3.101.30 \(\int \frac {1144 x^3+168 x^4-48 x^5-39754 x^6-29862 x^7-5292 x^8+504 x^9+144 x^{10}+(168 x-24 x^2-2288 x^3-18186 x^4-7794 x^5+216 x^7) \log (x)+(-336 x-2574 x^2-378 x^3+108 x^4) \log ^2(x)+(-126+18 x) \log ^3(x)}{3087 x^6+2646 x^7+756 x^8+72 x^9+(1323 x^4+756 x^5+108 x^6) \log (x)+(189 x^2+54 x^3) \log ^2(x)+9 \log ^3(x)} \, dx\)

Optimal. Leaf size=24 \[ \left (7-x-\frac {4}{3 \left (7+2 x+\frac {\log (x)}{x^2}\right )}\right )^2 \]

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Rubi [F]  time = 1.58, 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 {1144 x^3+168 x^4-48 x^5-39754 x^6-29862 x^7-5292 x^8+504 x^9+144 x^{10}+\left (168 x-24 x^2-2288 x^3-18186 x^4-7794 x^5+216 x^7\right ) \log (x)+\left (-336 x-2574 x^2-378 x^3+108 x^4\right ) \log ^2(x)+(-126+18 x) \log ^3(x)}{3087 x^6+2646 x^7+756 x^8+72 x^9+\left (1323 x^4+756 x^5+108 x^6\right ) \log (x)+\left (189 x^2+54 x^3\right ) \log ^2(x)+9 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1144*x^3 + 168*x^4 - 48*x^5 - 39754*x^6 - 29862*x^7 - 5292*x^8 + 504*x^9 + 144*x^10 + (168*x - 24*x^2 - 2
288*x^3 - 18186*x^4 - 7794*x^5 + 216*x^7)*Log[x] + (-336*x - 2574*x^2 - 378*x^3 + 108*x^4)*Log[x]^2 + (-126 +
18*x)*Log[x]^3)/(3087*x^6 + 2646*x^7 + 756*x^8 + 72*x^9 + (1323*x^4 + 756*x^5 + 108*x^6)*Log[x] + (189*x^2 + 5
4*x^3)*Log[x]^2 + 9*Log[x]^3),x]

[Out]

