3.86.88 \(\int \frac {15 x^3+12 x^4-9 x^5+12 x^6+e^5 (-10-2 x+x^3)+e^{2 x} (-36 x^3-6 x^4-3 x^5-6 x^6)}{x^3} \, dx\)

Optimal. Leaf size=34 \[ \left (\frac {5}{x^2}-\frac {-2+x}{x}+x\right ) \left (e^5+3 x^2 \left (-e^{2 x}+x\right )\right ) \]

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Rubi [B]  time = 0.14, antiderivative size = 74, normalized size of antiderivative = 2.18, number of steps used = 16, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14, 2196, 2194, 2176} \begin {gather*} 3 x^4-3 e^{2 x} x^3-3 x^3+3 e^{2 x} x^2+6 x^2+\frac {5 e^5}{x^2}-6 e^{2 x} x+\left (15+e^5\right ) x-15 e^{2 x}+\frac {2 e^5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15*x^3 + 12*x^4 - 9*x^5 + 12*x^6 + E^5*(-10 - 2*x + x^3) + E^(2*x)*(-36*x^3 - 6*x^4 - 3*x^5 - 6*x^6))/x^3
,x]

[Out]

-15*E^(2*x) + (5*E^5)/x^2 + (2*E^5)/x - 6*E^(2*x)*x + (15 + E^5)*x + 6*x^2 + 3*E^(2*x)*x^2 - 3*x^3 - 3*E^(2*x)
*x^3 + 3*x^4

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 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 2196

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3 e^{2 x} \left (12+2 x+x^2+2 x^3\right )+\frac {-10 e^5-2 e^5 x+15 \left (1+\frac {e^5}{15}\right ) x^3+12 x^4-9 x^5+12 x^6}{x^3}\right ) \, dx\\ &=-\left (3 \int e^{2 x} \left (12+2 x+x^2+2 x^3\right ) \, dx\right )+\int \frac {-10 e^5-2 e^5 x+15 \left (1+\frac {e^5}{15}\right ) x^3+12 x^4-9 x^5+12 x^6}{x^3} \, dx\\ &=-\left (3 \int \left (12 e^{2 x}+2 e^{2 x} x+e^{2 x} x^2+2 e^{2 x} x^3\right ) \, dx\right )+\int \left (15 \left (1+\frac {e^5}{15}\right )-\frac {10 e^5}{x^3}-\frac {2 e^5}{x^2}+12 x-9 x^2+12 x^3\right ) \, dx\\ &=\frac {5 e^5}{x^2}+\frac {2 e^5}{x}+\left (15+e^5\right ) x+6 x^2-3 x^3+3 x^4-3 \int e^{2 x} x^2 \, dx-6 \int e^{2 x} x \, dx-6 \int e^{2 x} x^3 \, dx-36 \int e^{2 x} \, dx\\ &=-18 e^{2 x}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-3 e^{2 x} x+\left (15+e^5\right ) x+6 x^2-\frac {3}{2} e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4+3 \int e^{2 x} \, dx+3 \int e^{2 x} x \, dx+9 \int e^{2 x} x^2 \, dx\\ &=-\frac {33 e^{2 x}}{2}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-\frac {3}{2} e^{2 x} x+\left (15+e^5\right ) x+6 x^2+3 e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4-\frac {3}{2} \int e^{2 x} \, dx-9 \int e^{2 x} x \, dx\\ &=-\frac {69 e^{2 x}}{4}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-6 e^{2 x} x+\left (15+e^5\right ) x+6 x^2+3 e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4+\frac {9}{2} \int e^{2 x} \, dx\\ &=-15 e^{2 x}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-6 e^{2 x} x+\left (15+e^5\right ) x+6 x^2+3 e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 60, normalized size = 1.76 \begin {gather*} \frac {5 e^5}{x^2}+\frac {2 e^5}{x}+15 x+e^5 x+6 x^2-3 x^3+3 x^4-3 e^{2 x} \left (5+2 x-x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15*x^3 + 12*x^4 - 9*x^5 + 12*x^6 + E^5*(-10 - 2*x + x^3) + E^(2*x)*(-36*x^3 - 6*x^4 - 3*x^5 - 6*x^6
))/x^3,x]

[Out]

(5*E^5)/x^2 + (2*E^5)/x + 15*x + E^5*x + 6*x^2 - 3*x^3 + 3*x^4 - 3*E^(2*x)*(5 + 2*x - x^2 + x^3)

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fricas [A]  time = 0.71, size = 61, normalized size = 1.79 \begin {gather*} \frac {3 \, x^{6} - 3 \, x^{5} + 6 \, x^{4} + 15 \, x^{3} + {\left (x^{3} + 2 \, x + 5\right )} e^{5} - 3 \, {\left (x^{5} - x^{4} + 2 \, x^{3} + 5 \, x^{2}\right )} e^{\left (2 \, x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^6-3*x^5-6*x^4-36*x^3)*exp(2*x)+(x^3-2*x-10)*exp(5)+12*x^6-9*x^5+12*x^4+15*x^3)/x^3,x, algorit
hm="fricas")

[Out]

(3*x^6 - 3*x^5 + 6*x^4 + 15*x^3 + (x^3 + 2*x + 5)*e^5 - 3*(x^5 - x^4 + 2*x^3 + 5*x^2)*e^(2*x))/x^2

