3.70.25 \(\int \frac {1}{15} (90 x+75 x^2+e^{\frac {-10+x^2}{5 x}} (10 x^2+20 x^3+x^4)) \, dx\)

Optimal. Leaf size=32 \[ x^2 \left (3+\frac {1}{3} \left (e^{-\frac {2}{x}+\frac {x}{5}}+\frac {5}{x}\right ) x^2\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 56, normalized size of antiderivative = 1.75, number of steps used = 4, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 1594, 2288} \begin {gather*} \frac {5 x^3}{3}+\frac {e^{-\frac {10-x^2}{5 x}} \left (x^2+10\right ) x^2}{3 \left (\frac {10-x^2}{x^2}+2\right )}+3 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(90*x + 75*x^2 + E^((-10 + x^2)/(5*x))*(10*x^2 + 20*x^3 + x^4))/15,x]

[Out]

3*x^2 + (5*x^3)/3 + (x^2*(10 + x^2))/(3*E^((10 - x^2)/(5*x))*(2 + (10 - x^2)/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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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}{15} \int \left (90 x+75 x^2+e^{\frac {-10+x^2}{5 x}} \left (10 x^2+20 x^3+x^4\right )\right ) \, dx\\ &=3 x^2+\frac {5 x^3}{3}+\frac {1}{15} \int e^{\frac {-10+x^2}{5 x}} \left (10 x^2+20 x^3+x^4\right ) \, dx\\ &=3 x^2+\frac {5 x^3}{3}+\frac {1}{15} \int e^{\frac {-10+x^2}{5 x}} x^2 \left (10+20 x+x^2\right ) \, dx\\ &=3 x^2+\frac {5 x^3}{3}+\frac {e^{-\frac {10-x^2}{5 x}} x^2 \left (10+x^2\right )}{3 \left (2+\frac {10-x^2}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 29, normalized size = 0.91 \begin {gather*} \frac {1}{3} x^2 \left (9+5 x+e^{-\frac {2}{x}+\frac {x}{5}} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(90*x + 75*x^2 + E^((-10 + x^2)/(5*x))*(10*x^2 + 20*x^3 + x^4))/15,x]

[Out]

(x^2*(9 + 5*x + E^(-2/x + x/5)*x^2))/3

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fricas [A]  time = 0.58, size = 27, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, x^{4} e^{\left (\frac {x^{2} - 10}{5 \, x}\right )} + \frac {5}{3} \, x^{3} + 3 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(x^4+20*x^3+10*x^2)*exp(1/5*(x^2-10)/x)+5*x^2+6*x,x, algorithm="fricas")

[Out]

1/3*x^4*e^(1/5*(x^2 - 10)/x) + 5/3*x^3 + 3*x^2

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giac [A]  time = 0.25, size = 27, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, x^{4} e^{\left (\frac {x^{2} - 10}{5 \, x}\right )} + \frac {5}{3} \, x^{3} + 3 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(x^4+20*x^3+10*x^2)*exp(1/5*(x^2-10)/x)+5*x^2+6*x,x, algorithm="giac")

[Out]

1/3*x^4*e^(1/5*(x^2 - 10)/x) + 5/3*x^3 + 3*x^2

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maple [A]  time = 0.05, size = 28, normalized size = 0.88

method	result	size
default	\(3 x^{2}+\frac {5 x^{3}}{3}+\frac {{\mathrm e}^{\frac {x^{2}-10}{5 x}} x^{4}}{3}\)	\(28\)
norman	\(3 x^{2}+\frac {5 x^{3}}{3}+\frac {{\mathrm e}^{\frac {x^{2}-10}{5 x}} x^{4}}{3}\)	\(28\)
risch	\(3 x^{2}+\frac {5 x^{3}}{3}+\frac {{\mathrm e}^{\frac {x^{2}-10}{5 x}} x^{4}}{3}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/15*(x^4+20*x^3+10*x^2)*exp(1/5*(x^2-10)/x)+5*x^2+6*x,x,method=_RETURNVERBOSE)

[Out]

3*x^2+5/3*x^3+1/3*exp(1/5*(x^2-10)/x)*x^4

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maxima [A]  time = 0.38, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, x^{4} e^{\left (\frac {1}{5} \, x - \frac {2}{x}\right )} + \frac {5}{3} \, x^{3} + 3 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(x^4+20*x^3+10*x^2)*exp(1/5*(x^2-10)/x)+5*x^2+6*x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.21, size = 26, normalized size = 0.81 \begin {gather*} \frac {x^4\,{\mathrm {e}}^{\frac {x}{5}-\frac {2}{x}}}{3}+3\,x^2+\frac {5\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x + (exp((x^2/5 - 2)/x)*(10*x^2 + 20*x^3 + x^4))/15 + 5*x^2,x)

[Out]

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

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sympy [A]  time = 0.12, size = 26, normalized size = 0.81 \begin {gather*} \frac {x^{4} e^{\frac {\frac {x^{2}}{5} - 2}{x}}}{3} + \frac {5 x^{3}}{3} + 3 x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(x**4+20*x**3+10*x**2)*exp(1/5*(x**2-10)/x)+5*x**2+6*x,x)

[Out]

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

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