3.37.20 \(\int \frac {e^9 (15+10 x)+e^4 (45+30 x)+(15 e^4+5 e^9) \log (3)}{9 x^2+6 x^3+x^4+(6 x^2+2 x^3) \log (3)+x^2 \log ^2(3)} \, dx\)
Optimal. Leaf size=20 \[ -\frac {5 e^4 \left (3+e^5\right )}{x (3+x+\log (3))} \]
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Rubi [A] time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used =
{6, 1680, 12, 261} \begin {gather*} -\frac {5 e^4 \left (3+e^5\right )}{x (x+3+\log (3))} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(E^9*(15 + 10*x) + E^4*(45 + 30*x) + (15*E^4 + 5*E^9)*Log[3])/(9*x^2 + 6*x^3 + x^4 + (6*x^2 + 2*x^3)*Log[3
] + x^2*Log[3]^2),x]
[Out]
(-5*E^4*(3 + E^5))/(x*(3 + x + Log[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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 261
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
Rule 1680
Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^9 (15+10 x)+e^4 (45+30 x)+\left (15 e^4+5 e^9\right ) \log (3)}{6 x^3+x^4+\left (6 x^2+2 x^3\right ) \log (3)+x^2 \left (9+\log ^2(3)\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {160 e^4 \left (3+e^5\right ) x}{\left (4 x^2-(3+\log (3))^2\right )^2} \, dx,x,x+\frac {1}{4} (6+2 \log (3))\right )\\ &=\left (160 e^4 \left (3+e^5\right )\right ) \operatorname {Subst}\left (\int \frac {x}{\left (4 x^2-(3+\log (3))^2\right )^2} \, dx,x,x+\frac {1}{4} (6+2 \log (3))\right )\\ &=-\frac {5 e^4 \left (3+e^5\right )}{x (3+x+\log (3))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} -\frac {5 e^4 \left (3+e^5\right )}{x (3+x+\log (3))} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^9*(15 + 10*x) + E^4*(45 + 30*x) + (15*E^4 + 5*E^9)*Log[3])/(9*x^2 + 6*x^3 + x^4 + (6*x^2 + 2*x^3)
*Log[3] + x^2*Log[3]^2),x]
[Out]
(-5*E^4*(3 + E^5))/(x*(3 + x + Log[3]))
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fricas [A] time = 0.72, size = 22, normalized size = 1.10 \begin {gather*} -\frac {5 \, {\left (e^{9} + 3 \, e^{4}\right )}}{x^{2} + x \log \relax (3) + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*exp(4)*exp(5)+15*exp(4))*log(3)+(10*x+15)*exp(4)*exp(5)+(30*x+45)*exp(4))/(x^2*log(3)^2+(2*x^3+6
*x^2)*log(3)+x^4+6*x^3+9*x^2),x, algorithm="fricas")
[Out]
-5*(e^9 + 3*e^4)/(x^2 + x*log(3) + 3*x)
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giac [B] time = 0.21, size = 48, normalized size = 2.40 \begin {gather*} -\frac {5 \, {\left (e^{18} + 6 \, e^{13} + 9 \, e^{8}\right )}}{x {\left (e^{9} + 3 \, e^{4}\right )} \log \relax (3) + {\left (x^{2} + 3 \, x\right )} e^{9} + 3 \, {\left (x^{2} + 3 \, x\right )} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*exp(4)*exp(5)+15*exp(4))*log(3)+(10*x+15)*exp(4)*exp(5)+(30*x+45)*exp(4))/(x^2*log(3)^2+(2*x^3+6
*x^2)*log(3)+x^4+6*x^3+9*x^2),x, algorithm="giac")
[Out]
-5*(e^18 + 6*e^13 + 9*e^8)/(x*(e^9 + 3*e^4)*log(3) + (x^2 + 3*x)*e^9 + 3*(x^2 + 3*x)*e^4)
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maple [A] time = 0.07, size = 19, normalized size = 0.95
