3.11.13 \(\int \frac {200 e^{\frac {50-30 x}{3 x}}+19 x^2}{3 x^2} \, dx\)

Optimal. Leaf size=30 \[ x-4 \left (e^{\frac {5 \left (\frac {10}{3}-2 x\right )}{x}}-\frac {4}{3} \left (-e^3+x\right )\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 0.63, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 14, 2209} \begin {gather*} \frac {19 x}{3}-4 e^{\frac {50}{3 x}-10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(200*E^((50 - 30*x)/(3*x)) + 19*x^2)/(3*x^2),x]

[Out]

-4*E^(-10 + 50/(3*x)) + (19*x)/3

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]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {200 e^{\frac {50-30 x}{3 x}}+19 x^2}{x^2} \, dx\\ &=\frac {1}{3} \int \left (19+\frac {200 e^{-10+\frac {50}{3 x}}}{x^2}\right ) \, dx\\ &=\frac {19 x}{3}+\frac {200}{3} \int \frac {e^{-10+\frac {50}{3 x}}}{x^2} \, dx\\ &=-4 e^{-10+\frac {50}{3 x}}+\frac {19 x}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.63 \begin {gather*} -4 e^{-10+\frac {50}{3 x}}+\frac {19 x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(200*E^((50 - 30*x)/(3*x)) + 19*x^2)/(3*x^2),x]

[Out]

-4*E^(-10 + 50/(3*x)) + (19*x)/3

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fricas [A]  time = 0.56, size = 17, normalized size = 0.57 \begin {gather*} \frac {19}{3} \, x - 4 \, e^{\left (-\frac {10 \, {\left (3 \, x - 5\right )}}{3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(200*exp(1/3*(-30*x+50)/x)+19*x^2)/x^2,x, algorithm="fricas")

[Out]

19/3*x - 4*e^(-10/3*(3*x - 5)/x)

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giac [B]  time = 0.33, size = 51, normalized size = 1.70 \begin {gather*} -\frac {\frac {12 \, {\left (3 \, x - 5\right )} e^{\left (-\frac {10 \, {\left (3 \, x - 5\right )}}{3 \, x}\right )}}{x} - 36 \, e^{\left (-\frac {10 \, {\left (3 \, x - 5\right )}}{3 \, x}\right )} + 95}{3 \, {\left (\frac {3 \, x - 5}{x} - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(200*exp(1/3*(-30*x+50)/x)+19*x^2)/x^2,x, algorithm="giac")

[Out]

-1/3*(12*(3*x - 5)*e^(-10/3*(3*x - 5)/x)/x - 36*e^(-10/3*(3*x - 5)/x) + 95)/((3*x - 5)/x - 3)

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maple [A]  time = 0.03, size = 15, normalized size = 0.50

method	result	size
derivativedivides	\(\frac {19 x}{3}-4 \,{\mathrm e}^{-10+\frac {50}{3 x}}\)	\(15\)
default	\(\frac {19 x}{3}-4 \,{\mathrm e}^{-10+\frac {50}{3 x}}\)	\(15\)
risch	\(\frac {19 x}{3}-4 \,{\mathrm e}^{-\frac {10 \left (3 x -5\right )}{3 x}}\)	\(18\)
norman	\(\frac {\frac {19 x^{2}}{3}-4 x \,{\mathrm e}^{\frac {-30 x +50}{3 x}}}{x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(200*exp(1/3*(-30*x+50)/x)+19*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

19/3*x-4*exp(-10+50/3/x)

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maxima [A]  time = 0.36, size = 14, normalized size = 0.47 \begin {gather*} \frac {19}{3} \, x - 4 \, e^{\left (\frac {50}{3 \, x} - 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(200*exp(1/3*(-30*x+50)/x)+19*x^2)/x^2,x, algorithm="maxima")

[Out]

19/3*x - 4*e^(50/3/x - 10)

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mupad [B]  time = 0.70, size = 14, normalized size = 0.47 \begin {gather*} \frac {19\,x}{3}-4\,{\mathrm {e}}^{\frac {50}{3\,x}-10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((200*exp(-(10*x - 50/3)/x))/3 + (19*x^2)/3)/x^2,x)

[Out]

(19*x)/3 - 4*exp(50/(3*x) - 10)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.50 \begin {gather*} \frac {19 x}{3} - 4 e^{\frac {\frac {50}{3} - 10 x}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(200*exp(1/3*(-30*x+50)/x)+19*x**2)/x**2,x)

[Out]

19*x/3 - 4*exp((50/3 - 10*x)/x)

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