3.53.94 \(\int \frac {-24 x^3 \log (20)+e^x (x^3-2 x^4) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}[phone] e^{2+3 x}-1003290624 e^{2+4 x}[phone] e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx\) [5294]

Optimal. Leaf size=19 \[ \frac {x^4 \log (20)}{4 e^2 \left (-24+e^x\right )^8} \]

[Out]

1/4*ln(20)/exp(1)^2*x^4/(-24+exp(x))^8

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Rubi [A]
time = 35.19, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 1980, number of rules used = 17, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6820, 12, 6874, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222, 2317, 2438, 36, 31, 29, 46} \begin {gather*} \frac {x^4 \log (20)}{4 e^2 \left (24-e^x\right )^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-24*x^3*Log[20] + E^x*(x^3 - 2*x^4)*Log[20])/(-2641807540224*E^2 + 990677827584*E^(2 + x) - 165112971264*
E^(2 + 2*x) + 16052649984*E^(2 + 3*x) - 1003290624*E^(2 + 4*x) + 41803776*E^(2 + 5*x) - 1161216*E^(2 + 6*x) +
20736*E^(2 + 7*x) - 216*E^(2 + 8*x) + E^(2 + 9*x)),x]

[Out]

(x^4*Log[20])/(4*E^2*(24 - E^x)^8)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2222

