3.103.52 \(\int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx\)

Optimal. Leaf size=23 \[ -4+e^{\frac {\log ^4(3)}{625 (-5+x)^4 x^2}}-x \]

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Rubi [B]  time = 1.52, antiderivative size = 76, normalized size of antiderivative = 3.30, number of steps used = 14, number of rules used = 6, integrand size = 110, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6741, 12, 6742, 43, 37, 6706} \begin {gather*} \frac {25 x^4}{2 (5-x)^4}+e^{\frac {\log ^4(3)}{625 (5-x)^4 x^2}}-x+\frac {250}{5-x}-\frac {1875}{(5-x)^2}+\frac {6250}{(5-x)^3}-\frac {15625}{2 (5-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1953125*x^3 - 1953125*x^4 + 781250*x^5 - 156250*x^6 + 15625*x^7 - 625*x^8 + E^(Log[3]^4/(390625*x^2 - 312
500*x^3 + 93750*x^4 - 12500*x^5 + 625*x^6))*(10 - 6*x)*Log[3]^4)/(-1953125*x^3 + 1953125*x^4 - 781250*x^5 + 15
6250*x^6 - 15625*x^7 + 625*x^8),x]

[Out]

E^(Log[3]^4/(625*(5 - x)^4*x^2)) - 15625/(2*(5 - x)^4) + 6250/(5 - x)^3 - 1875/(5 - x)^2 + 250/(5 - x) - x + (
25*x^4)/(2*(5 - x)^4)

Rule 12

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8-e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{625 (5-x)^5 x^3} \, dx\\ &=\frac {1}{625} \int \frac {-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8-e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{(5-x)^5 x^3} \, dx\\ &=\frac {1}{625} \int \left (\frac {1953125}{(-5+x)^5}-\frac {1953125 x}{(-5+x)^5}+\frac {781250 x^2}{(-5+x)^5}-\frac {156250 x^3}{(-5+x)^5}+\frac {15625 x^4}{(-5+x)^5}-\frac {625 x^5}{(-5+x)^5}-\frac {2 e^{\frac {\log ^4(3)}{625 (-5+x)^4 x^2}} (-5+3 x) \log ^4(3)}{(-5+x)^5 x^3}\right ) \, dx\\ &=-\frac {3125}{4 (5-x)^4}+25 \int \frac {x^4}{(-5+x)^5} \, dx-250 \int \frac {x^3}{(-5+x)^5} \, dx+1250 \int \frac {x^2}{(-5+x)^5} \, dx-3125 \int \frac {x}{(-5+x)^5} \, dx-\frac {1}{625} \left (2 \log ^4(3)\right ) \int \frac {e^{\frac {\log ^4(3)}{625 (-5+x)^4 x^2}} (-5+3 x)}{(-5+x)^5 x^3} \, dx-\int \frac {x^5}{(-5+x)^5} \, dx\\ &=e^{\frac {\log ^4(3)}{625 (5-x)^4 x^2}}-\frac {3125}{4 (5-x)^4}+\frac {25 x^4}{2 (5-x)^4}+25 \int \left (\frac {625}{(-5+x)^5}+\frac {500}{(-5+x)^4}+\frac {150}{(-5+x)^3}+\frac {20}{(-5+x)^2}+\frac {1}{-5+x}\right ) \, dx+1250 \int \left (\frac {25}{(-5+x)^5}+\frac {10}{(-5+x)^4}+\frac {1}{(-5+x)^3}\right ) \, dx-3125 \int \left (\frac {5}{(-5+x)^5}+\frac {1}{(-5+x)^4}\right ) \, dx-\int \left (1+\frac {3125}{(-5+x)^5}+\frac {3125}{(-5+x)^4}+\frac {1250}{(-5+x)^3}+\frac {250}{(-5+x)^2}+\frac {25}{-5+x}\right ) \, dx\\ &=e^{\frac {\log ^4(3)}{625 (5-x)^4 x^2}}-\frac {15625}{2 (5-x)^4}+\frac {6250}{(5-x)^3}-\frac {1875}{(5-x)^2}+\frac {250}{5-x}-x+\frac {25 x^4}{2 (5-x)^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 22, normalized size = 0.96 \begin {gather*} e^{\frac {\log ^4(3)}{625 (-5+x)^4 x^2}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1953125*x^3 - 1953125*x^4 + 781250*x^5 - 156250*x^6 + 15625*x^7 - 625*x^8 + E^(Log[3]^4/(390625*x^2
- 312500*x^3 + 93750*x^4 - 12500*x^5 + 625*x^6))*(10 - 6*x)*Log[3]^4)/(-1953125*x^3 + 1953125*x^4 - 781250*x^
5 + 156250*x^6 - 15625*x^7 + 625*x^8),x]

[Out]

E^(Log[3]^4/(625*(-5 + x)^4*x^2)) - x

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fricas [A]  time = 0.79, size = 37, normalized size = 1.61 \begin {gather*} -x + e^{\left (\frac {\log \relax (3)^{4}}{625 \, {\left (x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x+10)*log(3)^4*exp(log(3)^4/(625*x^6-12500*x^5+93750*x^4-312500*x^3+390625*x^2))-625*x^8+15625*
x^7-156250*x^6+781250*x^5-1953125*x^4+1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-195312
5*x^3),x, algorithm="fricas")

[Out]

