3.88.41 \(\int \frac {e^{-4-8 x} (-4860-6075 x+9720 x^2)}{x^5} \, dx\) [8741]

Optimal. Leaf size=22 \[ \frac {1215 e^{-16-12 (-1+x)+4 x} (1-x)}{x^4} \]

[Out]

1215*(1-x)/exp(1+2*x)^4/x^4

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Rubi [A]
time = 0.23, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2230, 2208, 2209} \begin {gather*} \frac {1215 e^{-8 x-4}}{x^4}-\frac {1215 e^{-8 x-4}}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-4 - 8*x)*(-4860 - 6075*x + 9720*x^2))/x^5,x]

[Out]

(1215*E^(-4 - 8*x))/x^4 - (1215*E^(-4 - 8*x))/x^3

Rule 2208

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2230

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4860 e^{-4-8 x}}{x^5}-\frac {6075 e^{-4-8 x}}{x^4}+\frac {9720 e^{-4-8 x}}{x^3}\right ) \, dx\\ &=-\left (4860 \int \frac {e^{-4-8 x}}{x^5} \, dx\right )-6075 \int \frac {e^{-4-8 x}}{x^4} \, dx+9720 \int \frac {e^{-4-8 x}}{x^3} \, dx\\ &=\frac {1215 e^{-4-8 x}}{x^4}+\frac {2025 e^{-4-8 x}}{x^3}-\frac {4860 e^{-4-8 x}}{x^2}+9720 \int \frac {e^{-4-8 x}}{x^4} \, dx+16200 \int \frac {e^{-4-8 x}}{x^3} \, dx-38880 \int \frac {e^{-4-8 x}}{x^2} \, dx\\ &=\frac {1215 e^{-4-8 x}}{x^4}-\frac {1215 e^{-4-8 x}}{x^3}-\frac {12960 e^{-4-8 x}}{x^2}+\frac {38880 e^{-4-8 x}}{x}-25920 \int \frac {e^{-4-8 x}}{x^3} \, dx-64800 \int \frac {e^{-4-8 x}}{x^2} \, dx+311040 \int \frac {e^{-4-8 x}}{x} \, dx\\ &=\frac {1215 e^{-4-8 x}}{x^4}-\frac {1215 e^{-4-8 x}}{x^3}+\frac {103680 e^{-4-8 x}}{x}+\frac {311040 \text {Ei}(-8 x)}{e^4}+103680 \int \frac {e^{-4-8 x}}{x^2} \, dx+518400 \int \frac {e^{-4-8 x}}{x} \, dx\\ &=\frac {1215 e^{-4-8 x}}{x^4}-\frac {1215 e^{-4-8 x}}{x^3}+\frac {829440 \text {Ei}(-8 x)}{e^4}-829440 \int \frac {e^{-4-8 x}}{x} \, dx\\ &=\frac {1215 e^{-4-8 x}}{x^4}-\frac {1215 e^{-4-8 x}}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.13, size = 15, normalized size = 0.68 \begin {gather*} -\frac {1215 e^{-4-8 x} (-1+x)}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 - 8*x)*(-4860 - 6075*x + 9720*x^2))/x^5,x]

[Out]

(-1215*E^(-4 - 8*x)*(-1 + x))/x^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(18)=36\).
time = 1.40, size = 81, normalized size = 3.68