|
|
|
| method |
result |
size |
|
|
|
| gosper |
\(-\frac {5 \left (3+{\mathrm e}^{5}\right ) {\mathrm e}^{4}}{x \left (x +3+\ln \relax (3)\right )}\) |
\(19\) |
| risch |
\(\frac {-5 \,{\mathrm e}^{9}-15 \,{\mathrm e}^{4}}{x \left (x +3+\ln \relax (3)\right )}\) |
\(21\) |
| norman |
\(\frac {-5 \,{\mathrm e}^{4} {\mathrm e}^{5}-15 \,{\mathrm e}^{4}}{x \left (x +3+\ln \relax (3)\right )}\) |
\(23\) |
| default |
\(5 \,{\mathrm e}^{4} \left (3+{\mathrm e}^{5}\right ) \left (-\frac {1}{\left (3+\ln \relax (3)\right ) x}+\frac {1}{\left (3+\ln \relax (3)\right ) \left (x +3+\ln \relax (3)\right )}\right )\) |
\(35\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((5*exp(4)*exp(5)+15*exp(4))*ln(3)+(10*x+15)*exp(4)*exp(5)+(30*x+45)*exp(4))/(x^2*ln(3)^2+(2*x^3+6*x^2)*ln
(3)+x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)
[Out]
-5*(3+exp(5))/x/(x+3+ln(3))*exp(4)
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maxima [A] time = 0.40, size = 21, normalized size = 1.05 \begin {gather*} -\frac {5 \, {\left (e^{9} + 3 \, e^{4}\right )}}{x^{2} + x {\left (\log \relax (3) + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*exp(4)*exp(5)+15*exp(4))*log(3)+(10*x+15)*exp(4)*exp(5)+(30*x+45)*exp(4))/(x^2*log(3)^2+(2*x^3+6
*x^2)*log(3)+x^4+6*x^3+9*x^2),x, algorithm="maxima")
[Out]
-5*(e^9 + 3*e^4)/(x^2 + x*(log(3) + 3))
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mupad [B] time = 6.82, size = 2430, normalized size = 121.50 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((log(3)*(15*exp(4) + 5*exp(9)) + exp(9)*(10*x + 15) + exp(4)*(30*x + 45))/(x^2*log(3)^2 + log(3)*(6*x^2 +
2*x^3) + 9*x^2 + 6*x^3 + x^4),x)
[Out]
(5*log(x + log(3) + 3)*(162*exp(4)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 54*exp(9)*(12*log(
9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 72*exp(4)*log(3)^3 + 24*exp(4)*log(3)^4 + 108*exp(4)*log(9)^2
- 24*exp(9)*log(3)^3 + 9*exp(4)*log(9)^3 + 8*exp(9)*log(3)^4 + 36*exp(9)*log(9)^2 + 3*exp(9)*log(9)^3 + 24*ex
p(4)*log(729)^2 + 8*exp(9)*log(729)^2 - 6*exp(4)*log(3)^2*log(9)^2 - 2*exp(9)*log(3)^2*log(9)^2 - 54*exp(4)*lo
g(3)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 18*exp(9)*log(3)*(12*log(9) - 4*log(729) - 4*log
(3)^2 + log(9)^2)^(1/2) - 54*exp(4)*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 18*exp(9)*
log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 54*exp(4)*log(729)*(12*log(9) - 4*log(729) - 4
*log(3)^2 + log(9)^2)^(1/2) + 18*exp(9)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 216*
exp(4)*log(3)*log(9) + 72*exp(9)*log(3)*log(9) - 72*exp(4)*log(3)*log(729) - 24*exp(9)*log(3)*log(729) - 108*e
xp(4)*log(9)*log(729) - 36*exp(9)*log(9)*log(729) + 54*exp(4)*log(3)^2*(12*log(9) - 4*log(729) - 4*log(3)^2 +
log(9)^2)^(1/2) + 6*exp(4)*log(3)^3*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 18*exp(9)*log(3)^
2*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 9*exp(4)*log(9)^2*(12*log(9) - 4*log(729) - 4*log(3
