3.71.91 \(\int \frac {30 x+e^5 (-1-6 x-12 x^2)}{e^5 x} \, dx\)

Optimal. Leaf size=20 \[ 19-6 \left (-5+x-\frac {5 x}{e^5}+x^2\right )-\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {12, 14} \begin {gather*} -6 x^2+\frac {6 \left (5-e^5\right ) x}{e^5}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(30*x + E^5*(-1 - 6*x - 12*x^2))/(E^5*x),x]

[Out]

(6*(5 - E^5)*x)/E^5 - 6*x^2 - Log[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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {30 x+e^5 \left (-1-6 x-12 x^2\right )}{x} \, dx}{e^5}\\ &=\frac {\int \left (6 \left (5-e^5\right )-\frac {e^5}{x}-12 e^5 x\right ) \, dx}{e^5}\\ &=\frac {6 \left (5-e^5\right ) x}{e^5}-6 x^2-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} -6 x+\frac {30 x}{e^5}-6 x^2-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30*x + E^5*(-1 - 6*x - 12*x^2))/(E^5*x),x]

[Out]

-6*x + (30*x)/E^5 - 6*x^2 - Log[x]

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fricas [A]  time = 0.61, size = 22, normalized size = 1.10 \begin {gather*} -{\left (6 \, {\left (x^{2} + x\right )} e^{5} + e^{5} \log \relax (x) - 30 \, x\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^2-6*x-1)*exp(5)+30*x)/x/exp(5),x, algorithm="fricas")

[Out]

-(6*(x^2 + x)*e^5 + e^5*log(x) - 30*x)*e^(-5)

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giac [A]  time = 0.15, size = 26, normalized size = 1.30 \begin {gather*} -{\left (6 \, x^{2} e^{5} + 6 \, x e^{5} + e^{5} \log \left ({\left | x \right |}\right ) - 30 \, x\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^2-6*x-1)*exp(5)+30*x)/x/exp(5),x, algorithm="giac")

[Out]

-(6*x^2*e^5 + 6*x*e^5 + e^5*log(abs(x)) - 30*x)*e^(-5)

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maple [A]  time = 0.03, size = 19, normalized size = 0.95

method	result	size
risch	\(-6 x^{2}-6 x +30 x \,{\mathrm e}^{-5}-\ln \relax (x )\)	\(19\)
norman	\(-6 x^{2}-6 \left ({\mathrm e}^{5}-5\right ) {\mathrm e}^{-5} x -\ln \relax (x )\)	\(22\)
default	\({\mathrm e}^{-5} \left (-6 x^{2} {\mathrm e}^{5}-6 x \,{\mathrm e}^{5}+30 x -{\mathrm e}^{5} \ln \relax (x )\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x^2-6*x-1)*exp(5)+30*x)/x/exp(5),x,method=_RETURNVERBOSE)

[Out]

-6*x^2-6*x+30*x*exp(-5)-ln(x)

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maxima [A]  time = 0.39, size = 24, normalized size = 1.20 \begin {gather*} -{\left (6 \, x^{2} e^{5} + 6 \, x {\left (e^{5} - 5\right )} + e^{5} \log \relax (x)\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^2-6*x-1)*exp(5)+30*x)/x/exp(5),x, algorithm="maxima")

[Out]

-(6*x^2*e^5 + 6*x*(e^5 - 5) + e^5*log(x))*e^(-5)

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mupad [B]  time = 4.11, size = 21, normalized size = 1.05 \begin {gather*} -\ln \relax (x)-6\,x^2-x\,{\mathrm {e}}^{-5}\,\left (6\,{\mathrm {e}}^5-30\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-5)*(30*x - exp(5)*(6*x + 12*x^2 + 1)))/x,x)

[Out]

- log(x) - 6*x^2 - x*exp(-5)*(6*exp(5) - 30)

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sympy [A]  time = 0.13, size = 27, normalized size = 1.35 \begin {gather*} \frac {- 6 x^{2} e^{5} - x \left (-30 + 6 e^{5}\right ) - e^{5} \log {\relax (x )}}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x**2-6*x-1)*exp(5)+30*x)/x/exp(5),x)

[Out]

(-6*x**2*exp(5) - x*(-30 + 6*exp(5)) - exp(5)*log(x))*exp(-5)

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