3.40.93 \(\int \frac {-50 x^4-50 x \log (x)+(-100-25 x-300 x^3-75 x^4) \log (4+x)}{(4 x^7+x^8+(8 x^4+2 x^5) \log (x)+(4 x+x^2) \log ^2(x)) \log ^3(4+x)} \, dx\)

Optimal. Leaf size=16 \[ \frac {25}{\left (x^3+\log (x)\right ) \log ^2(4+x)} \]

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Rubi [F]  time = 1.42, 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 {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-50*x^4 - 50*x*Log[x] + (-100 - 25*x - 300*x^3 - 75*x^4)*Log[4 + x])/((4*x^7 + x^8 + (8*x^4 + 2*x^5)*Log[
x] + (4*x + x^2)*Log[x]^2)*Log[4 + x]^3),x]

[Out]

-50*Defer[Int][1/((4 + x)*(x^3 + Log[x])*Log[4 + x]^3), x] - 25*Defer[Int][1/(x*(x^3 + Log[x])^2*Log[4 + x]^2)
, x] - 75*Defer[Int][x^2/((x^3 + Log[x])^2*Log[4 + x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (-2 x^4-2 x \log (x)-\left (4+x+12 x^3+3 x^4\right ) \log (4+x)\right )}{x (4+x) \left (x^3+\log (x)\right )^2 \log ^3(4+x)} \, dx\\ &=25 \int \frac {-2 x^4-2 x \log (x)-\left (4+x+12 x^3+3 x^4\right ) \log (4+x)}{x (4+x) \left (x^3+\log (x)\right )^2 \log ^3(4+x)} \, dx\\ &=25 \int \left (-\frac {2}{(4+x) \left (x^3+\log (x)\right ) \log ^3(4+x)}+\frac {-1-3 x^3}{x \left (x^3+\log (x)\right )^2 \log ^2(4+x)}\right ) \, dx\\ &=25 \int \frac {-1-3 x^3}{x \left (x^3+\log (x)\right )^2 \log ^2(4+x)} \, dx-50 \int \frac {1}{(4+x) \left (x^3+\log (x)\right ) \log ^3(4+x)} \, dx\\ &=25 \int \left (-\frac {1}{x \left (x^3+\log (x)\right )^2 \log ^2(4+x)}-\frac {3 x^2}{\left (x^3+\log (x)\right )^2 \log ^2(4+x)}\right ) \, dx-50 \int \frac {1}{(4+x) \left (x^3+\log (x)\right ) \log ^3(4+x)} \, dx\\ &=-\left (25 \int \frac {1}{x \left (x^3+\log (x)\right )^2 \log ^2(4+x)} \, dx\right )-50 \int \frac {1}{(4+x) \left (x^3+\log (x)\right ) \log ^3(4+x)} \, dx-75 \int \frac {x^2}{\left (x^3+\log (x)\right )^2 \log ^2(4+x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 16, normalized size = 1.00 \begin {gather*} \frac {25}{\left (x^3+\log (x)\right ) \log ^2(4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-50*x^4 - 50*x*Log[x] + (-100 - 25*x - 300*x^3 - 75*x^4)*Log[4 + x])/((4*x^7 + x^8 + (8*x^4 + 2*x^5
)*Log[x] + (4*x + x^2)*Log[x]^2)*Log[4 + x]^3),x]

[Out]

25/((x^3 + Log[x])*Log[4 + x]^2)

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fricas [A]  time = 0.58, size = 16, normalized size = 1.00 \begin {gather*} \frac {25}{{\left (x^{3} + \log \relax (x)\right )} \log \left (x + 4\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x^4-300*x^3-25*x-100)*log(4+x)-50*x*log(x)-50*x^4)/((x^2+4*x)*log(x)^2+(2*x^5+8*x^4)*log(x)+x^
8+4*x^7)/log(4+x)^3,x, algorithm="fricas")

[Out]

25/((x^3 + log(x))*log(x + 4)^2)

