∫ −25x4+60x5−21x6+2x7+(120x2−264x3+48x4) log(3)+(−144+288x) log2(3)+(16x4 log(3)−384x log2(3)) log(5)_______________________________________________________________________________ (25x4−10x5+x6+(−120x2+24x3) log(3)+144 log2(3)) log(5) dx [9563]

Optimal. Leaf size=38 \[ \frac {4}{-\frac {3}{x^2}+\frac {5-x}{4 \log (3)}}-\frac {-5+x-x^2}{\log (5)} \]

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Rubi [C] Result contains complex when optimal does not.
time = 76.53, antiderivative size = 4395, normalized size of antiderivative = 115.66, number of steps used = 21, number of rules used = 11, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {12, 2099, 2106, 2104, 814, 648, 632, 212, 642, 2126, 836} \begin {gather*} \text {Too large to display} \end {gather*}

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)} \, dx}{\log (5)}\\ &=\frac {\int \left (-1+2 x+\frac {16 (5+x) \log (3) \log (5)}{-5 x^2+x^3+12 \log (3)}-\frac {16 \log (3) \left (-25 x^2+60 \log (3)+36 x \log (3)\right ) \log (5)}{\left (-5 x^2+x^3+12 \log (3)\right )^2}\right ) \, dx}{\log (5)}\\ &=-\frac {x}{\log (5)}+\frac {x^2}{\log (5)}+(16 \log (3)) \int \frac {5+x}{-5 x^2+x^3+12 \log (3)} \, dx-(16 \log (3)) \int \frac {-25 x^2+60 \log (3)+36 x \log (3)}{\left (-5 x^2+x^3+12 \log (3)\right )^2} \, dx\\ &=\frac {400 \log (3)}{3 \left (5 x^2-x^3-12 \log (3)\right )}-\frac {x}{\log (5)}+\frac {x^2}{\log (5)}-\frac {1}{3} (16 \log (3)) \int \frac {-2 x (125-54 \log (3))+180 \log (3)}{\left (-5 x^2+x^3+12 \log (3)\right )^2} \, dx+(16 \log (3)) \text {Subst}\left (\int \frac {\frac {20}{3}+x}{-\frac {25 x}{3}+x^3-\frac {2}{27} (125-162 \log (3))} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {400 \log (3)}{3 \left (5 x^2-x^3-12 \log (3)\right )}-\frac {x}{\log (5)}+\frac {x^2}{\log (5)}-\frac {1}{3} (16 \log (3)) \text {Subst}\left (\int \frac {-2 x (125-54 \log (3))+\frac {1}{3} (-10 (125-54 \log (3))+540 \log (3))}{\left (-\frac {25 x}{3}+x^3-\frac {2}{27} (125-162 \log (3))\right )^2} \, dx,x,-\frac {5}{3}+x\right )+(16 \log (3)) \text {Subst}\left (\int \frac {\frac {20}{3}+x}{\left (x+\frac {25+\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}}{3 \sqrt [3]{-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}}}\right ) \left (x^2-\frac {x \left (25+\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}\right )}{3 \sqrt [3]{-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}}}+\frac {1}{9} \left (-25+\frac {625}{\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}}+\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}\right )\right )} \, dx,x,-\frac {5}{3}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(38)=76\).
time = 0.07, size = 90, normalized size = 2.37 \begin {gather*} \frac {x \left (x^3 (750-486 \log (3))+12 (125-81 \log (3)) \log (3)+5 x^2 (-125+81 \log (3))+x^4 (-125+81 \log (3))-4 x \log (3) (375-500 \log (5)+81 \log (3) (-3+\log (625)))\right )}{\left (-5 x^2+x^3+12 \log (3)\right ) (-125+81 \log (3)) \log (5)} \end {gather*}

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method	result	size

default	\(\frac {x^{2}-x -\frac {4 \ln \left (3\right ) \ln \left (5\right ) x^{2}}{3 \left (\frac {x^{3}}{12}-\frac {5 x^{2}}{12}+\ln \left (3\right )\right )}}{\ln \left (5\right )}\)	\(37\)
risch	\(\frac {x^{2}}{\ln \left (5\right )}-\frac {x}{\ln \left (5\right )}-\frac {4 \ln \left (3\right ) x^{2}}{3 \left (\frac {x^{3}}{12}-\frac {5 x^{2}}{12}+\ln \left (3\right )\right )}\)	\(39\)
gosper	\(-\frac {-x^{5}+16 x^{2} \ln \left (3\right ) \ln \left (5\right )+6 x^{4}-12 x^{2} \ln \left (3\right )+12 x \ln \left (3\right )-25 x^{2}+60 \ln \left (3\right )}{\ln \left (5\right ) \left (x^{3}-5 x^{2}+12 \ln \left (3\right )\right )}\)	\(63\)
norman	\(\frac {-\frac {\left (16 \ln \left (3\right ) \ln \left (5\right )-12 \ln \left (3\right )-25\right ) x^{2}}{\ln \left (5\right )}+\frac {x^{5}}{\ln \left (5\right )}-\frac {6 x^{4}}{\ln \left (5\right )}-\frac {12 \ln \left (3\right ) x}{\ln \left (5\right )}-\frac {60 \ln \left (3\right )}{\ln \left (5\right )}}{x^{3}-5 x^{2}+12 \ln \left (3\right )}\)	\(73\)

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Maxima [A]
time = 0.26, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\frac {16 \, x^{2} \log \left (5\right ) \log \left (3\right )}{x^{3} - 5 \, x^{2} + 12 \, \log \left (3\right )} - x^{2} + x}{\log \left (5\right )} \end {gather*}

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Fricas [A]
time = 0.37, size = 54, normalized size = 1.42 \begin {gather*} \frac {x^{5} - 6 \, x^{4} - 16 \, x^{2} \log \left (5\right ) \log \left (3\right ) + 5 \, x^{3} + 12 \, {\left (x^{2} - x\right )} \log \left (3\right )}{{\left (x^{3} - 5 \, x^{2} + 12 \, \log \left (3\right )\right )} \log \left (5\right )} \end {gather*}

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Sympy [A]
time = 1.01, size = 32, normalized size = 0.84 \begin {gather*} \frac {x^{2}}{\log {\left (5 \right )}} - \frac {16 x^{2} \log {\left (3 \right )}}{x^{3} - 5 x^{2} + 12 \log {\left (3 \right )}} - \frac {x}{\log {\left (5 \right )}} \end {gather*}

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Giac [A]
time = 0.40, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\frac {16 \, x^{2} \log \left (5\right ) \log \left (3\right )}{x^{3} - 5 \, x^{2} + 12 \, \log \left (3\right )} - x^{2} + x}{\log \left (5\right )} \end {gather*}

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Mupad [B]
time = 0.38, size = 38, normalized size = 1.00 \begin {gather*} \frac {x^2}{\ln \left (5\right )}-\frac {x}{\ln \left (5\right )}-\frac {16\,x^2\,\ln \left (3\right )}{x^3-5\,x^2+12\,\ln \left (3\right )} \end {gather*}

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3.96.63 \(\int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+(120 x^2-264 x^3+48 x^4) \log (3)+(-144+288 x) \log ^2(3)+(16 x^4 \log (3)-384 x \log ^2(3)) \log (5)}{(25 x^4-10 x^5+x^6+(-120 x^2+24 x^3) \log (3)+144 \log ^2(3)) \log (5)} \, dx\) [9563]