3.39.55 \(\int \frac {1464843750 x^2+e^8 (-46875000000 x+187500000 x^2-187500 x^3)+e^{16} (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4)+e^x (244140625 x-244140625 x^2+e^8 (3921875000 x-15656250 x^2+15625 x^3))}{244140625 x^3+e^8 (-7812500000 x^2+31250000 x^3-31250 x^4)+e^{16} (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5)} \, dx\) [3855]
Optimal. Leaf size=28 \[ \frac {e^x}{e^8 \left (4-\frac {x}{125}\right )^2-x}+6 \log (x) \]
[Out]
exp(x)/((4-1/125*x)^2*exp(4)^2-x)+6*ln(x)
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(295\) vs. \(2(28)=56\).
time = 1.58, antiderivative size = 295, normalized size of antiderivative = 10.54, number of steps
used = 15, number of rules used = 9, integrand size = 134, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6820, 6874,
736, 632, 212, 1660, 1642, 652, 2327} \begin {gather*} -\frac {1500 e^8 (500-x) (x+500)}{\left (125+16 e^8\right ) \left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )}+\frac {15625 e^x}{e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )}+6 \log (x)+\frac {60 \sqrt {5} \left (125+24 e^8\right ) \tanh ^{-1}\left (\frac {2 e^8 (500-x)+15625}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {60 \sqrt {5} \left (125+8 e^8\right ) \tanh ^{-1}\left (\frac {2 e^8 (500-x)+15625}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {960 \sqrt {5} e^8 \tanh ^{-1}\left (\frac {2 e^8 (500-x)+15625}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(1464843750*x^2 + E^8*(-46875000000*x + 187500000*x^2 - 187500*x^3) + E^16*(375000000000 - 3000000000*x +
9000000*x^2 - 12000*x^3 + 6*x^4) + E^x*(244140625*x - 244140625*x^2 + E^8*(3921875000*x - 15656250*x^2 + 15625
*x^3)))/(244140625*x^3 + E^8*(-7812500000*x^2 + 31250000*x^3 - 31250*x^4) + E^16*(62500000000*x - 500000000*x^
2 + 1500000*x^3 - 2000*x^4 + x^5)),x]
[Out]
(15625*E^x)/(250000*E^8 - 125*(125 + 8*E^8)*x + E^8*x^2) - (1500*E^8*(500 - x)*(500 + x))/((125 + 16*E^8)*(250
000*E^8 - 125*(125 + 8*E^8)*x + E^8*x^2)) + (187500*(4000*E^8 - (125 + 8*E^8)*x))/((125 + 16*E^8)*(250000*E^8
- 125*(125 + 8*E^8)*x + E^8*x^2)) - (960*Sqrt[5]*E^8*ArcTanh[(15625 + 2*E^8*(500 - x))/(625*Sqrt[5*(125 + 16*E
^8)])])/(125 + 16*E^8)^(3/2) - (60*Sqrt[5]*(125 + 8*E^8)*ArcTanh[(15625 + 2*E^8*(500 - x))/(625*Sqrt[5*(125 +
16*E^8)])])/(125 + 16*E^8)^(3/2) + (60*Sqrt[5]*(125 + 24*E^8)*ArcTanh[(15625 + 2*E^8*(500 - x))/(625*Sqrt[5*(1
25 + 16*E^8)])])/(125 + 16*E^8)^(3/2) + 6*Log[x]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 652
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Rule 736
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Dist[2*(2*p + 3)*((c*d
^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ
[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]
Rule 1642
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 1660
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Rule 2327
Int[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w
, x, Exponent[w, x]]/Coefficient[z, x, Exponent[z, x]])*F^u*v^(n + 1), x] /; EqQ[Exponent[w, x], Exponent[z, x
]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && Po
lynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]