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giac [A]  time = 0.14, size = 24, normalized size = 1.50 \begin {gather*} \frac {25}{x^{3} \log \left (x + 4\right )^{2} + \log \left (x + 4\right )^{2} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x^4-300*x^3-25*x-100)*log(4+x)-50*x*log(x)-50*x^4)/((x^2+4*x)*log(x)^2+(2*x^5+8*x^4)*log(x)+x^
8+4*x^7)/log(4+x)^3,x, algorithm="giac")

[Out]

25/(x^3*log(x + 4)^2 + log(x + 4)^2*log(x))

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maple [A]  time = 0.04, size = 17, normalized size = 1.06

method	result	size
risch	\(\frac {25}{\ln \left (4+x \right )^{2} \left (\ln \relax (x )+x^{3}\right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-75*x^4-300*x^3-25*x-100)*ln(4+x)-50*x*ln(x)-50*x^4)/((x^2+4*x)*ln(x)^2+(2*x^5+8*x^4)*ln(x)+x^8+4*x^7)/l
n(4+x)^3,x,method=_RETURNVERBOSE)

[Out]

25/ln(4+x)^2/(ln(x)+x^3)

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maxima [A]  time = 0.43, size = 16, normalized size = 1.00 \begin {gather*} \frac {25}{{\left (x^{3} + \log \relax (x)\right )} \log \left (x + 4\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x^4-300*x^3-25*x-100)*log(4+x)-50*x*log(x)-50*x^4)/((x^2+4*x)*log(x)^2+(2*x^5+8*x^4)*log(x)+x^
8+4*x^7)/log(4+x)^3,x, algorithm="maxima")

[Out]

25/((x^3 + log(x))*log(x + 4)^2)