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (24+e^x (-1+2 x)\right ) \log (20)}{e^2 \left (24-e^x\right )^9} \, dx\\ &=\frac {\log (20) \int \frac {x^3 \left (24+e^x (-1+2 x)\right )}{\left (24-e^x\right )^9} \, dx}{e^2}\\ &=\frac {\log (20) \int \left (-\frac {48 x^4}{\left (-24+e^x\right )^9}-\frac {x^3 (-1+2 x)}{\left (-24+e^x\right )^8}\right ) \, dx}{e^2}\\ &=-\frac {\log (20) \int \frac {x^3 (-1+2 x)}{\left (-24+e^x\right )^8} \, dx}{e^2}-\frac {(48 \log (20)) \int \frac {x^4}{\left (-24+e^x\right )^9} \, dx}{e^2}\\ &=-\frac {\log (20) \int \left (-\frac {x^3}{\left (-24+e^x\right )^8}+\frac {2 x^4}{\left (-24+e^x\right )^8}\right ) \, dx}{e^2}-\frac {(2 \log (20)) \int \frac {e^x x^4}{\left (-24+e^x\right )^9} \, dx}{e^2}+\frac {(2 \log (20)) \int \frac {x^4}{\left (-24+e^x\right )^8} \, dx}{e^2}\\ &=\frac {x^4 \log (20)}{4 e^2 \left (24-e^x\right )^8}+\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^8} \, dx}{12 e^2}-\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^7} \, dx}{12 e^2}-\frac {(2 \log (20)) \int \frac {x^4}{\left (-24+e^x\right )^8} \, dx}{e^2}\\ &=\frac {x^4 \log (20)}{4 e^2 \left (24-e^x\right )^8}+\frac {x^4 \log (20)}{84 e^2 \left (24-e^x\right )^7}-\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^7} \, dx}{288 e^2}+\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^6} \, dx}{288 e^2}+\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^7} \, dx}{21 e^2}-\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^8} \, dx}{12 e^2}+\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^7} \, dx}{12 e^2}\\ &=\frac {x^4 \log (20)}{4 e^2 \left (24-e^x\right )^8}+\frac {x^4 \log (20)}{1728 e^2 \left (24-e^x\right )^6}+\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^6} \, dx}{6912 e^2}-\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^5} \, dx}{6912 e^2}+\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^7} \, dx}{504 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^6} \, dx}{504 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^6} \, dx}{432 e^2}+\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^7} \, dx}{288 e^2}-\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^6} \, dx}{288 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^7} \, dx}{21 e^2}\\ &=-\frac {x^3 \log (20)}{3024 e^2 \left (24-e^x\right )^6}+\frac {x^4 \log (20)}{4 e^2 \left (24-e^x\right )^8}+\frac {x^4 \log (20)}{34560 e^2 \left (24-e^x\right )^5}-\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^5} \, dx}{165888 e^2}+\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^4} \, dx}{165888 e^2}-\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^6} \, dx}{12096 e^2}+\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^5} \, dx}{12096 e^2}-\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^6} \, dx}{10368 e^2}+\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^5} \, dx}{10368 e^2}+\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^5} \, dx}{8640 e^2}-\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^6} \, dx}{6912 e^2}+\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^5} \, dx}{6912 e^2}+\frac {\log (20) \int \frac {x^2}{\left (-24+e^x\right )^6} \, dx}{1008 e^2}-\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^7} \, dx}{504 e^2}+\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^6} \, dx}{504 e^2}+\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^6} \, dx}{432 e^2}\\ &=-\frac {13 x^3 \log (20)}{362880 e^2 \left (24-e^x\right )^5}+\frac {x^4 \log (20)}{4 e^2 \left (24-e^x\right )^8}+\frac {x^4 \log (20)}{663552 e^2 \left (24-e^x\right )^4}+\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^4} \, dx}{3981312 e^2}-\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^3} \, dx}{3981312 e^2}+\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^5} \, dx}{290304 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^4} \, dx}{290304 e^2}+\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^5} \, dx}{248832 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^4} \, dx}{248832 e^2}+\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^5} \, dx}{207360 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^4} \, dx}{207360 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^4} \, dx}{165888 e^2}+\frac {\log (20) \int \frac {e^x x^4}{\left (-24+e^x\right )^5} \, dx}{165888 e^2}-\frac {\log (20) \int \frac {x^4}{\left (-24+e^x\right )^4} \, dx}{165888 e^2}+\frac {\log (20) \int \frac {e^x x^2}{\left (-24+e^x\right )^6} \, dx}{24192 e^2}-\frac {\log (20) \int \frac {x^2}{\left (-24+e^x\right )^5} \, dx}{24192 e^2}-\frac {\log (20) \int \frac {x^2}{\left (-24+e^x\right )^5} \, dx}{20160 e^2}-\frac {\log (20) \int \frac {x^2}{\left (-24+e^x\right )^5} \, dx}{17280 e^2}+\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^6} \, dx}{12096 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^5} \, dx}{12096 e^2}+\frac {\log (20) \int \frac {e^x x^3}{\left (-24+e^x\right )^6} \, dx}{10368 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^5} \, dx}{10368 e^2}-\frac {\log (20) \int \frac {x^3}{\left (-24+e^x\right )^5} \, dx}{8640 e^2}-\frac {\log (20) \int \frac {x^2}{\left (-24+e^x\right )^6} \, dx}{1008 e^2}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.43, size = 19, normalized size = 1.00 \begin {gather*} \frac {x^4 \log (20)}{4 e^2 \left (-24+e^x\right )^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24*x^3*Log[20] + E^x*(x^3 - 2*x^4)*Log[20])/(-2641807540224*E^2 + 990677827584*E^(2 + x) - 1651129
71264*E^(2 + 2*x) + 16052649984*E^(2 + 3*x) - 1003290624*E^(2 + 4*x) + 41803776*E^(2 + 5*x) - 1161216*E^(2 + 6
*x) + 20736*E^(2 + 7*x) - 216*E^(2 + 8*x) + E^(2 + 9*x)),x]

[Out]

(x^4*Log[20])/(4*E^2*(-24 + E^x)^8)