(7 - x)^2 - (32*Defer[Int][x^3/(7*x^2 + 2*x^3 + Log[x])^3, x])/9 - (448*Defer[Int][x^5/(7*x^2 + 2*x^3 + Log[x]
)^3, x])/9 - (64*Defer[Int][x^6/(7*x^2 + 2*x^3 + Log[x])^3, x])/3 + (56*Defer[Int][x/(7*x^2 + 2*x^3 + Log[x])^
2, x])/3 - (8*Defer[Int][x^2/(7*x^2 + 2*x^3 + Log[x])^2, x])/3 + (2416*Defer[Int][x^3/(7*x^2 + 2*x^3 + Log[x])
^2, x])/9 + (224*Defer[Int][x^4/(7*x^2 + 2*x^3 + Log[x])^2, x])/3 - 16*Defer[Int][x^5/(7*x^2 + 2*x^3 + Log[x])
^2, x] - (112*Defer[Int][x/(7*x^2 + 2*x^3 + Log[x]), x])/3 + 8*Defer[Int][x^2/(7*x^2 + 2*x^3 + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x^3 \left (572+84 x-24 x^2-19877 x^3-14931 x^4-2646 x^5+252 x^6+72 x^7\right )+x \left (84-12 x-1144 x^2-9093 x^3-3897 x^4+108 x^6\right ) \log (x)+3 x \left (-56-429 x-63 x^2+18 x^3\right ) \log ^2(x)+9 (-7+x) \log ^3(x)\right )}{9 \left (x^2 (7+2 x)+\log (x)\right )^3} \, dx\\ &=\frac {2}{9} \int \frac {x^3 \left (572+84 x-24 x^2-19877 x^3-14931 x^4-2646 x^5+252 x^6+72 x^7\right )+x \left (84-12 x-1144 x^2-9093 x^3-3897 x^4+108 x^6\right ) \log (x)+3 x \left (-56-429 x-63 x^2+18 x^3\right ) \log ^2(x)+9 (-7+x) \log ^3(x)}{\left (x^2 (7+2 x)+\log (x)\right )^3} \, dx\\ &=\frac {2}{9} \int \left (9 (-7+x)+\frac {16 x^3 \left (-1-14 x^2-6 x^3\right )}{\left (7 x^2+2 x^3+\log (x)\right )^3}-\frac {4 x \left (-21+3 x-302 x^2-84 x^3+18 x^4\right )}{\left (7 x^2+2 x^3+\log (x)\right )^2}+\frac {12 x (-14+3 x)}{7 x^2+2 x^3+\log (x)}\right ) \, dx\\ &=(7-x)^2-\frac {8}{9} \int \frac {x \left (-21+3 x-302 x^2-84 x^3+18 x^4\right )}{\left (7 x^2+2 x^3+\log (x)\right )^2} \, dx+\frac {8}{3} \int \frac {x (-14+3 x)}{7 x^2+2 x^3+\log (x)} \, dx+\frac {32}{9} \int \frac {x^3 \left (-1-14 x^2-6 x^3\right )}{\left (7 x^2+2 x^3+\log (x)\right )^3} \, dx\\ &=(7-x)^2-\frac {8}{9} \int \left (-\frac {21 x}{\left (7 x^2+2 x^3+\log (x)\right )^2}+\frac {3 x^2}{\left (7 x^2+2 x^3+\log (x)\right )^2}-\frac {302 x^3}{\left (7 x^2+2 x^3+\log (x)\right )^2}-\frac {84 x^4}{\left (7 x^2+2 x^3+\log (x)\right )^2}+\frac {18 x^5}{\left (7 x^2+2 x^3+\log (x)\right )^2}\right ) \, dx+\frac {8}{3} \int \left (-\frac {14 x}{7 x^2+2 x^3+\log (x)}+\frac {3 x^2}{7 x^2+2 x^3+\log (x)}\right ) \, dx+\frac {32}{9} \int \left (-\frac {x^3}{\left (7 x^2+2 x^3+\log (x)\right )^3}-\frac {14 x^5}{\left (7 x^2+2 x^3+\log (x)\right )^3}-\frac {6 x^6}{\left (7 x^2+2 x^3+\log (x)\right )^3}\right ) \, dx\\ &=(7-x)^2-\frac {8}{3} \int \frac {x^2}{\left (7 x^2+2 x^3+\log (x)\right )^2} \, dx-\frac {32}{9} \int \frac {x^3}{\left (7 x^2+2 x^3+\log (x)\right )^3} \, dx+8 \int \frac {x^2}{7 x^2+2 x^3+\log (x)} \, dx-16 \int \frac {x^5}{\left (7 x^2+2 x^3+\log (x)\right )^2} \, dx+\frac {56}{3} \int \frac {x}{\left (7 x^2+2 x^3+\log (x)\right )^2} \, dx-\frac {64}{3} \int \frac {x^6}{\left (7 x^2+2 x^3+\log (x)\right )^3} \, dx-\frac {112}{3} \int \frac {x}{7 x^2+2 x^3+\log (x)} \, dx-\frac {448}{9} \int \frac {x^5}{\left (7 x^2+2 x^3+\log (x)\right )^3} \, dx+\frac {224}{3} \int \frac {x^4}{\left (7 x^2+2 x^3+\log (x)\right )^2} \, dx+\frac {2416}{9} \int \frac {x^3}{\left (7 x^2+2 x^3+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 49, normalized size = 2.04 \begin {gather*} \frac {1}{9} x \left (-126+9 x+\frac {16 x^3}{\left (x^2 (7+2 x)+\log (x)\right )^2}+\frac {24 (-7+x) x}{x^2 (7+2 x)+\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1144*x^3 + 168*x^4 - 48*x^5 - 39754*x^6 - 29862*x^7 - 5292*x^8 + 504*x^9 + 144*x^10 + (168*x - 24*x
^2 - 2288*x^3 - 18186*x^4 - 7794*x^5 + 216*x^7)*Log[x] + (-336*x - 2574*x^2 - 378*x^3 + 108*x^4)*Log[x]^2 + (-
126 + 18*x)*Log[x]^3)/(3087*x^6 + 2646*x^7 + 756*x^8 + 72*x^9 + (1323*x^4 + 756*x^5 + 108*x^6)*Log[x] + (189*x
^2 + 54*x^3)*Log[x]^2 + 9*Log[x]^3),x]

[Out]

(x*(-126 + 9*x + (16*x^3)/(x^2*(7 + 2*x) + Log[x])^2 + (24*(-7 + x)*x)/(x^2*(7 + 2*x) + Log[x])))/9