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 {6 e^{16} (-500+x)^4-187500 e^8 (-500+x)^2 x-244140625 e^x (-1+x) x+1464843750 x^2+15625 e^{8+x} x \left (251000-1002 x+x^2\right )}{x \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2} \, dx\\ &=\int \left (-\frac {187500 e^8 (500-x)^2}{\left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2}+\frac {6 e^{16} (500-x)^4}{x \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2}+\frac {1464843750 x}{\left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2}+\frac {15625 e^x \left (125 \left (125+2008 e^8\right )-\left (15625+1002 e^8\right ) x+e^8 x^2\right )}{\left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2}\right ) \, dx\\ &=15625 \int \frac {e^x \left (125 \left (125+2008 e^8\right )-\left (15625+1002 e^8\right ) x+e^8 x^2\right )}{\left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2} \, dx+1464843750 \int \frac {x}{\left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2} \, dx-\left (187500 e^8\right ) \int \frac {(500-x)^2}{\left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2} \, dx+\left (6 e^{16}\right ) \int \frac {(500-x)^4}{x \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )^2} \, dx\\ &=\frac {15625 e^x}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2}-\frac {1500 e^8 (500-x) (500+x)}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}-\frac {\left (1500000 e^8\right ) \int \frac {1}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2} \, dx}{125+16 e^8}-\frac {\left (6 e^{16}\right ) \int \frac {-\frac {488281250000 \left (125+16 e^8\right )}{e^8}+31250000000 x-1953125 \left (16+\frac {125}{e^8}\right ) x^2}{x \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )} \, dx}{1953125 \left (125+16 e^8\right )}-\frac {\left (93750 \left (125+8 e^8\right )\right ) \int \frac {1}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2} \, dx}{125+16 e^8}\\ &=\frac {15625 e^x}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2}-\frac {1500 e^8 (500-x) (500+x)}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}+\frac {\left (3000000 e^8\right ) \text {Subst}\left (\int \frac {1}{1953125 \left (125+16 e^8\right )-x^2} \, dx,x,-125 \left (125+8 e^8\right )+2 e^8 x\right )}{125+16 e^8}-\frac {\left (6 e^{16}\right ) \int \left (-\frac {1953125 \left (125+16 e^8\right )}{e^{16} x}+\frac {30517578125 \left (-125-24 e^8\right )}{e^{16} \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}\right ) \, dx}{1953125 \left (125+16 e^8\right )}+\frac {\left (187500 \left (125+8 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{1953125 \left (125+16 e^8\right )-x^2} \, dx,x,-125 \left (125+8 e^8\right )+2 e^8 x\right )}{125+16 e^8}\\ &=\frac {15625 e^x}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2}-\frac {1500 e^8 (500-x) (500+x)}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}-\frac {960 \sqrt {5} e^8 \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {60 \sqrt {5} \left (125+8 e^8\right ) \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}+6 \log (x)+\frac {\left (93750 \left (125+24 e^8\right )\right ) \int \frac {1}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2} \, dx}{125+16 e^8}\\ &=\frac {15625 e^x}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2}-\frac {1500 e^8 (500-x) (500+x)}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}-\frac {960 \sqrt {5} e^8 \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {60 \sqrt {5} \left (125+8 e^8\right ) \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}+6 \log (x)-\frac {\left (187500 \left (125+24 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{1953125 \left (125+16 e^8\right )-x^2} \, dx,x,-125 \left (125+8 e^8\right )+2 e^8 x\right )}{125+16 e^8}\\ &=\frac {15625 e^x}{250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2}-\frac {1500 e^8 (500-x) (500+x)}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (250000 e^8-125 \left (125+8 e^8\right ) x+e^8 x^2\right )}-\frac {960 \sqrt {5} e^8 \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {60 \sqrt {5} \left (125+8 e^8\right ) \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}+\frac {60 \sqrt {5} \left (125+24 e^8\right ) \tanh ^{-1}\left (\frac {15625+2 e^8 (500-x)}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}+6 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 2.61, size = 25, normalized size = 0.89 \begin {gather*} \frac {15625 e^x}{e^8 (-500+x)^2-15625 x}+6 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1464843750*x^2 + E^8*(-46875000000*x + 187500000*x^2 - 187500*x^3) + E^16*(375000000000 - 300000000
0*x + 9000000*x^2 - 12000*x^3 + 6*x^4) + E^x*(244140625*x - 244140625*x^2 + E^8*(3921875000*x - 15656250*x^2 +
15625*x^3)))/(244140625*x^3 + E^8*(-7812500000*x^2 + 31250000*x^3 - 31250*x^4) + E^16*(62500000000*x - 500000
000*x^2 + 1500000*x^3 - 2000*x^4 + x^5)),x]
[Out]
(15625*E^x)/(E^8*(-500 + x)^2 - 15625*x) + 6*Log[x]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.24, size = 4376, normalized size = 156.29
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {15625 \,{\mathrm e}^{x}}{x^{2} {\mathrm e}^{8}-1000 x \,{\mathrm e}^{8}+250000 \,{\mathrm e}^{8}-15625 x}+6 \ln \left (x \right )\) |
\(31\) |
| norman |
\(\frac {15625 \,{\mathrm e}^{x}}{x^{2} {\mathrm e}^{8}-1000 x \,{\mathrm e}^{8}+250000 \,{\mathrm e}^{8}-15625 x}+6 \ln \left (x \right )\) |
\(37\) |
| default |
\(\text {Expression too large to display}\) |
\(4376\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+244140625*x)*exp(x)+(6*x^4-12000*x^3+900000
0*x^2-3000000000*x+375000000000)*exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*x^2)/(
(x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-31250*x^4+31250000*x^3-7812500000*x^2)*exp(4
)^2+244140625*x^3),x,method=_RETURNVERBOSE)
[Out]
-732421875*sum(_R/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250000000*exp(16)+46875*_R^2*exp(8)-31
250000*_R*exp(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2000*exp(16))*_
Z^3-(-31250000*exp(8)-1500000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+500000000*exp(16))*_Z+62500000000*exp
(16)))+6*exp(4)^4/exp(16)*ln(x)-3*exp(4)^4*exp(-16)*sum((-_R^3*exp(16)+250*(125*exp(8)+8*exp(16))*_R^2+15625*(
-2000*exp(8)-96*exp(16)-15625)*_R+7812500000*exp(8)+500000000*exp(16))/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500
000*_R*exp(16)+250000000*exp(16)+46875*_R^2*exp(8)-31250000*_R*exp(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R)
,_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2000*exp(16))*_Z^3-(-31250000*exp(8)-1500000*exp(16)-244140625)*_Z^2-(78
12500000*exp(8)+500000000*exp(16))*_Z+62500000000*exp(16)))+23437500000*exp(4)^2*sum(1/(-2*_R^3*exp(16)+3000*_
R^2*exp(16)-1500000*_R*exp(16)+250000000*exp(16)+46875*_R^2*exp(8)-31250000*_R*exp(8)+3906250000*exp(8)-244140
625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2000*exp(16))*_Z^3-(-31250000*exp(8)-1500000*exp(16)-244
140625)*_Z^2-(7812500000*exp(8)+500000000*exp(16))*_Z+62500000000*exp(16)))+1500000000*exp(4)^4*sum(1/(-2*_R^3
*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250000000*exp(16)+46875*_R^2*exp(8)-31250000*_R*exp(8)+390625000
0*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2000*exp(16))*_Z^3-(-31250000*exp(8)-1500
000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+500000000*exp(16))*_Z+62500000000*exp(16)))-93750000*exp(4)^2*s
