3.85.30 \(\int \frac {-27+e^{\frac {1}{3} (9-42 x+49 x^2)} (-42 x^2+98 x^3)}{3 x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {1}{3} (3-7 x)^2}+\frac {9}{x} \]

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 14, 2209} \begin {gather*} e^{\frac {1}{3} (3-7 x)^2}+\frac {9}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-27 + E^((9 - 42*x + 49*x^2)/3)*(-42*x^2 + 98*x^3))/(3*x^2),x]

[Out]

E^((3 - 7*x)^2/3) + 9/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]]

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 {-27+e^{\frac {1}{3} \left (9-42 x+49 x^2\right )} \left (-42 x^2+98 x^3\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {27}{x^2}+14 e^{\frac {1}{3} (3-7 x)^2} (-3+7 x)\right ) \, dx\\ &=\frac {9}{x}+\frac {14}{3} \int e^{\frac {1}{3} (3-7 x)^2} (-3+7 x) \, dx\\ &=e^{\frac {1}{3} (3-7 x)^2}+\frac {9}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {1}{3} (3-7 x)^2}+\frac {9}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27 + E^((9 - 42*x + 49*x^2)/3)*(-42*x^2 + 98*x^3))/(3*x^2),x]

[Out]

E^((3 - 7*x)^2/3) + 9/x

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fricas [A]  time = 0.52, size = 19, normalized size = 1.00 \begin {gather*} \frac {x e^{\left (\frac {49}{3} \, x^{2} - 14 \, x + 3\right )} + 9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((98*x^3-42*x^2)*exp(49/3*x^2-14*x+3)-27)/x^2,x, algorithm="fricas")

[Out]

(x*e^(49/3*x^2 - 14*x + 3) + 9)/x

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giac [A]  time = 0.14, size = 19, normalized size = 1.00 \begin {gather*} \frac {x e^{\left (\frac {49}{3} \, x^{2} - 14 \, x + 3\right )} + 9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((98*x^3-42*x^2)*exp(49/3*x^2-14*x+3)-27)/x^2,x, algorithm="giac")

[Out]

(x*e^(49/3*x^2 - 14*x + 3) + 9)/x

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maple [A]  time = 0.03, size = 17, normalized size = 0.89

method	result	size
risch	\(\frac {9}{x}+{\mathrm e}^{\frac {\left (7 x -3\right )^{2}}{3}}\)	\(17\)
default	\(\frac {9}{x}+{\mathrm e}^{\frac {49}{3} x^{2}-14 x +3}\)	\(18\)
norman	\(\frac {9+x \,{\mathrm e}^{\frac {49}{3} x^{2}-14 x +3}}{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((98*x^3-42*x^2)*exp(49/3*x^2-14*x+3)-27)/x^2,x,method=_RETURNVERBOSE)

[Out]

9/x+exp(1/3*(7*x-3)^2)

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maxima [C]  time = 0.56, size = 92, normalized size = 4.84 \begin {gather*} i \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\frac {7}{3} i \, \sqrt {3} x - i \, \sqrt {3}\right ) + \frac {1}{3} \, \sqrt {3} {\left (\frac {3 \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (7 \, x - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {-{\left (7 \, x - 3\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (7 \, x - 3\right )}^{2}}} + \sqrt {3} e^{\left (\frac {1}{3} \, {\left (7 \, x - 3\right )}^{2}\right )}\right )} + \frac {9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((98*x^3-42*x^2)*exp(49/3*x^2-14*x+3)-27)/x^2,x, algorithm="maxima")

[Out]

I*sqrt(3)*sqrt(pi)*erf(7/3*I*sqrt(3)*x - I*sqrt(3)) + 1/3*sqrt(3)*(3*sqrt(3)*sqrt(1/3)*sqrt(pi)*(7*x - 3)*(erf
(sqrt(1/3)*sqrt(-(7*x - 3)^2)) - 1)/sqrt(-(7*x - 3)^2) + sqrt(3)*e^(1/3*(7*x - 3)^2)) + 9/x

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mupad [B]  time = 0.11, size = 17, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{\frac {49\,x^2}{3}-14\,x+3}+\frac {9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp((49*x^2)/3 - 14*x + 3)*(42*x^2 - 98*x^3))/3 + 9)/x^2,x)

[Out]

exp((49*x^2)/3 - 14*x + 3) + 9/x

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sympy [A]  time = 0.11, size = 15, normalized size = 0.79 \begin {gather*} e^{\frac {49 x^{2}}{3} - 14 x + 3} + \frac {9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((98*x**3-42*x**2)*exp(49/3*x**2-14*x+3)-27)/x**2,x)

[Out]

exp(49*x**2/3 - 14*x + 3) + 9/x

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