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fricas [B]  time = 1.15, size = 103, normalized size = 4.29 \begin {gather*} \frac {36 \, x^{8} - 252 \, x^{7} - 3039 \, x^{6} - 6342 \, x^{5} - 1160 \, x^{4} + 9 \, {\left (x^{2} - 14 \, x\right )} \log \relax (x)^{2} + 6 \, {\left (6 \, x^{5} - 63 \, x^{4} - 290 \, x^{3} - 28 \, x^{2}\right )} \log \relax (x)}{9 \, {\left (4 \, x^{6} + 28 \, x^{5} + 49 \, x^{4} + 2 \, {\left (2 \, x^{3} + 7 \, x^{2}\right )} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x-126)*log(x)^3+(108*x^4-378*x^3-2574*x^2-336*x)*log(x)^2+(216*x^7-7794*x^5-18186*x^4-2288*x^3-
24*x^2+168*x)*log(x)+144*x^10+504*x^9-5292*x^8-29862*x^7-39754*x^6-48*x^5+168*x^4+1144*x^3)/(9*log(x)^3+(54*x^
3+189*x^2)*log(x)^2+(108*x^6+756*x^5+1323*x^4)*log(x)+72*x^9+756*x^8+2646*x^7+3087*x^6),x, algorithm="fricas")

[Out]

1/9*(36*x^8 - 252*x^7 - 3039*x^6 - 6342*x^5 - 1160*x^4 + 9*(x^2 - 14*x)*log(x)^2 + 6*(6*x^5 - 63*x^4 - 290*x^3
- 28*x^2)*log(x))/(4*x^6 + 28*x^5 + 49*x^4 + 2*(2*x^3 + 7*x^2)*log(x) + log(x)^2)

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giac [B]  time = 0.32, size = 75, normalized size = 3.12 \begin {gather*} x^{2} - 14 \, x + \frac {8 \, {\left (6 \, x^{6} - 21 \, x^{5} - 145 \, x^{4} + 3 \, x^{3} \log \relax (x) - 21 \, x^{2} \log \relax (x)\right )}}{9 \, {\left (4 \, x^{6} + 28 \, x^{5} + 49 \, x^{4} + 4 \, x^{3} \log \relax (x) + 14 \, x^{2} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x-126)*log(x)^3+(108*x^4-378*x^3-2574*x^2-336*x)*log(x)^2+(216*x^7-7794*x^5-18186*x^4-2288*x^3-
24*x^2+168*x)*log(x)+144*x^10+504*x^9-5292*x^8-29862*x^7-39754*x^6-48*x^5+168*x^4+1144*x^3)/(9*log(x)^3+(54*x^
3+189*x^2)*log(x)^2+(108*x^6+756*x^5+1323*x^4)*log(x)+72*x^9+756*x^8+2646*x^7+3087*x^6),x, algorithm="giac")

[Out]

x^2 - 14*x + 8/9*(6*x^6 - 21*x^5 - 145*x^4 + 3*x^3*log(x) - 21*x^2*log(x))/(4*x^6 + 28*x^5 + 49*x^4 + 4*x^3*lo
g(x) + 14*x^2*log(x) + log(x)^2)

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maple [B]  time = 0.03, size = 53, normalized size = 2.21

method	result	size
risch	\(x^{2}-14 x +\frac {8 \left (6 x^{4}-21 x^{3}-145 x^{2}+3 x \ln \relax (x )-21 \ln \relax (x )\right ) x^{2}}{9 \left (2 x^{3}+7 x^{2}+\ln \relax (x )\right )^{2}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x-126)*ln(x)^3+(108*x^4-378*x^3-2574*x^2-336*x)*ln(x)^2+(216*x^7-7794*x^5-18186*x^4-2288*x^3-24*x^2+1
68*x)*ln(x)+144*x^10+504*x^9-5292*x^8-29862*x^7-39754*x^6-48*x^5+168*x^4+1144*x^3)/(9*ln(x)^3+(54*x^3+189*x^2)
*ln(x)^2+(108*x^6+756*x^5+1323*x^4)*ln(x)+72*x^9+756*x^8+2646*x^7+3087*x^6),x,method=_RETURNVERBOSE)

[Out]

x^2-14*x+8/9*(6*x^4-21*x^3-145*x^2+3*x*ln(x)-21*ln(x))*x^2/(2*x^3+7*x^2+ln(x))^2

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maxima [B]  time = 0.46, size = 103, normalized size = 4.29 \begin {gather*} \frac {36 \, x^{8} - 252 \, x^{7} - 3039 \, x^{6} - 6342 \, x^{5} - 1160 \, x^{4} + 9 \, {\left (x^{2} - 14 \, x\right )} \log \relax (x)^{2} + 6 \, {\left (6 \, x^{5} - 63 \, x^{4} - 290 \, x^{3} - 28 \, x^{2}\right )} \log \relax (x)}{9 \, {\left (4 \, x^{6} + 28 \, x^{5} + 49 \, x^{4} + 2 \, {\left (2 \, x^{3} + 7 \, x^{2}\right )} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x-126)*log(x)^3+(108*x^4-378*x^3-2574*x^2-336*x)*log(x)^2+(216*x^7-7794*x^5-18186*x^4-2288*x^3-
24*x^2+168*x)*log(x)+144*x^10+504*x^9-5292*x^8-29862*x^7-39754*x^6-48*x^5+168*x^4+1144*x^3)/(9*log(x)^3+(54*x^
3+189*x^2)*log(x)^2+(108*x^6+756*x^5+1323*x^4)*log(x)+72*x^9+756*x^8+2646*x^7+3087*x^6),x, algorithm="maxima")