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mupad [B]  time = 3.22, size = 956, normalized size = 59.75 \begin {gather*} \frac {\frac {25\,\left (27\,x^{11}+336\,x^{10}+960\,x^9+108\,x^8+1012\,x^7+2336\,x^6+54\,x^5+456\,x^4+960\,x^3+6\,x^2+48\,x+96\right )}{6\,x^2\,\left (3\,x^3+1\right )}+\frac {25\,{\ln \relax (x)}^2\,\left (27\,x^5+120\,x^4+96\,x^3+4\,x+32\right )}{6\,x^2\,\left (3\,x^3+1\right )}-\frac {25\,\ln \relax (x)\,\left (54\,x^8+420\,x^7+768\,x^6+27\,x^5+140\,x^4+112\,x^3-12\,x-48\right )}{3\,x^2\,\left (3\,x^3+1\right )}}{x^9+3\,x^6\,\ln \relax (x)+3\,x^3\,{\ln \relax (x)}^2+{\ln \relax (x)}^3}-\frac {\frac {25\,\ln \relax (x)\,\left (486\,x^{11}+2520\,x^{10}+2304\,x^9+243\,x^8+1320\,x^7+2112\,x^6+54\,x^5+292\,x^4+640\,x^3+8\,x+64\right )}{6\,x^2\,{\left (3\,x^3+1\right )}^3}-\frac {25\,\left (486\,x^{14}+5040\,x^{13}+11520\,x^{12}+891\,x^{11}+6240\,x^{10}+9120\,x^9+378\,x^8+2012\,x^7+1376\,x^6+54\,x^5+128\,x^4-416\,x^3-24\,x-96\right )}{12\,x^2\,{\left (3\,x^3+1\right )}^3}+\frac {25\,{\ln \relax (x)}^2\,\left (360\,x^7+576\,x^6-81\,x^5-192\,x^4+384\,x^3+4\,x+64\right )}{12\,x^2\,{\left (3\,x^3+1\right )}^3}}{x^6+2\,x^3\,\ln \relax (x)+{\ln \relax (x)}^2}-\frac {\frac {1750\,x^{13}}{27}+\frac {3200\,x^{12}}{27}-\frac {75\,x^{11}}{4}-\frac {1250\,x^{10}}{27}+\frac {3200\,x^9}{27}-\frac {25\,x^8}{12}-\frac {175\,x^7}{81}+\frac {400\,x^6}{9}-\frac {25\,x^5}{36}-\frac {1025\,x^4}{729}+\frac {5600\,x^3}{729}+\frac {25\,x}{729}+\frac {400}{729}}{x^{17}+\frac {5\,x^{14}}{3}+\frac {10\,x^{11}}{9}+\frac {10\,x^8}{27}+\frac {5\,x^5}{81}+\frac {x^2}{243}}+\frac {\frac {25}{\ln \relax (x)+x^3}+\frac {25\,\ln \left (x+4\right )\,\left (3\,x^3+1\right )\,\left (x+4\right )}{2\,x\,{\left (\ln \relax (x)+x^3\right )}^2}}{{\ln \left (x+4\right )}^2}+\frac {\frac {25\,\left (4374\,x^{17}+30240\,x^{16}+34560\,x^{15}+2916\,x^{14}+21600\,x^{13}+46656\,x^{12}+2187\,x^{11}+12816\,x^{10}+25920\,x^9+486\,x^8+2980\,x^7+7424\,x^6+54\,x^5+344\,x^4+1088\,x^3+8\,x+64\right )}{12\,x^2\,{\left (3\,x^3+1\right )}^5}+\frac {25\,{\ln \relax (x)}^2\,\left (4320\,x^{10}+8640\,x^9-1458\,x^8-5832\,x^7+6912\,x^6+243\,x^5+504\,x^4+1728\,x^3+4\,x+128\right )}{12\,x^2\,{\left (3\,x^3+1\right )}^5}+\frac {25\,\ln \relax (x)\,\left (7560\,x^{13}+13824\,x^{12}-2187\,x^{11}-5400\,x^{10}+13824\,x^9-243\,x^8-252\,x^7+5184\,x^6-81\,x^5-164\,x^4+896\,x^3+4\,x+64\right )}{6\,x^2\,{\left (3\,x^3+1\right )}^5}}{\ln \relax (x)+x^3}-\frac {\frac {25\,\left (3\,x^3+1\right )\,\left (x+4\right )}{2\,x\,{\left (\ln \relax (x)+x^3\right )}^2}+\frac {25\,\ln \left (x+4\right )\,\left (x+4\right )\,\left (2\,x+4\,\ln \relax (x)-24\,x^3\,\ln \relax (x)-9\,x^4\,\ln \relax (x)+52\,x^3+12\,x^4+48\,x^6+9\,x^7+8\right )}{2\,x^2\,{\left (\ln \relax (x)+x^3\right )}^3}}{\ln \left (x+4\right )}-\frac {\ln \relax (x)\,\left (\frac {1000\,x^{10}}{27}+\frac {2000\,x^9}{27}-\frac {25\,x^8}{2}-50\,x^7+\frac {1600\,x^6}{27}+\frac {25\,x^5}{12}+\frac {350\,x^4}{81}+\frac {400\,x^3}{27}+\frac {25\,x}{729}+\frac {800}{729}\right )}{x^{17}+\frac {5\,x^{14}}{3}+\frac {10\,x^{11}}{9}+\frac {10\,x^8}{27}+\frac {5\,x^5}{81}+\frac {x^2}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + 4)*(25*x + 300*x^3 + 75*x^4 + 100) + 50*x*log(x) + 50*x^4)/(log(x + 4)^3*(log(x)*(8*x^4 + 2*x^5)
+ log(x)^2*(4*x + x^2) + 4*x^7 + x^8)),x)

[Out]