um(_R/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250000000*exp(16)+46875*_R^2*exp(8)-31250000*_R*ex
p(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2000*exp(16))*_Z^3-(-312500
00*exp(8)-1500000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+500000000*exp(16))*_Z+62500000000*exp(16)))-45000
00*exp(4)^4*sum(_R/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250000000*exp(16)+46875*_R^2*exp(8)-3
1250000*_R*exp(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2000*exp(16))*
_Z^3-(-31250000*exp(8)-1500000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+500000000*exp(16))*_Z+62500000000*ex
p(16)))+93750*exp(4)^2*sum(_R^2/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250000000*exp(16)+46875*
_R^2*exp(8)-31250000*_R*exp(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31250*exp(8)+2
000*exp(16))*_Z^3-(-31250000*exp(8)-1500000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+500000000*exp(16))*_Z+6
2500000000*exp(16)))+6000*exp(4)^4*sum(_R^2/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250000000*ex
p(16)+46875*_R^2*exp(8)-31250000*_R*exp(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*exp(16)-(31
250*exp(8)+2000*exp(16))*_Z^3-(-31250000*exp(8)-1500000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+500000000*e
xp(16))*_Z+62500000000*exp(16)))-3*exp(4)^4*sum(_R^3/(-2*_R^3*exp(16)+3000*_R^2*exp(16)-1500000*_R*exp(16)+250
000000*exp(16)+46875*_R^2*exp(8)-31250000*_R*exp(8)+3906250000*exp(8)-244140625*_R)*ln(x-_R),_R=RootOf(_Z^4*ex
p(16)-(31250*exp(8)+2000*exp(16))*_Z^3-(-31250000*exp(8)-1500000*exp(16)-244140625)*_Z^2-(7812500000*exp(8)+50
0000000*exp(16))*_Z+62500000000*exp(16)))-125*exp(x)*(2*x*exp(4)^2-1000*exp(4)^2-15625)/(16*exp(4)^2+125)/(x^2
*exp(4)^2-1000*x*exp(4)^2+250000*exp(4)^2-15625*x)-1/5*(625*exp(125/2*(8*exp(8)+125+5*(80*exp(8)+625)^(1/2))/e
xp(8))*Ei(1,1/2*(-2*x*exp(8)+625*(80*exp(8)+625)^(1/2)+1000*exp(8)+15625)/exp(8))*(80*exp(8)+625)^(1/2)+625*ex
p(-125/2*(-8*exp(8)-125+5*(80*exp(8)+625)^(1/2))/exp(8))*Ei(1,-1/2*(2*x*exp(8)+625*(80*exp(8)+625)^(1/2)-1000*
exp(8)-15625)/exp(8))*(80*exp(8)+625)^(1/2)-2*exp(125/2*(8*exp(8)+125+5*(80*exp(8)+625)^(1/2))/exp(8))*Ei(1,1/
2*(-2*x*exp(8)+625*(80*exp(8)+625)^(1/2)+1000*exp(8)+15625)/exp(8))*exp(8)+2*exp(-125/2*(-8*exp(8)-125+5*(80*e
xp(8)+625)^(1/2))/exp(8))*Ei(1,-1/2*(2*x*exp(8)+625*(80*exp(8)+625)^(1/2)-1000*exp(8)-15625)/exp(8))*exp(8))/(
16*exp(8)+125)/(80*exp(8)+625)^(1/2)+15625*exp(x)*(8*x*exp(4)^2-4000*exp(4)^2+125*x)/(16*exp(4)^2+125)/(x^2*ex
p(4)^2-1000*x*exp(4)^2+250000*exp(4)^2-15625*x)+25/2*(5000*(80*exp(8)+625)^(1/2)*exp(8)*exp(125/2*(8*exp(8)+12
5+5*(80*exp(8)+625)^(1/2))/exp(8))*Ei(1,1/2*(-2*x*exp(8)+625*(80*exp(8)+625)^(1/2)+1000*exp(8)+15625)/exp(8))+
5000*(80*exp(8)+625)^(1/2)*exp(8)*exp(-125/2*(-8*exp(8)-125+5*(80*exp(8)+625)^(1/2))/exp(8))*Ei(1,-1/2*(2*x*ex
p(8)+625*(80*exp(8)+625)^(1/2)-1000*exp(8)-15625)/exp(8))-16*exp(8)^2*exp(125/2*(8*exp(8)+125+5*(80*exp(8)+625
)^(1/2))/exp(8))*Ei(1,1/2*(-2*x*exp(8)+625*(80*exp(8)+625)^(1/2)+1000*exp(8)+15625)/exp(8))+16*exp(8)^2*exp(-1
25/2*(-8*exp(8)-125+5*(80*exp(8)+625)^(1/2))/exp(8))*Ei(1,-1/2*(2*x*exp(8)+625*(80*exp(8)+625)^(1/2)-1000*exp(
8)-15625)/exp(8))+78125*exp(125/2*(8*exp(8)+125+5*(80*exp(8)+625)^(1/2))/exp(8))*Ei(1,1/2*(-2*x*exp(8)+625*(80
*exp(8)+625)^(1/2)+1000*exp(8)+15625)/exp(8))*(80*exp(8)+625)^(1/2)+78125*exp(-125/2*(-8*exp(8)-125+5*(80*exp(
8)+625)^(1/2))/exp(8))*Ei(1,-1/2*(2*x*exp(8)+62...
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1272 vs.