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Maple [A]
time = 6.67, size = 21, normalized size = 1.11

method	result	size
risch	\(\frac {x^{4} \left (2 \ln \left (2\right )+\ln \left (5\right )\right ) {\mathrm e}^{-2}}{4 \left (-24+{\mathrm e}^{x}\right )^{8}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^4+x^3)*ln(20)*exp(x)-24*x^3*ln(20))/(exp(1)^2*exp(x)^9-216*exp(1)^2*exp(x)^8+20736*exp(1)^2*exp(x)^
7-1161216*exp(1)^2*exp(x)^6+41803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)^2*exp(x
)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641807540224*exp(1)^2),x,method=_RETURNVERBOS
E)

[Out]

1/4*x^4*(2*ln(2)+ln(5))*exp(-2)/(-24+exp(x))^8

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (15) = 30\).
time = 0.55, size = 79, normalized size = 4.16 \begin {gather*} \frac {x^{4} {\left (\log \left (5\right ) + 2 \, \log \left (2\right )\right )}}{4 \, {\left (110075314176 \, e^{2} + e^{\left (8 \, x + 2\right )} - 192 \, e^{\left (7 \, x + 2\right )} + 16128 \, e^{\left (6 \, x + 2\right )} - 774144 \, e^{\left (5 \, x + 2\right )} + 23224320 \, e^{\left (4 \, x + 2\right )} - 445906944 \, e^{\left (3 \, x + 2\right )} + 5350883328 \, e^{\left (2 \, x + 2\right )} - 36691771392 \, e^{\left (x + 2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^3)*log(20)*exp(x)-24*x^3*log(20))/(exp(1)^2*exp(x)^9-216*exp(1)^2*exp(x)^8+20736*exp(1)^2
*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)
^2*exp(x)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641807540224*exp(1)^2),x, algorithm="
maxima")

[Out]

1/4*x^4*(log(5) + 2*log(2))/(110075314176*e^2 + e^(8*x + 2) - 192*e^(7*x + 2) + 16128*e^(6*x + 2) - 774144*e^(
5*x + 2) + 23224320*e^(4*x + 2) - 445906944*e^(3*x + 2) + 5350883328*e^(2*x + 2) - 36691771392*e^(x + 2))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (15) = 30\).
time = 0.36, size = 76, normalized size = 4.00 \begin {gather*} \frac {x^{4} e^{14} \log \left (20\right )}{4 \, {\left (110075314176 \, e^{16} + e^{\left (8 \, x + 16\right )} - 192 \, e^{\left (7 \, x + 16\right )} + 16128 \, e^{\left (6 \, x + 16\right )} - 774144 \, e^{\left (5 \, x + 16\right )} + 23224320 \, e^{\left (4 \, x + 16\right )} - 445906944 \, e^{\left (3 \, x + 16\right )} + 5350883328 \, e^{\left (2 \, x + 16\right )} - 36691771392 \, e^{\left (x + 16\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^3)*log(20)*exp(x)-24*x^3*log(20))/(exp(1)^2*exp(x)^9-216*exp(1)^2*exp(x)^8+20736*exp(1)^2
*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)
^2*exp(x)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641807540224*exp(1)^2),x, algorithm="
fricas")

[Out]

1/4*x^4*e^14*log(20)/(110075314176*e^16 + e^(8*x + 16) - 192*e^(7*x + 16) + 16128*e^(6*x + 16) - 774144*e^(5*x
+ 16) + 23224320*e^(4*x + 16) - 445906944*e^(3*x + 16) + 5350883328*e^(2*x + 16) - 36691771392*e^(x + 16))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).
time = 0.17, size = 90, normalized size = 4.74 \begin {gather*} \frac {x^{4} \log {\left (20 \right )}}{4 e^{2} e^{8 x} - 768 e^{2} e^{7 x} + 64512 e^{2} e^{6 x} - 3096576 e^{2} e^{5 x} + 92897280 e^{2} e^{4 x} - 1783627776 e^{2} e^{3 x} + 21403533312 e^{2} e^{2 x} - 146767085568 e^{2} e^{x} + 440301256704 e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**4+x**3)*ln(20)*exp(x)-24*x**3*ln(20))/(exp(1)**2*exp(x)**9-216*exp(1)**2*exp(x)**8+20736*exp
(1)**2*exp(x)**7-1161216*exp(1)**2*exp(x)**6+41803776*exp(1)**2*exp(x)**5-1003290624*exp(1)**2*exp(x)**4+16052
649984*exp(1)**2*exp(x)**3-165112971264*exp(1)**2*exp(x)**2+990677827584*exp(1)**2*exp(x)-2641807540224*exp(1)
**2),x)