((25*(48*x + 6*x^2 + 960*x^3 + 456*x^4 + 54*x^5 + 2336*x^6 + 1012*x^7 + 108*x^8 + 960*x^9 + 336*x^10 + 27*x^11
+ 96))/(6*x^2*(3*x^3 + 1)) + (25*log(x)^2*(4*x + 96*x^3 + 120*x^4 + 27*x^5 + 32))/(6*x^2*(3*x^3 + 1)) - (25*l
og(x)*(112*x^3 - 12*x + 140*x^4 + 27*x^5 + 768*x^6 + 420*x^7 + 54*x^8 - 48))/(3*x^2*(3*x^3 + 1)))/(3*x^6*log(x
) + log(x)^3 + 3*x^3*log(x)^2 + x^9) - ((25*log(x)*(8*x + 640*x^3 + 292*x^4 + 54*x^5 + 2112*x^6 + 1320*x^7 + 2
43*x^8 + 2304*x^9 + 2520*x^10 + 486*x^11 + 64))/(6*x^2*(3*x^3 + 1)^3) - (25*(128*x^4 - 416*x^3 - 24*x + 54*x^5
+ 1376*x^6 + 2012*x^7 + 378*x^8 + 9120*x^9 + 6240*x^10 + 891*x^11 + 11520*x^12 + 5040*x^13 + 486*x^14 - 96))/
(12*x^2*(3*x^3 + 1)^3) + (25*log(x)^2*(4*x + 384*x^3 - 192*x^4 - 81*x^5 + 576*x^6 + 360*x^7 + 64))/(12*x^2*(3*
x^3 + 1)^3))/(2*x^3*log(x) + log(x)^2 + x^6) - ((25*x)/729 + (5600*x^3)/729 - (1025*x^4)/729 - (25*x^5)/36 + (
400*x^6)/9 - (175*x^7)/81 - (25*x^8)/12 + (3200*x^9)/27 - (1250*x^10)/27 - (75*x^11)/4 + (3200*x^12)/27 + (175
0*x^13)/27 + 400/729)/(x^2/243 + (5*x^5)/81 + (10*x^8)/27 + (10*x^11)/9 + (5*x^14)/3 + x^17) + (25/(log(x) + x
^3) + (25*log(x + 4)*(3*x^3 + 1)*(x + 4))/(2*x*(log(x) + x^3)^2))/log(x + 4)^2 + ((25*(8*x + 1088*x^3 + 344*x^
4 + 54*x^5 + 7424*x^6 + 2980*x^7 + 486*x^8 + 25920*x^9 + 12816*x^10 + 2187*x^11 + 46656*x^12 + 21600*x^13 + 29
16*x^14 + 34560*x^15 + 30240*x^16 + 4374*x^17 + 64))/(12*x^2*(3*x^3 + 1)^5) + (25*log(x)^2*(4*x + 1728*x^3 + 5
04*x^4 + 243*x^5 + 6912*x^6 - 5832*x^7 - 1458*x^8 + 8640*x^9 + 4320*x^10 + 128))/(12*x^2*(3*x^3 + 1)^5) + (25*
log(x)*(4*x + 896*x^3 - 164*x^4 - 81*x^5 + 5184*x^6 - 252*x^7 - 243*x^8 + 13824*x^9 - 5400*x^10 - 2187*x^11 +
13824*x^12 + 7560*x^13 + 64))/(6*x^2*(3*x^3 + 1)^5))/(log(x) + x^3) - ((25*(3*x^3 + 1)*(x + 4))/(2*x*(log(x) +
x^3)^2) + (25*log(x + 4)*(x + 4)*(2*x + 4*log(x) - 24*x^3*log(x) - 9*x^4*log(x) + 52*x^3 + 12*x^4 + 48*x^6 +
9*x^7 + 8))/(2*x^2*(log(x) + x^3)^3))/log(x + 4) - (log(x)*((25*x)/729 + (400*x^3)/27 + (350*x^4)/81 + (25*x^5
)/12 + (1600*x^6)/27 - 50*x^7 - (25*x^8)/2 + (2000*x^9)/27 + (1000*x^10)/27 + 800/729))/(x^2/243 + (5*x^5)/81
+ (10*x^8)/27 + (10*x^11)/9 + (5*x^14)/3 + x^17)

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sympy [A]  time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{\left (x^{3} + \log {\relax (x )}\right ) \log {\left (x + 4 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x**4-300*x**3-25*x-100)*ln(4+x)-50*x*ln(x)-50*x**4)/((x**2+4*x)*ln(x)**2+(2*x**5+8*x**4)*ln(x)
+x**8+4*x**7)/ln(4+x)**3,x)

[Out]

25/((x**3 + log(x))*log(x + 4)**2)

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