3.22.34 \(\int (12 x^3+3 e x^3-4 x^4+(-24 x^3-4 e x^3+5 x^4) \log (x)-4 x^3 \log ^2(x)+16 x^3 \log ^3(x)) \, dx\)

Optimal. Leaf size=20 \[ x^4 (-1+\log (x)) \left (-4-e+x+4 \log ^2(x)\right ) \]

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Rubi [B]  time = 0.16, antiderivative size = 75, normalized size of antiderivative = 3.75, number of steps used = 12, number of rules used = 6, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {6, 1593, 43, 2334, 2305, 2304} \begin {gather*} -x^5+\frac {1}{4} (6+e) x^4+\frac {3}{4} (4+e) x^4-\frac {x^4}{2}+4 x^4 \log ^3(x)-4 x^4 \log ^2(x)+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[12*x^3 + 3*E*x^3 - 4*x^4 + (-24*x^3 - 4*E*x^3 + 5*x^4)*Log[x] - 4*x^3*Log[x]^2 + 16*x^3*Log[x]^3,x]

[Out]

-1/2*x^4 + (3*(4 + E)*x^4)/4 + ((6 + E)*x^4)/4 - x^5 + 2*x^4*Log[x] - ((6 + E)*x^4 - x^5)*Log[x] - 4*x^4*Log[x
]^2 + 4*x^4*Log[x]^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left ((12+3 e) x^3-4 x^4+\left (-24 x^3-4 e x^3+5 x^4\right ) \log (x)-4 x^3 \log ^2(x)+16 x^3 \log ^3(x)\right ) \, dx\\ &=\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}-4 \int x^3 \log ^2(x) \, dx+16 \int x^3 \log ^3(x) \, dx+\int \left (-24 x^3-4 e x^3+5 x^4\right ) \log (x) \, dx\\ &=\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}-x^4 \log ^2(x)+4 x^4 \log ^3(x)+2 \int x^3 \log (x) \, dx-12 \int x^3 \log ^2(x) \, dx+\int \left ((-24-4 e) x^3+5 x^4\right ) \log (x) \, dx\\ &=-\frac {x^4}{8}+\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}+\frac {1}{2} x^4 \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)+6 \int x^3 \log (x) \, dx+\int x^3 (-24-4 e+5 x) \log (x) \, dx\\ &=-\frac {x^4}{2}+\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)-\int x^3 (-6-e+x) \, dx\\ &=-\frac {x^4}{2}+\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)-\int \left (-\left ((6+e) x^3\right )+x^4\right ) \, dx\\ &=-\frac {x^4}{2}+\frac {3}{4} (4+e) x^4+\frac {1}{4} (6+e) x^4-x^5+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 55, normalized size = 2.75 \begin {gather*} 4 x^4+e x^4-x^5-4 x^4 \log (x)-e x^4 \log (x)+x^5 \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[12*x^3 + 3*E*x^3 - 4*x^4 + (-24*x^3 - 4*E*x^3 + 5*x^4)*Log[x] - 4*x^3*Log[x]^2 + 16*x^3*Log[x]^3,x]

[Out]

4*x^4 + E*x^4 - x^5 - 4*x^4*Log[x] - E*x^4*Log[x] + x^5*Log[x] - 4*x^4*Log[x]^2 + 4*x^4*Log[x]^3

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fricas [B]  time = 0.67, size = 54, normalized size = 2.70 \begin {gather*} 4 \, x^{4} \log \relax (x)^{3} - 4 \, x^{4} \log \relax (x)^{2} - x^{5} + x^{4} e + 4 \, x^{4} + {\left (x^{5} - x^{4} e - 4 \, x^{4}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(x)^3-4*x^3*log(x)^2+(-4*x^3*exp(1)+5*x^4-24*x^3)*log(x)+3*x^3*exp(1)-4*x^4+12*x^3,x, algo
rithm="fricas")

[Out]

4*x^4*log(x)^3 - 4*x^4*log(x)^2 - x^5 + x^4*e + 4*x^4 + (x^5 - x^4*e - 4*x^4)*log(x)

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giac [B]  time = 0.25, size = 57, normalized size = 2.85 \begin {gather*} 4 \, x^{4} \log \relax (x)^{3} + x^{5} \log \relax (x) - x^{4} e \log \relax (x) - 4 \, x^{4} \log \relax (x)^{2} - x^{5} + x^{4} e - 4 \, x^{4} \log \relax (x) + 4 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(x)^3-4*x^3*log(x)^2+(-4*x^3*exp(1)+5*x^4-24*x^3)*log(x)+3*x^3*exp(1)-4*x^4+12*x^3,x, algo
rithm="giac")