\(2 (23) = 46\).
time = 0.62, size = 1272, normalized size = 45.43 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+244140625*x)*exp(x)+(6*x^4-12000*x^3+
9000000*x^2-3000000000*x+375000000000)*exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*
x^2)/((x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-31250*x^4+31250000*x^3-7812500000*x^2)
*exp(4)^2+244140625*x^3),x, algorithm="maxima")
[Out]
-3/625*(625*e^(-16)*log(x^2*e^8 - 125*x*(8*e^8 + 125) + 250000*e^8) - 1250*e^(-16)*log(x) + (256*e^24 - 12000*
e^16 - 375000*e^8 - 1953125)*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*
e^8 + 625) - 1000*e^8 - 15625))/((16*e^24 + 125*e^16)*sqrt(80*e^8 + 625)) + 20000*(x*(8*e^16 + 125*e^8) - 4000
*e^16 - 250000*e^8 - 1953125)/(x^2*(16*e^24 + 125*e^16) - 125*x*(128*e^24 + 3000*e^16 + 15625*e^8) + 4000000*e
^24 + 31250000*e^16))*e^16 + 3/625*(625*e^(-16)*log(x^2*e^8 - 125*x*(8*e^8 + 125) + 250000*e^8) - (256*e^24 -
12000*e^16 - 375000*e^8 - 1953125)*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sq
rt(80*e^8 + 625) - 1000*e^8 - 15625))/((16*e^24 + 125*e^16)*sqrt(80*e^8 + 625)) - 1250*(x*(128*e^24 + 18000*e^
16 + 375000*e^8 + 1953125) - 64000*e^24 - 4000000*e^16 - 31250000*e^8)/(x^2*(16*e^32 + 125*e^24) - 125*x*(128*
e^32 + 3000*e^24 + 15625*e^16) + 4000000*e^32 + 31250000*e^24))*e^16 - 576/625*((8*e^8 + 125)*log((2*x*e^8 - 6
25*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(sqrt(80*e^8
+ 625)*(16*e^8 + 125)) + 625*(x*(8*e^8 + 125) - 4000*e^8)/(x^2*(16*e^16 + 125*e^8) - 125*x*(128*e^16 + 3000*e^
8 + 15625) + 4000000*e^16 + 31250000*e^8))*e^16 + 96/625*(32*e^8*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*
e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(sqrt(80*e^8 + 625)*(16*e^8 + 125)) + 625*
(x*(32*e^16 + 2000*e^8 + 15625) - 16000*e^16 - 250000*e^8)/(x^2*(16*e^24 + 125*e^16) - 125*x*(128*e^24 + 3000*
e^16 + 15625*e^8) + 4000000*e^24 + 31250000*e^16))*e^16 + 1536/625*(2*e^8*log((2*x*e^8 - 625*sqrt(80*e^8 + 625
) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(sqrt(80*e^8 + 625)*(16*e^8 + 125
)) + 625*(2*x*e^8 - 1000*e^8 - 15625)/(x^2*(16*e^16 + 125*e^8) - 125*x*(128*e^16 + 3000*e^8 + 15625) + 4000000
*e^16 + 31250000*e^8))*e^16 - 96/5*((8*e^8 + 125)*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2
*x*e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(sqrt(80*e^8 + 625)*(16*e^8 + 125)) + 625*(x*(8*e^8 + 125
) - 4000*e^8)/(x^2*(16*e^16 + 125*e^8) - 125*x*(128*e^16 + 3000*e^8 + 15625) + 4000000*e^16 + 31250000*e^8))*e
^8 + 12/5*(32*e^8*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^8 + 625)
- 1000*e^8 - 15625))/(sqrt(80*e^8 + 625)*(16*e^8 + 125)) + 625*(x*(32*e^16 + 2000*e^8 + 15625) - 16000*e^16 -
250000*e^8)/(x^2*(16*e^24 + 125*e^16) - 125*x*(128*e^24 + 3000*e^16 + 15625*e^8) + 4000000*e^24 + 31250000*e^1
6))*e^8 + 192/5*(2*e^8*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^8 +
625) - 1000*e^8 - 15625))/(sqrt(80*e^8 + 625)*(16*e^8 + 125)) + 625*(2*x*e^8 - 1000*e^8 - 15625)/(x^2*(16*e^16
+ 125*e^8) - 125*x*(128*e^16 + 3000*e^8 + 15625) + 4000000*e^16 + 31250000*e^8))*e^8 - 150*(8*e^8 + 125)*log(
(2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(
sqrt(80*e^8 + 625)*(16*e^8 + 125)) - 93750*(x*(8*e^8 + 125) - 4000*e^8)/(x^2*(16*e^16 + 125*e^8) - 125*x*(128*
