3.28.12 \(\int \frac {e^{\frac {1+30 x}{x}} (-1-4 x)}{45 x^6} \, dx\)

Optimal. Leaf size=14 \[ \frac {e^{30+\frac {1}{x}}}{45 x^4} \]

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Rubi [B]  time = 0.04, antiderivative size = 36, normalized size of antiderivative = 2.57, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 2288} \begin {gather*} -\frac {e^{\frac {30 x+1}{x}}}{45 x^6 \left (\frac {30}{x}-\frac {30 x+1}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((1 + 30*x)/x)*(-1 - 4*x))/(45*x^6),x]

[Out]

-1/45*E^((1 + 30*x)/x)/(x^6*(30/x - (1 + 30*x)/x^2))

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{45} \int \frac {e^{\frac {1+30 x}{x}} (-1-4 x)}{x^6} \, dx\\ &=-\frac {e^{\frac {1+30 x}{x}}}{45 x^6 \left (\frac {30}{x}-\frac {1+30 x}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {e^{30+\frac {1}{x}}}{45 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 + 30*x)/x)*(-1 - 4*x))/(45*x^6),x]

[Out]

E^(30 + x^(-1))/(45*x^4)

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fricas [A]  time = 0.54, size = 15, normalized size = 1.07 \begin {gather*} \frac {e^{\left (\frac {30 \, x + 1}{x}\right )}}{45 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-4*x-1)*exp((30*x+1)/x)/x^6,x, algorithm="fricas")

[Out]

1/45*e^((30*x + 1)/x)/x^4

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giac [B]  time = 0.21, size = 99, normalized size = 7.07 \begin {gather*} \frac {{\left (30 \, x + 1\right )}^{4} e^{\left (\frac {30 \, x + 1}{x}\right )}}{45 \, x^{4}} - \frac {8 \, {\left (30 \, x + 1\right )}^{3} e^{\left (\frac {30 \, x + 1}{x}\right )}}{3 \, x^{3}} + \frac {120 \, {\left (30 \, x + 1\right )}^{2} e^{\left (\frac {30 \, x + 1}{x}\right )}}{x^{2}} - \frac {2400 \, {\left (30 \, x + 1\right )} e^{\left (\frac {30 \, x + 1}{x}\right )}}{x} + 18000 \, e^{\left (\frac {30 \, x + 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-4*x-1)*exp((30*x+1)/x)/x^6,x, algorithm="giac")

[Out]

1/45*(30*x + 1)^4*e^((30*x + 1)/x)/x^4 - 8/3*(30*x + 1)^3*e^((30*x + 1)/x)/x^3 + 120*(30*x + 1)^2*e^((30*x + 1
)/x)/x^2 - 2400*(30*x + 1)*e^((30*x + 1)/x)/x + 18000*e^((30*x + 1)/x)

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maple [A]  time = 0.07, size = 16, normalized size = 1.14

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {30 x +1}{x}}}{45 x^{4}}\)	\(16\)
norman	\(\frac {{\mathrm e}^{\frac {30 x +1}{x}}}{45 x^{4}}\)	\(16\)
risch	\(\frac {{\mathrm e}^{\frac {30 x +1}{x}}}{45 x^{4}}\)	\(16\)
meijerg	\(-\frac {{\mathrm e}^{30} \left (24-\frac {\left (\frac {5}{x^{4}}-\frac {20}{x^{3}}+\frac {60}{x^{2}}-\frac {120}{x}+120\right ) {\mathrm e}^{\frac {1}{x}}}{5}\right )}{45}+\frac {4 \,{\mathrm e}^{30} \left (6-\frac {\left (-\frac {4}{x^{3}}+\frac {12}{x^{2}}-\frac {24}{x}+24\right ) {\mathrm e}^{\frac {1}{x}}}{4}\right )}{45}\)	\(65\)
derivativedivides	\(-2400 \,{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )+120 \,{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )^{2}-\frac {8 \,{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )^{3}}{3}+\frac {{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )^{4}}{45}+18000 \,{\mathrm e}^{30+\frac {1}{x}}\)	\(68\)
default	\(-2400 \,{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )+120 \,{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )^{2}-\frac {8 \,{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )^{3}}{3}+\frac {{\mathrm e}^{30+\frac {1}{x}} \left (30+\frac {1}{x}\right )^{4}}{45}+18000 \,{\mathrm e}^{30+\frac {1}{x}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/45*(-4*x-1)*exp((30*x+1)/x)/x^6,x,method=_RETURNVERBOSE)

[Out]

1/45*exp((30*x+1)/x)/x^4

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maxima [C]  time = 0.79, size = 23, normalized size = 1.64 \begin {gather*} \frac {1}{45} \, e^{30} \Gamma \left (5, -\frac {1}{x}\right ) - \frac {4}{45} \, e^{30} \Gamma \left (4, -\frac {1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-4*x-1)*exp((30*x+1)/x)/x^6,x, algorithm="maxima")

[Out]

1/45*e^30*gamma(5, -1/x) - 4/45*e^30*gamma(4, -1/x)

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mupad [B]  time = 1.49, size = 11, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{30}}{45\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((30*x + 1)/x)*(4*x + 1))/(45*x^6),x)

[Out]

(exp(1/x)*exp(30))/(45*x^4)

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sympy [A]  time = 0.10, size = 12, normalized size = 0.86 \begin {gather*} \frac {e^{\frac {30 x + 1}{x}}}{45 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-4*x-1)*exp((30*x+1)/x)/x**6,x)

[Out]

exp((30*x + 1)/x)/(45*x**4)

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