3.80.23 $\int \frac{-3x^2+e^2(-x^2+3x^4)+e^x(-30+30x+e^2(-2x^3-x^4))}{x^2}dx$

Optimal. Leaf size=36 $$-e^2(x+(e^x-x)x^2)+3(-x+\frac{10e^x+x}{x})$$

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 34, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 6, integrand size = 49, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.122, Rules used = {14, 2199, 2177, 2178, 2176, 2194} $$\begin{array}{c}{e}^2{x}^3-e^{x+2}{x}^2-(3+e^2)x+\frac{30e^x}{x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-3*x^2 + E^2*(-x^2 + 3*x^4) + E^x*(-30 + 30*x + E^2*(-2*x^3 - x^4)))/x^2,x]

[Out]

(30*E^x)/x - (3 + E^2)*x - E^(2 + x)*x^2 + E^2*x^3

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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-3(1+\frac{e^2}{3})+3e^2{x}^2-\frac{e^x(30-30x+2e^2{x}^3+e^2{x}^4)}{x^2})dx\\ & =-\left((3+e^2)x\right)+e^2{x}^3-\int \frac{e^x(30-30x+2e^2{x}^3+e^2{x}^4)}{x^2}dx\\ & =-\left((3+e^2)x\right)+e^2{x}^3-\int (\frac{30e^x}{x^2}-\frac{30e^x}{x}+2e^{2+x}x+e^{2+x}{x}^2)dx\\ & =-\left((3+e^2)x\right)+e^2{x}^3-2\int e^{2+x}xdx-30\int \frac{e^x}{x^2}dx+30\int \frac{e^x}{x}dx-\int e^{2+x}{x}^{2}dx\\ & =\frac{30e^x}{x}-2e^{2+x}x-(3+e^2)x-e^{2+x}{x}^2+e^2{x}^3+30\text{Ei}(x)+2\int e^{2+x}dx+2\int e^{2+x}xdx-30\int \frac{e^x}{x}dx\\ & =2e^{2+x}+\frac{30e^x}{x}-(3+e^2)x-e^{2+x}{x}^2+e^2{x}^3-2\int e^{2+x}dx\\ & =\frac{30e^x}{x}-(3+e^2)x-e^{2+x}{x}^2+e^2{x}^3\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 35, normalized size = 0.97 $$\begin{array}{c}\frac{30e^x}{x}-3x-e^{2}x-e^{2+x}{x}^2+e^2{x}^3\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^2 + E^2*(-x^2 + 3*x^4) + E^x*(-30 + 30*x + E^2*(-2*x^3 - x^4)))/x^2,x]

[Out]

(30*E^x)/x - 3*x - E^2*x - E^(2 + x)*x^2 + E^2*x^3

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 35, normalized size = 0.97 $$\begin{array}{c}-\frac{3x^2-(x^4-x^2)e^2+(x^3{e}^2-30)e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-2*x^3)*exp(2)+30*x-30)*exp(x)+(3*x^4-x^2)*exp(2)-3*x^2)/x^2,x, algorithm="fricas")

[Out]

-(3*x^2 - (x^4 - x^2)*e^2 + (x^3*e^2 - 30)*e^x)/x

________________________________________________________________________________________

giac [A]  time = 0.24, size = 36, normalized size = 1.00 $$\begin{array}{c}\frac{x^4{e}^2-x^3{e}^{(x+2)}-x^2{e}^2-3x^2+30e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-2*x^3)*exp(2)+30*x-30)*exp(x)+(3*x^4-x^2)*exp(2)-3*x^2)/x^2,x, algorithm="giac")

[Out]

(x^4*e^2 - x^3*e^(x + 2) - x^2*e^2 - 3*x^2 + 30*e^x)/x

________________________________________________________________________________________

maple [A]  time = 0.06, size = 31, normalized size = 0.86

method	result	size
risch	$x^3{\mathrm{e}}^2-{\mathrm{e}}^{2}x-3x-\frac{(x^3{\mathrm{e}}^2-30){\mathrm{e}}^x}{x}$	$31$
norman	$\frac{x^4{\mathrm{e}}^2+(-{\mathrm{e}}^2-3)x^2-x^3{\mathrm{e}}^2{\mathrm{e}}^x+30{\mathrm{e}}^x}{x}$	$35$
default	$-3x+x^3{\mathrm{e}}^2+\frac{30{\mathrm{e}}^x}{x}-2{\mathrm{e}}^2({\mathrm{e}}^{x}x-{\mathrm{e}}^x)-{\mathrm{e}}^2({\mathrm{e}}^x{x}^2-2{\mathrm{e}}^{x}x+2{\mathrm{e}}^x)-{\mathrm{e}}^{2}x$	$56$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^4-2*x^3)*exp(2)+30*x-30)*exp(x)+(3*x^4-x^2)*exp(2)-3*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^3*exp(2)-exp(2)*x-3*x-(x^3*exp(2)-30)/x*exp(x)

________________________________________________________________________________________

maxima [C]  time = 0.39, size = 59, normalized size = 1.64 $$\begin{array}{c}{x}^3{e}^2-x{e}^2-(x^2{e}^2-2x{e}^2+2e^2)e^x-2(x{e}^2-e^2)e^x-3x+30\mathrm{E}\mathrm{i}(x)-30\mathrm{\Gamma }(-1,-x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-2*x^3)*exp(2)+30*x-30)*exp(x)+(3*x^4-x^2)*exp(2)-3*x^2)/x^2,x, algorithm="maxima")

[Out]

x^3*e^2 - x*e^2 - (x^2*e^2 - 2*x*e^2 + 2*e^2)*e^x - 2*(x*e^2 - e^2)*e^x - 3*x + 30*Ei(x) - 30*gamma(-1, -x)

________________________________________________________________________________________

mupad [B]  time = 5.02, size = 30, normalized size = 0.83 $$\begin{array}{c}\frac{30{\mathrm{e}}^x}{x}-x^2{\mathrm{e}}^{x+2}-x({\mathrm{e}}^2+3)+x^3{\mathrm{e}}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2)*(x^2 - 3*x^4) + exp(x)*(exp(2)*(2*x^3 + x^4) - 30*x + 30) + 3*x^2)/x^2,x)

[Out]

(30*exp(x))/x - x^2*exp(x + 2) - x*(exp(2) + 3) + x^3*exp(2)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 27, normalized size = 0.75 $$\begin{array}{c}{x}^3{e}^2+x(-e^2-3)+\frac{(-x^3{e}^2+30)e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**4-2*x**3)*exp(2)+30*x-30)*exp(x)+(3*x**4-x**2)*exp(2)-3*x**2)/x**2,x)

[Out]

x**3*exp(2) + x*(-exp(2) - 3) + (-x**3*exp(2) + 30)*exp(x)/x

________________________________________________________________________________________