\(\int \frac {1}{25} (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} (200 x-300 x^2+e^5 (4 x^2-4 x^3))) \, dx\) [1841]

Optimal result

Integrand size = 53, antiderivative size = 26 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=x \left (4-4 \left (2+e^{\frac {e^5 x}{25}}\right )+x\right ) \left (-x+x^2\right ) \]

[Out]

x*(x^2-x)*(x-4*exp(1/25*x*exp(5))-4)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 18, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 2227, 2207, 2225} \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=x^4-4 e^{\frac {e^5 x}{25}} x^3-5 x^3+4 e^{\frac {e^5 x}{25}} x^2+4 x^2 \]

[In]

Int[(200*x - 375*x^2 + 100*x^3 + E^((E^5*x)/25)*(200*x - 300*x^2 + E^5*(4*x^2 - 4*x^3)))/25,x]

[Out]

4*x^2 + 4*E^((E^5*x)/25)*x^2 - 5*x^3 - 4*E^((E^5*x)/25)*x^3 + 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 2207

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

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx \\ & = 4 x^2-5 x^3+x^4+\frac {1}{25} \int e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right ) \, dx \\ & = 4 x^2-5 x^3+x^4+\frac {1}{25} \int \left (200 e^{\frac {e^5 x}{25}} x-300 e^{\frac {e^5 x}{25}} x^2-4 e^{5+\frac {e^5 x}{25}} (-1+x) x^2\right ) \, dx \\ & = 4 x^2-5 x^3+x^4-\frac {4}{25} \int e^{5+\frac {e^5 x}{25}} (-1+x) x^2 \, dx+8 \int e^{\frac {e^5 x}{25}} x \, dx-12 \int e^{\frac {e^5 x}{25}} x^2 \, dx \\ & = 200 e^{-5+\frac {e^5 x}{25}} x+4 x^2-300 e^{-5+\frac {e^5 x}{25}} x^2-5 x^3+x^4-\frac {4}{25} \int \left (-e^{5+\frac {e^5 x}{25}} x^2+e^{5+\frac {e^5 x}{25}} x^3\right ) \, dx-\frac {200 \int e^{\frac {e^5 x}{25}} \, dx}{e^5}+\frac {600 \int e^{\frac {e^5 x}{25}} x \, dx}{e^5} \\ & = -5000 e^{-10+\frac {e^5 x}{25}}+15000 e^{-10+\frac {e^5 x}{25}} x+200 e^{-5+\frac {e^5 x}{25}} x+4 x^2-300 e^{-5+\frac {e^5 x}{25}} x^2-5 x^3+x^4+\frac {4}{25} \int e^{5+\frac {e^5 x}{25}} x^2 \, dx-\frac {4}{25} \int e^{5+\frac {e^5 x}{25}} x^3 \, dx-\frac {15000 \int e^{\frac {e^5 x}{25}} \, dx}{e^{10}} \\ & = -375000 e^{-15+\frac {e^5 x}{25}}-5000 e^{-10+\frac {e^5 x}{25}}+15000 e^{-10+\frac {e^5 x}{25}} x+200 e^{-5+\frac {e^5 x}{25}} x+4 x^2+4 e^{\frac {e^5 x}{25}} x^2-300 e^{-5+\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4-\frac {8 \int e^{5+\frac {e^5 x}{25}} x \, dx}{e^5}+\frac {12 \int e^{5+\frac {e^5 x}{25}} x^2 \, dx}{e^5} \\ & = -375000 e^{-15+\frac {e^5 x}{25}}-5000 e^{-10+\frac {e^5 x}{25}}+15000 e^{-10+\frac {e^5 x}{25}} x+4 x^2+4 e^{\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4+\frac {200 \int e^{5+\frac {e^5 x}{25}} \, dx}{e^{10}}-\frac {600 \int e^{5+\frac {e^5 x}{25}} x \, dx}{e^{10}} \\ & = -375000 e^{-15+\frac {e^5 x}{25}}+4 x^2+4 e^{\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4+\frac {15000 \int e^{5+\frac {e^5 x}{25}} \, dx}{e^{15}} \\ & = 4 x^2+4 e^{\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=(-1+x) x^2 \left (-4-4 e^{\frac {e^5 x}{25}}+x\right ) \]

[In]

Integrate[(200*x - 375*x^2 + 100*x^3 + E^((E^5*x)/25)*(200*x - 300*x^2 + E^5*(4*x^2 - 4*x^3)))/25,x]

[Out]