[Out]

4*x^4*log(x)^3 + x^5*log(x) - x^4*e*log(x) - 4*x^4*log(x)^2 - x^5 + x^4*e - 4*x^4*log(x) + 4*x^4

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maple [B]  time = 0.05, size = 51, normalized size = 2.55

method	result	size
norman	\(x^{5} \ln \relax (x )+\left ({\mathrm e}+4\right ) x^{4}+\left (-{\mathrm e}-4\right ) x^{4} \ln \relax (x )-x^{5}-4 x^{4} \ln \relax (x )^{2}+4 x^{4} \ln \relax (x )^{3}\)	\(51\)
default	\(-{\mathrm e} \ln \relax (x ) x^{4}+x^{4} {\mathrm e}+x^{5} \ln \relax (x )-x^{5}-4 x^{4} \ln \relax (x )+4 x^{4}-4 x^{4} \ln \relax (x )^{2}+4 x^{4} \ln \relax (x )^{3}\)	\(58\)
risch	\(-{\mathrm e} \ln \relax (x ) x^{4}+x^{4} {\mathrm e}+x^{5} \ln \relax (x )-x^{5}-4 x^{4} \ln \relax (x )+4 x^{4}-4 x^{4} \ln \relax (x )^{2}+4 x^{4} \ln \relax (x )^{3}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x^3*ln(x)^3-4*x^3*ln(x)^2+(-4*x^3*exp(1)+5*x^4-24*x^3)*ln(x)+3*x^3*exp(1)-4*x^4+12*x^3,x,method=_RETURN
VERBOSE)

[Out]

x^5*ln(x)+(exp(1)+4)*x^4+(-exp(1)-4)*x^4*ln(x)-x^5-4*x^4*ln(x)^2+4*x^4*ln(x)^3

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maxima [B]  time = 0.34, size = 86, normalized size = 4.30 \begin {gather*} \frac {1}{8} \, {\left (32 \, \log \relax (x)^{3} - 24 \, \log \relax (x)^{2} + 12 \, \log \relax (x) - 3\right )} x^{4} - \frac {1}{8} \, {\left (8 \, \log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} x^{4} - x^{5} + \frac {1}{4} \, x^{4} {\left (e + 6\right )} + \frac {3}{4} \, x^{4} e + 3 \, x^{4} + {\left (x^{5} - x^{4} e - 6 \, x^{4}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(x)^3-4*x^3*log(x)^2+(-4*x^3*exp(1)+5*x^4-24*x^3)*log(x)+3*x^3*exp(1)-4*x^4+12*x^3,x, algo
rithm="maxima")

[Out]

1/8*(32*log(x)^3 - 24*log(x)^2 + 12*log(x) - 3)*x^4 - 1/8*(8*log(x)^2 - 4*log(x) + 1)*x^4 - x^5 + 1/4*x^4*(e +
6) + 3/4*x^4*e + 3*x^4 + (x^5 - x^4*e - 6*x^4)*log(x)

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mupad [B]  time = 1.26, size = 21, normalized size = 1.05 \begin {gather*} x^4\,\left (\ln \relax (x)-1\right )\,\left (4\,{\ln \relax (x)}^2+x-\mathrm {e}-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x^3*log(x)^3 - 4*x^3*log(x)^2 + 3*x^3*exp(1) - log(x)*(4*x^3*exp(1) + 24*x^3 - 5*x^4) + 12*x^3 - 4*x^4,
x)

[Out]

x^4*(log(x) - 1)*(x - exp(1) + 4*log(x)^2 - 4)

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sympy [B]  time = 0.16, size = 49, normalized size = 2.45 \begin {gather*} - x^{5} + 4 x^{4} \log {\relax (x )}^{3} - 4 x^{4} \log {\relax (x )}^{2} + x^{4} \left (e + 4\right ) + \left (x^{5} - 4 x^{4} - e x^{4}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x**3*ln(x)**3-4*x**3*ln(x)**2+(-4*x**3*exp(1)+5*x**4-24*x**3)*ln(x)+3*x**3*exp(1)-4*x**4+12*x**3,
x)

[Out]

-x**5 + 4*x**4*log(x)**3 - 4*x**4*log(x)**2 + x**4*(E + 4) + (x**5 - 4*x**4 - E*x**4)*log(x)

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