)^2 + log(9)^2)^(1/2) + 2*exp(9)*log(3)^3*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 3*exp(9)*lo
g(9)^2*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 54*exp(4)*log(3)*log(9)^2 - 108*exp(4)*log(3)^
2*log(9) + 3*exp(4)*log(3)*log(9)^3 - 12*exp(4)*log(3)^3*log(9) + 18*exp(9)*log(3)*log(9)^2 - 36*exp(9)*log(3)
^2*log(9) + exp(9)*log(3)*log(9)^3 - 4*exp(9)*log(3)^3*log(9) + 48*exp(4)*log(3)^2*log(729) + 16*exp(9)*log(3)
^2*log(729) - 6*exp(4)*log(9)^2*log(729) - 2*exp(9)*log(9)^2*log(729) - 12*exp(4)*log(3)*log(9)*log(729) - 4*e
xp(9)*log(3)*log(9)*log(729) - 3*exp(4)*log(3)*log(9)^2*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2)
+ 6*exp(4)*log(3)^2*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - exp(9)*log(3)*log(9)^2*(1
2*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 2*exp(9)*log(3)^2*log(9)*(12*log(9) - 4*log(729) - 4*lo
g(3)^2 + log(9)^2)^(1/2) - 36*exp(4)*log(3)*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 12
*exp(9)*log(3)*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 6*exp(4)*log(3)*log(729)*(12*lo
g(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 2*exp(9)*log(3)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)
^2 + log(9)^2)^(1/2) + 6*exp(4)*log(9)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 2*exp
(9)*log(9)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2)))/(2*(972*log(9) - 324*log(729) + 2
16*log(9)*log(729) + 216*log(3)^2*log(9) + 12*log(3)^4*log(9) - 144*log(3)^2*log(729) - 12*log(3)^4*log(729) +
12*log(9)*log(729)^2 + 18*log(9)^2*log(729) - 324*log(3)^2 - 72*log(3)^4 - 4*log(3)^6 + 81*log(9)^2 - 72*log(
729)^2 - 4*log(729)^3 + 18*log(3)^2*log(9)^2 + log(3)^4*log(9)^2 - 12*log(3)^2*log(729)^2 + log(9)^2*log(729)^
2 + 24*log(3)^2*log(9)*log(729) + 2*log(3)^2*log(9)^2*log(729))) - (log(x)*(log(3)*(90*exp(4) + 30*exp(9) + 15
*exp(4)*log(9) + 5*exp(9)*log(9)) - log(3)^2*(30*exp(4) + 10*exp(9)) + 45*exp(4)*log(9) + 15*exp(9)*log(9) - l
og(729)*(30*exp(4) + 10*exp(9))))/(18*log(729) + 2*log(3)^2*log(729) + 18*log(3)^2 + log(3)^4 + log(729)^2 + 8
1) - (5*(9*exp(4) + 3*exp(9) + 3*exp(4)*log(3) + exp(9)*log(3)))/(x*(log(729) + log(3)^2 + 9)) - (5*log(x + lo
g(3) + 3)*(162*exp(4)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 54*exp(9)*(12*log(9) - 4*log(72
9) - 4*log(3)^2 + log(9)^2)^(1/2) + 72*exp(4)*log(3)^3 - 24*exp(4)*log(3)^4 - 108*exp(4)*log(9)^2 + 24*exp(9)*
log(3)^3 - 9*exp(4)*log(9)^3 - 8*exp(9)*log(3)^4 - 36*exp(9)*log(9)^2 - 3*exp(9)*log(9)^3 - 24*exp(4)*log(729)
^2 - 8*exp(9)*log(729)^2 + 6*exp(4)*log(3)^2*log(9)^2 + 2*exp(9)*log(3)^2*log(9)^2 - 54*exp(4)*log(3)*(12*log(
9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 18*exp(9)*log(3)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9
)^2)^(1/2) - 54*exp(4)*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 18*exp(9)*log(9)*(12*lo