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giac [B]  time = 0.16, size = 76, normalized size = 2.24 \begin {gather*} \frac {3 \, x^{6} - 3 \, x^{5} e^{\left (2 \, x\right )} - 3 \, x^{5} + 3 \, x^{4} e^{\left (2 \, x\right )} + 6 \, x^{4} + x^{3} e^{5} - 6 \, x^{3} e^{\left (2 \, x\right )} + 15 \, x^{3} - 15 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{5} + 5 \, e^{5}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^6-3*x^5-6*x^4-36*x^3)*exp(2*x)+(x^3-2*x-10)*exp(5)+12*x^6-9*x^5+12*x^4+15*x^3)/x^3,x, algorit
hm="giac")

[Out]

(3*x^6 - 3*x^5*e^(2*x) - 3*x^5 + 3*x^4*e^(2*x) + 6*x^4 + x^3*e^5 - 6*x^3*e^(2*x) + 15*x^3 - 15*x^2*e^(2*x) + 2
*x*e^5 + 5*e^5)/x^2

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maple [A]  time = 0.06, size = 58, normalized size = 1.71

method	result	size
risch	\(3 x^{4}-3 x^{3}+x \,{\mathrm e}^{5}+6 x^{2}+15 x +\frac {2 x \,{\mathrm e}^{5}+5 \,{\mathrm e}^{5}}{x^{2}}+\left (-3 x^{3}+3 x^{2}-6 x -15\right ) {\mathrm e}^{2 x}\)	\(58\)
derivativedivides	\(-3 x^{3}+6 x^{2}+15 x +3 x^{4}+\frac {5 \,{\mathrm e}^{5}}{x^{2}}+\frac {2 \,{\mathrm e}^{5}}{x}-6 x \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x} x^{2}-3 \,{\mathrm e}^{2 x} x^{3}+x \,{\mathrm e}^{5}\)	\(69\)
default	\(-3 x^{3}+6 x^{2}+15 x +3 x^{4}+\frac {5 \,{\mathrm e}^{5}}{x^{2}}+\frac {2 \,{\mathrm e}^{5}}{x}-6 x \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x} x^{2}-3 \,{\mathrm e}^{2 x} x^{3}+x \,{\mathrm e}^{5}\)	\(69\)
norman	\(\frac {\left ({\mathrm e}^{5}+15\right ) x^{3}+6 x^{4}-3 x^{5}+3 x^{6}+2 x \,{\mathrm e}^{5}-3 x^{5} {\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 x} x^{2}-6 \,{\mathrm e}^{2 x} x^{3}+3 \,{\mathrm e}^{2 x} x^{4}+5 \,{\mathrm e}^{5}}{x^{2}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^6-3*x^5-6*x^4-36*x^3)*exp(2*x)+(x^3-2*x-10)*exp(5)+12*x^6-9*x^5+12*x^4+15*x^3)/x^3,x,method=_RETURN
VERBOSE)

[Out]

3*x^4-3*x^3+x*exp(5)+6*x^2+15*x+(2*x*exp(5)+5*exp(5))/x^2+(-3*x^3+3*x^2-6*x-15)*exp(2*x)

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maxima [B]  time = 0.35, size = 91, normalized size = 2.68 \begin {gather*} 3 \, x^{4} - 3 \, x^{3} + 6 \, x^{2} + x e^{5} - \frac {3}{4} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} - \frac {3}{4} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - \frac {3}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 15 \, x + \frac {2 \, e^{5}}{x} + \frac {5 \, e^{5}}{x^{2}} - 18 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^6-3*x^5-6*x^4-36*x^3)*exp(2*x)+(x^3-2*x-10)*exp(5)+12*x^6-9*x^5+12*x^4+15*x^3)/x^3,x, algorit
hm="maxima")

[Out]

3*x^4 - 3*x^3 + 6*x^2 + x*e^5 - 3/4*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) - 3/4*(2*x^2 - 2*x + 1)*e^(2*x) - 3/2*(2
*x - 1)*e^(2*x) + 15*x + 2*e^5/x + 5*e^5/x^2 - 18*e^(2*x)

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mupad [B]  time = 6.05, size = 63, normalized size = 1.85 \begin {gather*} x^2\,\left (3\,{\mathrm {e}}^{2\,x}+6\right )-x^3\,\left (3\,{\mathrm {e}}^{2\,x}+3\right )-15\,{\mathrm {e}}^{2\,x}+x\,\left ({\mathrm {e}}^5-6\,{\mathrm {e}}^{2\,x}+15\right )+\frac {5\,{\mathrm {e}}^5+2\,x\,{\mathrm {e}}^5}{x^2}+3\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5)*(2*x - x^3 + 10) + exp(2*x)*(36*x^3 + 6*x^4 + 3*x^5 + 6*x^6) - 15*x^3 - 12*x^4 + 9*x^5 - 12*x^6)/
x^3,x)

[Out]

x^2*(3*exp(2*x) + 6) - x^3*(3*exp(2*x) + 3) - 15*exp(2*x) + x*(exp(5) - 6*exp(2*x) + 15) + (5*exp(5) + 2*x*exp
(5))/x^2 + 3*x^4

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sympy [B]  time = 0.22, size = 56, normalized size = 1.65 \begin {gather*} 3 x^{4} - 3 x^{3} + 6 x^{2} + x \left (15 + e^{5}\right ) + \left (- 3 x^{3} + 3 x^{2} - 6 x - 15\right ) e^{2 x} + \frac {2 x e^{5} + 5 e^{5}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**6-3*x**5-6*x**4-36*x**3)*exp(2*x)+(x**3-2*x-10)*exp(5)+12*x**6-9*x**5+12*x**4+15*x**3)/x**3,
x)

[Out]

3*x**4 - 3*x**3 + 6*x**2 + x*(15 + exp(5)) + (-3*x**3 + 3*x**2 - 6*x - 15)*exp(2*x) + (2*x*exp(5) + 5*exp(5))/
x**2

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