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

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

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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} $$\begin{array}{c}{e}^{\frac{1}{3}(3-7x{)}^2}+\frac{9}{x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int \frac{-27+e^{\frac{1}{3}(9-42x+49x^2)}(-42x^2+98x^3)}{x^2}dx\\ & =\frac{1}{3}\int (-\frac{27}{x^2}+14e^{\frac{1}{3}(3-7x{)}^2}(-3+7x))dx\\ & =\frac{9}{x}+\frac{14}{3}\int e^{\frac{1}{3}(3-7x{)}^2}(-3+7x)dx\\ & =e^{\frac{1}{3}(3-7x{)}^2}+\frac{9}{x}\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{x{e}^{(\frac{49}{3}{x}^2-14x+3)}+9}{x}\end{array}$$

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 $$\begin{array}{c}\frac{x{e}^{(\frac{49}{3}{x}^2-14x+3)}+9}{x}\end{array}$$

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	$\frac{9}{x}+{\mathrm{e}}^{\frac{{(7x-3)}^2}{3}}$	$17$
default	$\frac{9}{x}+{\mathrm{e}}^{\frac{49}{3}{x}^2-14x+3}$	$18$
norman	$\frac{9+x{\mathrm{e}}^{\frac{49}{3}{x}^2-14x+3}}{x}$	$20$

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 $$\begin{array}{c}i\sqrt{3}\sqrt{\pi }\mathrm{erf}(\frac{7}{3}i\sqrt{3}x-i\sqrt{3})+\frac{1}{3}\sqrt{3}(\frac{3\sqrt{3}\sqrt{\frac{1}{3}}\sqrt{\pi }(7x-3)(\mathrm{erf}\left(\sqrt{\frac{1}{3}}\sqrt{-{(7x-3)}^2}\right)-1)}{\sqrt{-{(7x-3)}^2}}+\sqrt{3}{e}^{\left(\frac{1}{3}{(7x-3)}^2\right)})+\frac{9}{x}\end{array}$$

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 $$\begin{array}{c}{\mathrm{e}}^{\frac{49x^2}{3}-14x+3}+\frac{9}{x}\end{array}$$

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 $$\begin{array}{c}{e}^{\frac{49x^2}{3}-14x+3}+\frac{9}{x}\end{array}$$

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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