3.73.46 \(\int (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+(-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5) \log (x)+(18 x-144 x^2+376 x^3-400 x^4+150 x^5) \log ^2(x)) \, dx\)

Optimal. Leaf size=21 \[ (2+x-(-1+x) x (x+(-3+5 x) \log (x)))^2 \]

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Rubi [B]  time = 0.21, antiderivative size = 112, normalized size of antiderivative = 5.33, number of steps used = 21, number of rules used = 4, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {2356, 2295, 2304, 2305} \begin {gather*} x^6+25 x^6 \log ^2(x)+10 x^6 \log (x)-2 x^5-80 x^5 \log ^2(x)-26 x^5 \log (x)-x^4+94 x^4 \log ^2(x)+12 x^4 \log (x)-2 x^3-48 x^3 \log ^2(x)-10 x^3 \log (x)+5 x^2+9 x^2 \log ^2(x)+26 x^2 \log (x)+4 x-12 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-8 + 36*x - 16*x^2 + 8*x^3 - 36*x^4 + 16*x^5 + (-12 + 70*x - 126*x^2 + 236*x^3 - 290*x^4 + 110*x^5)*Log[x]
+ (18*x - 144*x^2 + 376*x^3 - 400*x^4 + 150*x^5)*Log[x]^2,x]

[Out]

4*x + 5*x^2 - 2*x^3 - x^4 - 2*x^5 + x^6 - 12*x*Log[x] + 26*x^2*Log[x] - 10*x^3*Log[x] + 12*x^4*Log[x] - 26*x^5
*Log[x] + 10*x^6*Log[x] + 9*x^2*Log[x]^2 - 48*x^3*Log[x]^2 + 94*x^4*Log[x]^2 - 80*x^5*Log[x]^2 + 25*x^6*Log[x]
^2

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-8 x+18 x^2-\frac {16 x^3}{3}+2 x^4-\frac {36 x^5}{5}+\frac {8 x^6}{3}+\int \left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x) \, dx+\int \left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x) \, dx\\ &=-8 x+18 x^2-\frac {16 x^3}{3}+2 x^4-\frac {36 x^5}{5}+\frac {8 x^6}{3}+\int \left (-12 \log (x)+70 x \log (x)-126 x^2 \log (x)+236 x^3 \log (x)-290 x^4 \log (x)+110 x^5 \log (x)\right ) \, dx+\int \left (18 x \log ^2(x)-144 x^2 \log ^2(x)+376 x^3 \log ^2(x)-400 x^4 \log ^2(x)+150 x^5 \log ^2(x)\right ) \, dx\\ &=-8 x+18 x^2-\frac {16 x^3}{3}+2 x^4-\frac {36 x^5}{5}+\frac {8 x^6}{3}-12 \int \log (x) \, dx+18 \int x \log ^2(x) \, dx+70 \int x \log (x) \, dx+110 \int x^5 \log (x) \, dx-126 \int x^2 \log (x) \, dx-144 \int x^2 \log ^2(x) \, dx+150 \int x^5 \log ^2(x) \, dx+236 \int x^3 \log (x) \, dx-290 \int x^4 \log (x) \, dx+376 \int x^3 \log ^2(x) \, dx-400 \int x^4 \log ^2(x) \, dx\\ &=4 x+\frac {x^2}{2}+\frac {26 x^3}{3}-\frac {51 x^4}{4}+\frac {22 x^5}{5}-\frac {7 x^6}{18}-12 x \log (x)+35 x^2 \log (x)-42 x^3 \log (x)+59 x^4 \log (x)-58 x^5 \log (x)+\frac {55}{3} x^6 \log (x)+9 x^2 \log ^2(x)-48 x^3 \log ^2(x)+94 x^4 \log ^2(x)-80 x^5 \log ^2(x)+25 x^6 \log ^2(x)-18 \int x \log (x) \, dx-50 \int x^5 \log (x) \, dx+96 \int x^2 \log (x) \, dx+160 \int x^4 \log (x) \, dx-188 \int x^3 \log (x) \, dx\\ &=4 x+5 x^2-2 x^3-x^4-2 x^5+x^6-12 x \log (x)+26 x^2 \log (x)-10 x^3 \log (x)+12 x^4 \log (x)-26 x^5 \log (x)+10 x^6 \log (x)+9 x^2 \log ^2(x)-48 x^3 \log ^2(x)+94 x^4 \log ^2(x)-80 x^5 \log ^2(x)+25 x^6 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 112, normalized size = 5.33 \begin {gather*} 4 x+5 x^2-2 x^3-x^4-2 x^5+x^6-12 x \log (x)+26 x^2 \log (x)-10 x^3 \log (x)+12 x^4 \log (x)-26 x^5 \log (x)+10 x^6 \log (x)+9 x^2 \log ^2(x)-48 x^3 \log ^2(x)+94 x^4 \log ^2(x)-80 x^5 \log ^2(x)+25 x^6 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-8 + 36*x - 16*x^2 + 8*x^3 - 36*x^4 + 16*x^5 + (-12 + 70*x - 126*x^2 + 236*x^3 - 290*x^4 + 110*x^5)*
Log[x] + (18*x - 144*x^2 + 376*x^3 - 400*x^4 + 150*x^5)*Log[x]^2,x]