[Out]

1/9*(36*x^8 - 252*x^7 - 3039*x^6 - 6342*x^5 - 1160*x^4 + 9*(x^2 - 14*x)*log(x)^2 + 6*(6*x^5 - 63*x^4 - 290*x^3
- 28*x^2)*log(x))/(4*x^6 + 28*x^5 + 49*x^4 + 2*(2*x^3 + 7*x^2)*log(x) + log(x)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\ln \relax (x)}^2\,\left (-108\,x^4+378\,x^3+2574\,x^2+336\,x\right )+\ln \relax (x)\,\left (-216\,x^7+7794\,x^5+18186\,x^4+2288\,x^3+24\,x^2-168\,x\right )-1144\,x^3-168\,x^4+48\,x^5+39754\,x^6+29862\,x^7+5292\,x^8-504\,x^9-144\,x^{10}-{\ln \relax (x)}^3\,\left (18\,x-126\right )}{9\,{\ln \relax (x)}^3+\ln \relax (x)\,\left (108\,x^6+756\,x^5+1323\,x^4\right )+{\ln \relax (x)}^2\,\left (54\,x^3+189\,x^2\right )+3087\,x^6+2646\,x^7+756\,x^8+72\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(336*x + 2574*x^2 + 378*x^3 - 108*x^4) + log(x)*(24*x^2 - 168*x + 2288*x^3 + 18186*x^4 + 7794*x
^5 - 216*x^7) - 1144*x^3 - 168*x^4 + 48*x^5 + 39754*x^6 + 29862*x^7 + 5292*x^8 - 504*x^9 - 144*x^10 - log(x)^3
*(18*x - 126))/(9*log(x)^3 + log(x)*(1323*x^4 + 756*x^5 + 108*x^6) + log(x)^2*(189*x^2 + 54*x^3) + 3087*x^6 +
2646*x^7 + 756*x^8 + 72*x^9),x)

[Out]

int(-(log(x)^2*(336*x + 2574*x^2 + 378*x^3 - 108*x^4) + log(x)*(24*x^2 - 168*x + 2288*x^3 + 18186*x^4 + 7794*x
^5 - 216*x^7) - 1144*x^3 - 168*x^4 + 48*x^5 + 39754*x^6 + 29862*x^7 + 5292*x^8 - 504*x^9 - 144*x^10 - log(x)^3
*(18*x - 126))/(9*log(x)^3 + log(x)*(1323*x^4 + 756*x^5 + 108*x^6) + log(x)^2*(189*x^2 + 54*x^3) + 3087*x^6 +
2646*x^7 + 756*x^8 + 72*x^9), x)

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sympy [B]  time = 0.23, size = 70, normalized size = 2.92 \begin {gather*} x^{2} - 14 x + \frac {48 x^{6} - 168 x^{5} - 1160 x^{4} + \left (24 x^{3} - 168 x^{2}\right ) \log {\relax (x )}}{36 x^{6} + 252 x^{5} + 441 x^{4} + \left (36 x^{3} + 126 x^{2}\right ) \log {\relax (x )} + 9 \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x-126)*ln(x)**3+(108*x**4-378*x**3-2574*x**2-336*x)*ln(x)**2+(216*x**7-7794*x**5-18186*x**4-228
8*x**3-24*x**2+168*x)*ln(x)+144*x**10+504*x**9-5292*x**8-29862*x**7-39754*x**6-48*x**5+168*x**4+1144*x**3)/(9*
ln(x)**3+(54*x**3+189*x**2)*ln(x)**2+(108*x**6+756*x**5+1323*x**4)*ln(x)+72*x**9+756*x**8+2646*x**7+3087*x**6)
,x)

[Out]

x**2 - 14*x + (48*x**6 - 168*x**5 - 1160*x**4 + (24*x**3 - 168*x**2)*log(x))/(36*x**6 + 252*x**5 + 441*x**4 +
(36*x**3 + 126*x**2)*log(x) + 9*log(x)**2)

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