g(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 54*exp(4)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + l
og(9)^2)^(1/2) + 18*exp(9)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 216*exp(4)*log(3)
*log(9) - 72*exp(9)*log(3)*log(9) + 72*exp(4)*log(3)*log(729) + 24*exp(9)*log(3)*log(729) + 108*exp(4)*log(9)*
log(729) + 36*exp(9)*log(9)*log(729) + 54*exp(4)*log(3)^2*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/
2) + 6*exp(4)*log(3)^3*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 18*exp(9)*log(3)^2*(12*log(9)
- 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 9*exp(4)*log(9)^2*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^
2)^(1/2) + 2*exp(9)*log(3)^3*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 3*exp(9)*log(9)^2*(12*lo
g(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 54*exp(4)*log(3)*log(9)^2 + 108*exp(4)*log(3)^2*log(9) - 3*
exp(4)*log(3)*log(9)^3 + 12*exp(4)*log(3)^3*log(9) - 18*exp(9)*log(3)*log(9)^2 + 36*exp(9)*log(3)^2*log(9) - e
xp(9)*log(3)*log(9)^3 + 4*exp(9)*log(3)^3*log(9) - 48*exp(4)*log(3)^2*log(729) - 16*exp(9)*log(3)^2*log(729) +
6*exp(4)*log(9)^2*log(729) + 2*exp(9)*log(9)^2*log(729) + 12*exp(4)*log(3)*log(9)*log(729) + 4*exp(9)*log(3)*
log(9)*log(729) - 3*exp(4)*log(3)*log(9)^2*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 6*exp(4)*l
og(3)^2*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - exp(9)*log(3)*log(9)^2*(12*log(9) - 4*
log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 2*exp(9)*log(3)^2*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(
9)^2)^(1/2) - 36*exp(4)*log(3)*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) - 12*exp(9)*log(3
)*log(9)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 6*exp(4)*log(3)*log(729)*(12*log(9) - 4*log(
729) - 4*log(3)^2 + log(9)^2)^(1/2) + 2*exp(9)*log(3)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2
)^(1/2) + 6*exp(4)*log(9)*log(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2) + 2*exp(9)*log(9)*lo
g(729)*(12*log(9) - 4*log(729) - 4*log(3)^2 + log(9)^2)^(1/2)))/(2*(972*log(9) - 324*log(729) + 216*log(9)*log
(729) + 216*log(3)^2*log(9) + 12*log(3)^4*log(9) - 144*log(3)^2*log(729) - 12*log(3)^4*log(729) + 12*log(9)*lo
g(729)^2 + 18*log(9)^2*log(729) - 324*log(3)^2 - 72*log(3)^4 - 4*log(3)^6 + 81*log(9)^2 - 72*log(729)^2 - 4*lo
g(729)^3 + 18*log(3)^2*log(9)^2 + log(3)^4*log(9)^2 - 12*log(3)^2*log(729)^2 + log(9)^2*log(729)^2 + 24*log(3)
^2*log(9)*log(729) + 2*log(3)^2*log(9)^2*log(729)))
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sympy [A] time = 0.81, size = 20, normalized size = 1.00 \begin {gather*} \frac {- 5 e^{9} - 15 e^{4}}{x^{2} + x \left (\log {\relax (3 )} + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*exp(4)*exp(5)+15*exp(4))*ln(3)+(10*x+15)*exp(4)*exp(5)+(30*x+45)*exp(4))/(x**2*ln(3)**2+(2*x**3+
6*x**2)*ln(3)+x**4+6*x**3+9*x**2),x)
[Out]
(-5*exp(9) - 15*exp(4))/(x**2 + x*(log(3) + 3))
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