method	result	size
risch	\(-\frac {1215 \left (x -1\right ) {\mathrm e}^{-8 x -4}}{x^{4}}\)	\(15\)
gosper	\(-\frac {1215 \left (x -1\right ) {\mathrm e}^{-8 x -4}}{x^{4}}\)	\(17\)
norman	\(\frac {\left (1215-1215 x \right ) {\mathrm e}^{-8 x -4}}{x^{4}}\)	\(18\)
derivativedivides	\(\frac {405 \,{\mathrm e}^{-8 x -4} \left (32 \left (2 x +1\right )^{3}-104 \left (2 x +1\right )^{2}+232 x +69\right )}{8 x^{4}}+\frac {15795 \,{\mathrm e}^{-8 x -4}}{8 x^{4}}-\frac {405 \,{\mathrm e}^{-8 x -4} \left (8 \left (2 x +1\right )^{3}-26 \left (2 x +1\right )^{2}+64 x +21\right )}{2 x^{4}}\)	\(81\)
default	\(\frac {405 \,{\mathrm e}^{-8 x -4} \left (32 \left (2 x +1\right )^{3}-104 \left (2 x +1\right )^{2}+232 x +69\right )}{8 x^{4}}+\frac {15795 \,{\mathrm e}^{-8 x -4}}{8 x^{4}}-\frac {405 \,{\mathrm e}^{-8 x -4} \left (8 \left (2 x +1\right )^{3}-26 \left (2 x +1\right )^{2}+64 x +21\right )}{2 x^{4}}\)	\(81\)
meijerg	\(622080 \,{\mathrm e}^{-8 x -12+8 x \,{\mathrm e}^{-4}} \left (\frac {{\mathrm e}^{8} \left (576 x^{2} {\mathrm e}^{-8}-96 x \,{\mathrm e}^{-4}+6\right )}{768 x^{2}}-\frac {{\mathrm e}^{8-8 x \,{\mathrm e}^{-4}} \left (3-24 x \,{\mathrm e}^{-4}\right )}{384 x^{2}}-\frac {\ln \left (8 x \,{\mathrm e}^{-4}\right )}{2}-\frac {\expIntegral \left (1, 8 x \,{\mathrm e}^{-4}\right )}{2}-\frac {11}{4}+\frac {\ln \left (x \right )}{2}+\frac {3 \ln \left (2\right )}{2}-\frac {{\mathrm e}^{8}}{128 x^{2}}+\frac {{\mathrm e}^{4}}{8 x}\right )-3110400 \,{\mathrm e}^{-8 x -16+8 x \,{\mathrm e}^{-4}} \left (\frac {{\mathrm e}^{12} \left (-11264 x^{3} {\mathrm e}^{-12}+2304 x^{2} {\mathrm e}^{-8}-288 x \,{\mathrm e}^{-4}+24\right )}{36864 x^{3}}-\frac {{\mathrm e}^{12-8 x \,{\mathrm e}^{-4}} \left (256 x^{2} {\mathrm e}^{-8}-32 x \,{\mathrm e}^{-4}+8\right )}{12288 x^{3}}+\frac {\ln \left (8 x \,{\mathrm e}^{-4}\right )}{6}+\frac {\expIntegral \left (1, 8 x \,{\mathrm e}^{-4}\right )}{6}+\frac {35}{36}-\frac {\ln \left (x \right )}{6}-\frac {\ln \left (2\right )}{2}-\frac {{\mathrm e}^{12}}{1536 x^{3}}+\frac {{\mathrm e}^{8}}{128 x^{2}}-\frac {{\mathrm e}^{4}}{16 x}\right )-19906560 \,{\mathrm e}^{-8 x -20+8 x \,{\mathrm e}^{-4}} \left (\frac {{\mathrm e}^{16} \left (512000 x^{4} {\mathrm e}^{-16}-122880 x^{3} {\mathrm e}^{-12}+23040 x^{2} {\mathrm e}^{-8}-3840 x \,{\mathrm e}^{-4}+360\right )}{5898240 x^{4}}-\frac {{\mathrm e}^{16-8 x \,{\mathrm e}^{-4}} \left (-2560 x^{3} {\mathrm e}^{-12}+320 x^{2} {\mathrm e}^{-8}-80 x \,{\mathrm e}^{-4}+30\right )}{491520 x^{4}}-\frac {\ln \left (8 x \,{\mathrm e}^{-4}\right )}{24}-\frac {\expIntegral \left (1, 8 x \,{\mathrm e}^{-4}\right )}{24}-\frac {73}{288}+\frac {\ln \left (x \right )}{24}+\frac {\ln \left (2\right )}{8}-\frac {{\mathrm e}^{16}}{16384 x^{4}}+\frac {{\mathrm e}^{12}}{1536 x^{3}}-\frac {{\mathrm e}^{8}}{256 x^{2}}+\frac {{\mathrm e}^{4}}{48 x}\right )\)	\(350\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9720*x^2-6075*x-4860)/x^5/exp(2*x+1)^4,x,method=_RETURNVERBOSE)

[Out]

405/8*exp(-8*x-4)*(32*(2*x+1)^3-104*(2*x+1)^2+232*x+69)/x^4+15795/8*exp(-8*x-4)/x^4-405/2*exp(-8*x-4)*(8*(2*x+
1)^3-26*(2*x+1)^2+64*x+21)/x^4

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.30, size = 28, normalized size = 1.27 \begin {gather*} -622080 \, e^{\left (-4\right )} \Gamma \left (-2, 8 \, x\right ) + 3110400 \, e^{\left (-4\right )} \Gamma \left (-3, 8 \, x\right ) + 19906560 \, e^{\left (-4\right )} \Gamma \left (-4, 8 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9720*x^2-6075*x-4860)/x^5/exp(1+2*x)^4,x, algorithm="maxima")

[Out]

-622080*e^(-4)*gamma(-2, 8*x) + 3110400*e^(-4)*gamma(-3, 8*x) + 19906560*e^(-4)*gamma(-4, 8*x)

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Fricas [A]
time = 0.36, size = 14, normalized size = 0.64 \begin {gather*} -\frac {1215 \, {\left (x - 1\right )} e^{\left (-8 \, x - 4\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9720*x^2-6075*x-4860)/x^5/exp(1+2*x)^4,x, algorithm="fricas")

[Out]

-1215*(x - 1)*e^(-8*x - 4)/x^4

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Sympy [A]
time = 0.03, size = 15, normalized size = 0.68 \begin {gather*} \frac {\left (1215 - 1215 x\right ) e^{- 8 x - 4}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9720*x**2-6075*x-4860)/x**5/exp(1+2*x)**4,x)

[Out]

(1215 - 1215*x)*exp(-8*x - 4)/x**4

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Giac [A]
time = 0.40, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1215 \, {\left (x e^{\left (-8 \, x\right )} - e^{\left (-8 \, x\right )}\right )} e^{\left (-4\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9720*x^2-6075*x-4860)/x^5/exp(1+2*x)^4,x, algorithm="giac")

[Out]

-1215*(x*e^(-8*x) - e^(-8*x))*e^(-4)/x^4

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Mupad [B]
time = 5.40, size = 14, normalized size = 0.64 \begin {gather*} -\frac {1215\,{\mathrm {e}}^{-8\,x-4}\,\left (x-1\right )}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- 8*x - 4)*(6075*x - 9720*x^2 + 4860))/x^5,x)

[Out]

-(1215*exp(- 8*x - 4)*(x - 1))/x^4

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