[Out]

4*x + 5*x^2 - 2*x^3 - x^4 - 2*x^5 + x^6 - 12*x*Log[x] + 26*x^2*Log[x] - 10*x^3*Log[x] + 12*x^4*Log[x] - 26*x^5
*Log[x] + 10*x^6*Log[x] + 9*x^2*Log[x]^2 - 48*x^3*Log[x]^2 + 94*x^4*Log[x]^2 - 80*x^5*Log[x]^2 + 25*x^6*Log[x]
^2

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fricas [B]  time = 0.65, size = 91, normalized size = 4.33 \begin {gather*} x^{6} - 2 \, x^{5} - x^{4} - 2 \, x^{3} + {\left (25 \, x^{6} - 80 \, x^{5} + 94 \, x^{4} - 48 \, x^{3} + 9 \, x^{2}\right )} \log \relax (x)^{2} + 5 \, x^{2} + 2 \, {\left (5 \, x^{6} - 13 \, x^{5} + 6 \, x^{4} - 5 \, x^{3} + 13 \, x^{2} - 6 \, x\right )} \log \relax (x) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4+236*x^3-126*x^2+70*x-12)*log(x)+16*
x^5-36*x^4+8*x^3-16*x^2+36*x-8,x, algorithm="fricas")

[Out]

x^6 - 2*x^5 - x^4 - 2*x^3 + (25*x^6 - 80*x^5 + 94*x^4 - 48*x^3 + 9*x^2)*log(x)^2 + 5*x^2 + 2*(5*x^6 - 13*x^5 +
6*x^4 - 5*x^3 + 13*x^2 - 6*x)*log(x) + 4*x

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giac [B]  time = 0.17, size = 112, normalized size = 5.33 \begin {gather*} 25 \, x^{6} \log \relax (x)^{2} + 10 \, x^{6} \log \relax (x) - 80 \, x^{5} \log \relax (x)^{2} + x^{6} - 26 \, x^{5} \log \relax (x) + 94 \, x^{4} \log \relax (x)^{2} - 2 \, x^{5} + 12 \, x^{4} \log \relax (x) - 48 \, x^{3} \log \relax (x)^{2} - x^{4} - 10 \, x^{3} \log \relax (x) + 9 \, x^{2} \log \relax (x)^{2} - 2 \, x^{3} + 26 \, x^{2} \log \relax (x) + 5 \, x^{2} - 12 \, x \log \relax (x) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4+236*x^3-126*x^2+70*x-12)*log(x)+16*
x^5-36*x^4+8*x^3-16*x^2+36*x-8,x, algorithm="giac")

[Out]

25*x^6*log(x)^2 + 10*x^6*log(x) - 80*x^5*log(x)^2 + x^6 - 26*x^5*log(x) + 94*x^4*log(x)^2 - 2*x^5 + 12*x^4*log
(x) - 48*x^3*log(x)^2 - x^4 - 10*x^3*log(x) + 9*x^2*log(x)^2 - 2*x^3 + 26*x^2*log(x) + 5*x^2 - 12*x*log(x) + 4
*x

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maple [B]  time = 0.03, size = 113, normalized size = 5.38

method	result	size
default	\(4 x +26 x^{2} \ln \relax (x )+9 x^{2} \ln \relax (x )^{2}+x^{6}-2 x^{5}-x^{4}-2 x^{3}+5 x^{2}-10 x^{3} \ln \relax (x )+25 x^{6} \ln \relax (x )^{2}+10 x^{6} \ln \relax (x )-26 x^{5} \ln \relax (x )+94 x^{4} \ln \relax (x )^{2}+12 x^{4} \ln \relax (x )-12 x \ln \relax (x )-80 x^{5} \ln \relax (x )^{2}-48 x^{3} \ln \relax (x )^{2}\)	\(113\)
norman	\(4 x +26 x^{2} \ln \relax (x )+9 x^{2} \ln \relax (x )^{2}+x^{6}-2 x^{5}-x^{4}-2 x^{3}+5 x^{2}-10 x^{3} \ln \relax (x )+25 x^{6} \ln \relax (x )^{2}+10 x^{6} \ln \relax (x )-26 x^{5} \ln \relax (x )+94 x^{4} \ln \relax (x )^{2}+12 x^{4} \ln \relax (x )-12 x \ln \relax (x )-80 x^{5} \ln \relax (x )^{2}-48 x^{3} \ln \relax (x )^{2}\)	\(113\)
risch	\(\left (25 x^{6}-80 x^{5}+94 x^{4}-48 x^{3}+9 x^{2}\right ) \ln \relax (x )^{2}+\left (-\frac {25}{3} x^{6}+32 x^{5}-47 x^{4}+32 x^{3}-9 x^{2}\right ) \ln \relax (x )+x^{6}-2 x^{5}-x^{4}-2 x^{3}+5 x^{2}+\left (\frac {55}{3} x^{6}-58 x^{5}+59 x^{4}-42 x^{3}+35 x^{2}-12 x \right ) \ln \relax (x )+4 x\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*ln(x)^2+(110*x^5-290*x^4+236*x^3-126*x^2+70*x-12)*ln(x)+16*x^5-36*x
^4+8*x^3-16*x^2+36*x-8,x,method=_RETURNVERBOSE)