[Out]

x**4*log(20)/(4*exp(2)*exp(8*x) - 768*exp(2)*exp(7*x) + 64512*exp(2)*exp(6*x) - 3096576*exp(2)*exp(5*x) + 9289
7280*exp(2)*exp(4*x) - 1783627776*exp(2)*exp(3*x) + 21403533312*exp(2)*exp(2*x) - 146767085568*exp(2)*exp(x) +
440301256704*exp(2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (15) = 30\).
time = 0.39, size = 74, normalized size = 3.89 \begin {gather*} \frac {x^{4} \log \left (20\right )}{4 \, {\left (110075314176 \, e^{2} + e^{\left (8 \, x + 2\right )} - 192 \, e^{\left (7 \, x + 2\right )} + 16128 \, e^{\left (6 \, x + 2\right )} - 774144 \, e^{\left (5 \, x + 2\right )} + 23224320 \, e^{\left (4 \, x + 2\right )} - 445906944 \, e^{\left (3 \, x + 2\right )} + 5350883328 \, e^{\left (2 \, x + 2\right )} - 36691771392 \, e^{\left (x + 2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^3)*log(20)*exp(x)-24*x^3*log(20))/(exp(1)^2*exp(x)^9-216*exp(1)^2*exp(x)^8+20736*exp(1)^2
*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)
^2*exp(x)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641807540224*exp(1)^2),x, algorithm="
giac")

[Out]

1/4*x^4*log(20)/(110075314176*e^2 + e^(8*x + 2) - 192*e^(7*x + 2) + 16128*e^(6*x + 2) - 774144*e^(5*x + 2) + 2
3224320*e^(4*x + 2) - 445906944*e^(3*x + 2) + 5350883328*e^(2*x + 2) - 36691771392*e^(x + 2))

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Mupad [B]
time = 3.73, size = 58, normalized size = 3.05 \begin {gather*} \frac {x^4\,{\mathrm {e}}^{-2}\,\ln \left (20\right )}{4\,\left (5350883328\,{\mathrm {e}}^{2\,x}-445906944\,{\mathrm {e}}^{3\,x}[phone]\,{\mathrm {e}}^{4\,x}-774144\,{\mathrm {e}}^{5\,x}+16128\,{\mathrm {e}}^{6\,x}-192\,{\mathrm {e}}^{7\,x}+{\mathrm {e}}^{8\,x}-36691771392\,{\mathrm {e}}^x+110075314176\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^3*log(20) - exp(x)*log(20)*(x^3 - 2*x^4))/(2641807540224*exp(2) + 165112971264*exp(2*x)*exp(2) - 160
52649984*exp(3*x)*exp(2) + 1003290624*exp(4*x)*exp(2) - 41803776*exp(5*x)*exp(2) + 1161216*exp(6*x)*exp(2) - 2
0736*exp(7*x)*exp(2) + 216*exp(8*x)*exp(2) - exp(9*x)*exp(2) - 990677827584*exp(2)*exp(x)),x)

[Out]

(x^4*exp(-2)*log(20))/(4*(5350883328*exp(2*x) - 445906944*exp(3*x) + 23224320*exp(4*x) - 774144*exp(5*x) + 161
28*exp(6*x) - 192*exp(7*x) + exp(8*x) - 36691771392*exp(x) + 110075314176))

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