(-1 + x)*x^2*(-4 - 4*E^((E^5*x)/25) + x)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31

method	result	size
risch	\(\frac {\left (-100 x^{3}+100 x^{2}\right ) {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}}{25}+x^{4}-5 x^{3}+4 x^{2}\)	\(34\)
norman	\(x^{4}+4 x^{2}-5 x^{3}-4 x^{3} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}+4 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}\)	\(37\)
parallelrisch	\(x^{4}+4 x^{2}-5 x^{3}-4 x^{3} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}+4 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}\)	\(37\)
default	\({\mathrm e}^{-5} \left (5000 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-187500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+2500 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-62500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{3} {\mathrm e}^{15}}{15625}-\frac {3 \,{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}+\frac {6 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-6 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )\right )+4 x^{2}-5 x^{3}+x^{4}\)	\(185\)
parts	\(2500 \,{\mathrm e}^{-5} \left ({\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+2 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-75 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-25 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{3} {\mathrm e}^{15}}{15625}-\frac {3 \,{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}+\frac {6 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-6 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )\right )+4 x^{2}-5 x^{3}+x^{4}\)	\(185\)
derivativedivides	\({\mathrm e}^{-5} \left (5000 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-187500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+2500 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-62500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{3} {\mathrm e}^{15}}{15625}-\frac {3 \,{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}+\frac {6 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-6 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+4 x^{2} {\mathrm e}^{5}-5 x^{3} {\mathrm e}^{5}+x^{4} {\mathrm e}^{5}\right )\)	\(191\)

[In]

int(1/25*((-4*x^3+4*x^2)*exp(5)-300*x^2+200*x)*exp(1/25*x*exp(5))+4*x^3-15*x^2+8*x,x,method=_RETURNVERBOSE)

[Out]

1/25*(-100*x^3+100*x^2)*exp(1/25*x*exp(5))+x^4-5*x^3+4*x^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=x^{4} - 5 \, x^{3} + 4 \, x^{2} - 4 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{25} \, x e^{5}\right )} \]

[In]

integrate(1/25*((-4*x^3+4*x^2)*exp(5)-300*x^2+200*x)*exp(1/25*x*exp(5))+4*x^3-15*x^2+8*x,x, algorithm="fricas"
)

[Out]

x^4 - 5*x^3 + 4*x^2 - 4*(x^3 - x^2)*e^(1/25*x*e^5)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=x^{4} - 5 x^{3} + 4 x^{2} + \left (- 4 x^{3} + 4 x^{2}\right ) e^{\frac {x e^{5}}{25}} \]

[In]

integrate(1/25*((-4*x**3+4*x**2)*exp(5)-300*x**2+200*x)*exp(1/25*x*exp(5))+4*x**3-15*x**2+8*x,x)

[Out]

x**4 - 5*x**3 + 4*x**2 + (-4*x**3 + 4*x**2)*exp(x*exp(5)/25)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=x^{4} - 5 \, x^{3} + 4 \, x^{2} - 4 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{25} \, x e^{5}\right )} \]

[In]

integrate(1/25*((-4*x^3+4*x^2)*exp(5)-300*x^2+200*x)*exp(1/25*x*exp(5))+4*x^3-15*x^2+8*x,x, algorithm="maxima"
)

[Out]

x^4 - 5*x^3 + 4*x^2 - 4*(x^3 - x^2)*e^(1/25*x*e^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (20) = 40\).

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.58 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=x^{4} - 5 \, x^{3} + 4 \, x^{2} - 4 \, {\left (x^{3} e^{15} - x^{2} e^{15} - 75 \, x^{2} e^{10} + 50 \, x e^{10} + 3750 \, x e^{5} - 1250 \, e^{5} - 93750\right )} e^{\left (\frac {1}{25} \, x e^{5} - 15\right )} - 100 \, {\left (3 \, x^{2} e^{10} - 2 \, x e^{10} - 150 \, x e^{5} + 50 \, e^{5} + 3750\right )} e^{\left (\frac {1}{25} \, x e^{5} - 15\right )} \]

[In]

integrate(1/25*((-4*x^3+4*x^2)*exp(5)-300*x^2+200*x)*exp(1/25*x*exp(5))+4*x^3-15*x^2+8*x,x, algorithm="giac")

[Out]

x^4 - 5*x^3 + 4*x^2 - 4*(x^3*e^15 - x^2*e^15 - 75*x^2*e^10 + 50*x*e^10 + 3750*x*e^5 - 1250*e^5 - 93750)*e^(1/2
5*x*e^5 - 15) - 100*(3*x^2*e^10 - 2*x*e^10 - 150*x*e^5 + 50*e^5 + 3750)*e^(1/25*x*e^5 - 15)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {1}{25} \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx=-x^2\,\left (x-1\right )\,\left (4\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{25}}-x+4\right ) \]

[In]

int(8*x + (exp((x*exp(5))/25)*(200*x + exp(5)*(4*x^2 - 4*x^3) - 300*x^2))/25 - 15*x^2 + 4*x^3,x)

[Out]

-x^2*(x - 1)*(4*exp((x*exp(5))/25) - x + 4)