e^16 + 3000*e^8 + 15625) + 4000000*e^16 + 31250000*e^8) + 15625*e^x/(x^2*e^8 - 125*x*(8*e^8 + 125) + 250000*e^
8)
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Fricas [A]
time = 0.38, size = 42, normalized size = 1.50 \begin {gather*} \frac {6 \, {\left ({\left (x^{2} - 1000 \, x + 250000\right )} e^{8} - 15625 \, x\right )} \log \left (x\right ) + 15625 \, e^{x}}{{\left (x^{2} - 1000 \, x + 250000\right )} e^{8} - 15625 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+244140625*x)*exp(x)+(6*x^4-12000*x^3+
9000000*x^2-3000000000*x+375000000000)*exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*
x^2)/((x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-31250*x^4+31250000*x^3-7812500000*x^2)
*exp(4)^2+244140625*x^3),x, algorithm="fricas")
[Out]
(6*((x^2 - 1000*x + 250000)*e^8 - 15625*x)*log(x) + 15625*e^x)/((x^2 - 1000*x + 250000)*e^8 - 15625*x)
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Sympy [A]
time = 0.11, size = 31, normalized size = 1.11 \begin {gather*} 6 \log {\left (x \right )} + \frac {15625 e^{x}}{x^{2} e^{8} - 1000 x e^{8} - 15625 x + 250000 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((15625*x**3-15656250*x**2+3921875000*x)*exp(4)**2-244140625*x**2+244140625*x)*exp(x)+(6*x**4-12000
*x**3+9000000*x**2-3000000000*x+375000000000)*exp(4)**4+(-187500*x**3+187500000*x**2-46875000000*x)*exp(4)**2+
1464843750*x**2)/((x**5-2000*x**4+1500000*x**3-500000000*x**2+62500000000*x)*exp(4)**4+(-31250*x**4+31250000*x
**3-7812500000*x**2)*exp(4)**2+244140625*x**3),x)
[Out]
6*log(x) + 15625*exp(x)/(x**2*exp(8) - 1000*x*exp(8) - 15625*x + 250000*exp(8))
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (23) = 46\).
time = 0.45, size = 55, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (3 \, x^{2} e^{8} \log \left (x\right ) - 3000 \, x e^{8} \log \left (x\right ) - 46875 \, x \log \left (x\right ) + 750000 \, e^{8} \log \left (x\right ) + 15625 \, e^{x}\right )}}{x^{2} e^{8} - 1000 \, x e^{8} - 15625 \, x + 250000 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+244140625*x)*exp(x)+(6*x^4-12000*x^3+
9000000*x^2-3000000000*x+375000000000)*exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*
x^2)/((x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-31250*x^4+31250000*x^3-7812500000*x^2)
*exp(4)^2+244140625*x^3),x, algorithm="giac")
[Out]
2*(3*x^2*e^8*log(x) - 3000*x*e^8*log(x) - 46875*x*log(x) + 750000*e^8*log(x) + 15625*e^x)/(x^2*e^8 - 1000*x*e^
8 - 15625*x + 250000*e^8)
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Mupad [B]
time = 3.03, size = 29, normalized size = 1.04 \begin {gather*} 6\,\ln \left (x\right )+\frac {15625\,{\mathrm {e}}^{x-8}}{x^2-{\mathrm {e}}^{-8}\,\left (1000\,{\mathrm {e}}^8+15625\right )\,x+250000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(x)*(244140625*x + exp(8)*(3921875000*x - 15656250*x^2 + 15625*x^3) - 244140625*x^2) - exp(8)*(4687500
0000*x - 187500000*x^2 + 187500*x^3) + exp(16)*(9000000*x^2 - 3000000000*x - 12000*x^3 + 6*x^4 + 375000000000)
+ 1464843750*x^2)/(exp(16)*(62500000000*x - 500000000*x^2 + 1500000*x^3 - 2000*x^4 + x^5) - exp(8)*(781250000
0*x^2 - 31250000*x^3 + 31250*x^4) + 244140625*x^3),x)
[Out]
6*log(x) + (15625*exp(x - 8))/(x^2 - x*exp(-8)*(1000*exp(8) + 15625) + 250000)
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