3.6.36 $\int \frac{e^x{x}^5+e^{-6x+56x^2}(-4-6x+112x^2)+e^{-6x+29x^2}(6x+12x^2-116x^3)+e^{-6x+2x^2}(-2x^2-6x^3+4x^4)}{x^5}dx$

Optimal. Leaf size=29 $$e^x+\frac{e^{2(-3+x)x}{(-e^{27x^2}+x)}^2}{x^4}$$

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Rubi [B]  time = 0.24, antiderivative size = 96, normalized size of antiderivative = 3.31, number of steps used = 6, number of rules used = 3, integrand size = 88, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.034, Rules used = {14, 2194, 2288} $$\begin{array}{c}\frac{e^{-2(3-28x)x}(3x-56x^2)}{(3-56x)x^5}-\frac{2e^{29x^2-6x}(3x-29x^2)}{(3-29x)x^4}+\frac{e^{-2(3-x)x}(3x-2x^2)}{(3-2x)x^3}+e^x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^x*x^5 + E^(-6*x + 56*x^2)*(-4 - 6*x + 112*x^2) + E^(-6*x + 29*x^2)*(6*x + 12*x^2 - 116*x^3) + E^(-6*x +
2*x^2)*(-2*x^2 - 6*x^3 + 4*x^4))/x^5,x]

[Out]

E^x + (3*x - 56*x^2)/(E^(2*(3 - 28*x)*x)*(3 - 56*x)*x^5) - (2*E^(-6*x + 29*x^2)*(3*x - 29*x^2))/((3 - 29*x)*x^
4) + (3*x - 2*x^2)/(E^(2*(3 - x)*x)*(3 - 2*x)*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 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 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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (e^x+\frac{2e^{-6x+29x^2}(3+6x-58x^2)}{x^4}+\frac{2e^{2(-3+x)x}(-1-3x+2x^2)}{x^3}+\frac{2e^{2x(-3+28x)}(-2-3x+56x^2)}{x^5})dx\\ & =2\int \frac{e^{-6x+29x^2}(3+6x-58x^2)}{x^4}dx+2\int \frac{e^{2(-3+x)x}(-1-3x+2x^2)}{x^3}dx+2\int \frac{e^{2x(-3+28x)}(-2-3x+56x^2)}{x^5}dx+\int e^{x}dx\\ & =e^x+\frac{e^{-2(3-28x)x}(3x-56x^2)}{(3-56x)x^5}-\frac{2e^{-6x+29x^2}(3x-29x^2)}{(3-29x)x^4}+\frac{e^{-2(3-x)x}(3x-2x^2)}{(3-2x)x^3}\end{array}\end{array}$$

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Mathematica [A]  time = 0.05, size = 47, normalized size = 1.62 $$\begin{array}{c}\frac{e^{-6x}(e^{56x^2}-2e^{29x^2}x+e^{2x^2}{x}^2+e^{7x}{x}^4)}{x^4}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x^5 + E^(-6*x + 56*x^2)*(-4 - 6*x + 112*x^2) + E^(-6*x + 29*x^2)*(6*x + 12*x^2 - 116*x^3) + E^(
-6*x + 2*x^2)*(-2*x^2 - 6*x^3 + 4*x^4))/x^5,x]

[Out]

(E^(56*x^2) - 2*E^(29*x^2)*x + E^(2*x^2)*x^2 + E^(7*x)*x^4)/(E^(6*x)*x^4)

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fricas [A]  time = 0.65, size = 48, normalized size = 1.66 $$\begin{array}{c}\frac{x^4{e}^x+x^2{e}^{(2x^2-6x)}-2x{e}^{(29x^2-6x)}+e^{(56x^2-6x)}}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^
4-6*x^3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x, algorithm="fricas")

[Out]

(x^4*e^x + x^2*e^(2*x^2 - 6*x) - 2*x*e^(29*x^2 - 6*x) + e^(56*x^2 - 6*x))/x^4

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giac [A]  time = 0.30, size = 48, normalized size = 1.66 $$\begin{array}{c}\frac{x^4{e}^x+x^2{e}^{(2x^2-6x)}-2x{e}^{(29x^2-6x)}+e^{(56x^2-6x)}}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^
4-6*x^3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x, algorithm="giac")

[Out]

(x^4*e^x + x^2*e^(2*x^2 - 6*x) - 2*x*e^(29*x^2 - 6*x) + e^(56*x^2 - 6*x))/x^4

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maple [A]  time = 0.12, size = 41, normalized size = 1.41

method	result	size
risch	${\mathrm{e}}^x+\frac{{\mathrm{e}}^{2x(28x-3)}}{x^4}-\frac{2{\mathrm{e}}^{x(29x-6)}}{x^3}+\frac{{\mathrm{e}}^{2x(x-3)}}{x^2}$	$41$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^4-6*x^
3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x,method=_RETURNVERBOSE)

[Out]

exp(x)+1/x^4*exp(2*x*(28*x-3))-2/x^3*exp(x*(29*x-6))+1/x^2*exp(2*x*(x-3))

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maxima [A]  time = 0.61, size = 37, normalized size = 1.28 $$\begin{array}{c}\frac{(x^2{e}^{\left(2x^2\right)}-2x{e}^{\left(29x^2\right)}+e^{\left(56x^2\right)})e^{(-6x)}}{x^4}+e^x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^
4-6*x^3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x, algorithm="maxima")

[Out]

(x^2*e^(2*x^2) - 2*x*e^(29*x^2) + e^(56*x^2))*e^(-6*x)/x^4 + e^x

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mupad [B]  time = 0.67, size = 46, normalized size = 1.59 $$\begin{array}{c}{\mathrm{e}}^x+\frac{{\mathrm{e}}^{2x^2-6x}}{x^2}-\frac{2{\mathrm{e}}^{29x^2-6x}}{x^3}+\frac{{\mathrm{e}}^{56x^2-6x}}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*exp(x) - exp(2*x^2 - 6*x)*(2*x^2 + 6*x^3 - 4*x^4) - exp(54*x^2)*exp(2*x^2 - 6*x)*(6*x - 112*x^2 + 4)
+ exp(27*x^2)*exp(2*x^2 - 6*x)*(6*x + 12*x^2 - 116*x^3))/x^5,x)

[Out]

exp(x) + exp(2*x^2 - 6*x)/x^2 - (2*exp(29*x^2 - 6*x))/x^3 + exp(56*x^2 - 6*x)/x^4

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sympy [B]  time = 1.14, size = 68, normalized size = 2.34 $$\begin{array}{c}{e}^x+\frac{(x^7{e}^{12x}{\left(e^{27x^2}\right)}^{\frac{2}{27}}-2x^6{e}^{12x}{\left(e^{27x^2}\right)}^{\frac{29}{27}}+x^5{e}^{12x}{\left(e^{27x^2}\right)}^{\frac{56}{27}})e^{-18x}}{x^9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x**2-6*x-4)*exp(x**2-3*x)**2*exp(27*x**2)**2+(-116*x**3+12*x**2+6*x)*exp(x**2-3*x)**2*exp(27*x
**2)+(4*x**4-6*x**3-2*x**2)*exp(x**2-3*x)**2+x**5*exp(x))/x**5,x)

[Out]

exp(x) + (x**7*exp(12*x)*exp(27*x**2)**(2/27) - 2*x**6*exp(12*x)*exp(27*x**2)**(29/27) + x**5*exp(12*x)*exp(27
*x**2)**(56/27))*exp(-18*x)/x**9

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