-x + e^(1/625*log(3)^4/(x^6 - 20*x^5 + 150*x^4 - 500*x^3 + 625*x^2))

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giac [A]  time = 0.21, size = 37, normalized size = 1.61 \begin {gather*} -x + e^{\left (\frac {\log \relax (3)^{4}}{625 \, {\left (x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x+10)*log(3)^4*exp(log(3)^4/(625*x^6-12500*x^5+93750*x^4-312500*x^3+390625*x^2))-625*x^8+15625*
x^7-156250*x^6+781250*x^5-1953125*x^4+1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-195312
5*x^3),x, algorithm="giac")

[Out]

-x + e^(1/625*log(3)^4/(x^6 - 20*x^5 + 150*x^4 - 500*x^3 + 625*x^2))

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maple [A]  time = 0.42, size = 20, normalized size = 0.87

method	result	size
risch	\(-x +{\mathrm e}^{\frac {\ln \relax (3)^{4}}{625 x^{2} \left (x -5\right )^{4}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x+10)*ln(3)^4*exp(ln(3)^4/(625*x^6-12500*x^5+93750*x^4-312500*x^3+390625*x^2))-625*x^8+15625*x^7-1562
50*x^6+781250*x^5-1953125*x^4+1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-1953125*x^3),x
,method=_RETURNVERBOSE)

[Out]

-x+exp(1/625*ln(3)^4/x^2/(x-5)^4)

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maxima [B]  time = 0.62, size = 290, normalized size = 12.61 \begin {gather*} -x - \frac {125 \, {\left (48 \, x^{3} - 540 \, x^{2} + 2200 \, x - 3125\right )}}{12 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + \frac {125 \, {\left (24 \, x^{3} - 300 \, x^{2} + 1300 \, x - 1925\right )}}{12 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + \frac {125 \, {\left (4 \, x^{3} - 30 \, x^{2} + 100 \, x - 125\right )}}{2 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} - \frac {625 \, {\left (6 \, x^{2} - 20 \, x + 25\right )}}{6 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + \frac {3125 \, {\left (4 \, x - 5\right )}}{12 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} - \frac {3125}{4 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + e^{\left (\frac {\log \relax (3)^{4}}{15625 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} - \frac {2 \, \log \relax (3)^{4}}{78125 \, {\left (x^{3} - 15 \, x^{2} + 75 \, x - 125\right )}} + \frac {3 \, \log \relax (3)^{4}}{390625 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {4 \, \log \relax (3)^{4}}{1953125 \, {\left (x - 5\right )}} + \frac {4 \, \log \relax (3)^{4}}{1953125 \, x} + \frac {\log \relax (3)^{4}}{390625 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x+10)*log(3)^4*exp(log(3)^4/(625*x^6-12500*x^5+93750*x^4-312500*x^3+390625*x^2))-625*x^8+15625*
x^7-156250*x^6+781250*x^5-1953125*x^4+1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-195312
5*x^3),x, algorithm="maxima")

[Out]

-x - 125/12*(48*x^3 - 540*x^2 + 2200*x - 3125)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 125/12*(24*x^3 - 300*x
^2 + 1300*x - 1925)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 125/2*(4*x^3 - 30*x^2 + 100*x - 125)/(x^4 - 20*x^
3 + 150*x^2 - 500*x + 625) - 625/6*(6*x^2 - 20*x + 25)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 3125/12*(4*x -
5)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 3125/4/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + e^(1/15625*log(3)
^4/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 2/78125*log(3)^4/(x^3 - 15*x^2 + 75*x - 125) + 3/390625*log(3)^4/(
x^2 - 10*x + 25) - 4/1953125*log(3)^4/(x - 5) + 4/1953125*log(3)^4/x + 1/390625*log(3)^4/x^2)

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mupad [B]  time = 6.00, size = 38, normalized size = 1.65 \begin {gather*} {\mathrm {e}}^{\frac {{\ln \relax (3)}^4}{625\,x^6-12500\,x^5+93750\,x^4-312500\,x^3+390625\,x^2}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1953125*x^4 - 1953125*x^3 - 781250*x^5 + 156250*x^6 - 15625*x^7 + 625*x^8 + exp(log(3)^4/(390625*x^2 - 31
2500*x^3 + 93750*x^4 - 12500*x^5 + 625*x^6))*log(3)^4*(6*x - 10))/(1953125*x^3 - 1953125*x^4 + 781250*x^5 - 15
6250*x^6 + 15625*x^7 - 625*x^8),x)

[Out]

exp(log(3)^4/(390625*x^2 - 312500*x^3 + 93750*x^4 - 12500*x^5 + 625*x^6)) - x

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sympy [A]  time = 0.82, size = 32, normalized size = 1.39 \begin {gather*} - x + e^{\frac {\log {\relax (3 )}^{4}}{625 x^{6} - 12500 x^{5} + 93750 x^{4} - 312500 x^{3} + 390625 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x+10)*ln(3)**4*exp(ln(3)**4/(625*x**6-12500*x**5+93750*x**4-312500*x**3+390625*x**2))-625*x**8+
15625*x**7-156250*x**6+781250*x**5-1953125*x**4+1953125*x**3)/(625*x**8-15625*x**7+156250*x**6-781250*x**5+195
3125*x**4-1953125*x**3),x)

[Out]

-x + exp(log(3)**4/(625*x**6 - 12500*x**5 + 93750*x**4 - 312500*x**3 + 390625*x**2))

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