[Out]

4*x+26*x^2*ln(x)+9*x^2*ln(x)^2+x^6-2*x^5-x^4-2*x^3+5*x^2-10*x^3*ln(x)+25*x^6*ln(x)^2+10*x^6*ln(x)-26*x^5*ln(x)
+94*x^4*ln(x)^2+12*x^4*ln(x)-12*x*ln(x)-80*x^5*ln(x)^2-48*x^3*ln(x)^2

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maxima [B]  time = 0.35, size = 147, normalized size = 7.00 \begin {gather*} \frac {25}{18} \, {\left (18 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 1\right )} x^{6} - \frac {16}{5} \, {\left (25 \, \log \relax (x)^{2} - 10 \, \log \relax (x) + 2\right )} x^{5} - \frac {7}{18} \, x^{6} + \frac {47}{4} \, {\left (8 \, \log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} x^{4} + \frac {22}{5} \, x^{5} - \frac {16}{3} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} - \frac {51}{4} \, x^{4} + \frac {9}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + \frac {26}{3} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {1}{3} \, {\left (55 \, x^{6} - 174 \, x^{5} + 177 \, x^{4} - 126 \, x^{3} + 105 \, x^{2} - 36 \, x\right )} \log \relax (x) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4+236*x^3-126*x^2+70*x-12)*log(x)+16*
x^5-36*x^4+8*x^3-16*x^2+36*x-8,x, algorithm="maxima")

[Out]

25/18*(18*log(x)^2 - 6*log(x) + 1)*x^6 - 16/5*(25*log(x)^2 - 10*log(x) + 2)*x^5 - 7/18*x^6 + 47/4*(8*log(x)^2
- 4*log(x) + 1)*x^4 + 22/5*x^5 - 16/3*(9*log(x)^2 - 6*log(x) + 2)*x^3 - 51/4*x^4 + 9/2*(2*log(x)^2 - 2*log(x)
+ 1)*x^2 + 26/3*x^3 + 1/2*x^2 + 1/3*(55*x^6 - 174*x^5 + 177*x^4 - 126*x^3 + 105*x^2 - 36*x)*log(x) + 4*x

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mupad [B]  time = 4.63, size = 112, normalized size = 5.33 \begin {gather*} 25\,x^6\,{\ln \relax (x)}^2+10\,x^6\,\ln \relax (x)+x^6-80\,x^5\,{\ln \relax (x)}^2-26\,x^5\,\ln \relax (x)-2\,x^5+94\,x^4\,{\ln \relax (x)}^2+12\,x^4\,\ln \relax (x)-x^4-48\,x^3\,{\ln \relax (x)}^2-10\,x^3\,\ln \relax (x)-2\,x^3+9\,x^2\,{\ln \relax (x)}^2+26\,x^2\,\ln \relax (x)+5\,x^2-12\,x\,\ln \relax (x)+4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(36*x + log(x)*(70*x - 126*x^2 + 236*x^3 - 290*x^4 + 110*x^5 - 12) + log(x)^2*(18*x - 144*x^2 + 376*x^3 - 4
00*x^4 + 150*x^5) - 16*x^2 + 8*x^3 - 36*x^4 + 16*x^5 - 8,x)

[Out]

4*x + 26*x^2*log(x) - 10*x^3*log(x) + 12*x^4*log(x) - 26*x^5*log(x) + 10*x^6*log(x) + 9*x^2*log(x)^2 - 48*x^3*
log(x)^2 + 94*x^4*log(x)^2 - 80*x^5*log(x)^2 + 25*x^6*log(x)^2 - 12*x*log(x) + 5*x^2 - 2*x^3 - x^4 - 2*x^5 + x
^6

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sympy [B]  time = 0.20, size = 87, normalized size = 4.14 \begin {gather*} x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + 5 x^{2} + 4 x + \left (25 x^{6} - 80 x^{5} + 94 x^{4} - 48 x^{3} + 9 x^{2}\right ) \log {\relax (x )}^{2} + \left (10 x^{6} - 26 x^{5} + 12 x^{4} - 10 x^{3} + 26 x^{2} - 12 x\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*x**5-400*x**4+376*x**3-144*x**2+18*x)*ln(x)**2+(110*x**5-290*x**4+236*x**3-126*x**2+70*x-12)*ln
(x)+16*x**5-36*x**4+8*x**3-16*x**2+36*x-8,x)

[Out]

x**6 - 2*x**5 - x**4 - 2*x**3 + 5*x**2 + 4*x + (25*x**6 - 80*x**5 + 94*x**4 - 48*x**3 + 9*x**2)*log(x)**2 + (1
0*x**6 - 26*x**5 + 12*x**4 - 10*x**3 + 26*x**2 - 12*x)*log(x)

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