (44928*x^2)/117649 + (960*x^3)/16807 + 506880/823543)/((6912*x)/343 + (864*x^2)/49 + (48*x^3
)/7 + x^4 + 20736/2401)

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sympy [A]  time = 0.23, size = 34, normalized size = 1.13 \begin {gather*} \frac {- 4 x^{4} - 16 x^{3} - 16 x^{2}}{\left (49 x^{2} + 168 x + 144\right ) \log {\left (\frac {3}{x + 2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-56*x**4-304*x**3-576*x**2-384*x)*ln(3/(2+x))-56*x**4-208*x**3-192*x**2)/(343*x**3+1764*x**2+3024*
x+1728)/ln(3/(2+x))**3,x)

[Out]

(-4*x**4 - 16*x**3 - 16*x**2)/((49*x**2 + 168*x + 144)*log(3/(x + 2))**2)

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