∫ ex(−7+x+9x2+x3−x4)+ex(7−9x−2x2+x3) log(x)___________________________________

3.50.55 $\int \frac {e^x (-7+x+9 x^2+x^3-x^4)+e^x (7-9 x-2 x^2+x^3) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx$

Optimal. Leaf size=25 \[ \frac {e^x (x-\log (x))}{x \left (8+x-(1+x)^2\right )} \]

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Rubi [C] time = 10.68, antiderivative size = 1291, normalized size of antiderivative = 51.64, number of steps used = 189, number of rules used = 14, integrand size = 67, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.209, Rules used = {6741, 6742, 2199, 2194, 2177, 2178, 2176, 2554, 14, 6483, 6475, 2268, 6728, 12}

result too large to display

Antiderivative was successfully veriﬁed.

[In]

Int[(E^x*(-7 + x + 9*x^2 + x^3 - x^4) + E^x*(7 - 9*x - 2*x^2 + x^3)*Log[x])/(49*x^2 - 14*x^3 - 13*x^4 + 2*x^5

+ x^6),x]

[Out]

(-2*E^x)/(29*(1 - Sqrt[29] + 2*x)) + (2*(1 - Sqrt[29])*E^x)/(29*(1 - Sqrt[29] + 2*x)) - (2*E^x)/(29*(1 + Sqrt[

29] + 2*x)) + (2*(1 + Sqrt[29])*E^x)/(29*(1 + Sqrt[29] + 2*x)) + (E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqr

t[29] + 2*x)/2])/29 + (E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/(49*Sqrt[29]) - ((1 - Sqrt

[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/29 - ((13 + Sqrt[29])*E^((-1 + Sqrt[29])/2)

*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/98 + ((87 + 4*Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[

29] + 2*x)/2])/1421 + (9*(29 + 13*Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/2842

- ((29 + 15*Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/2842 - ((29 + 181*Sqrt[29])

*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/2842 + (E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 +

Sqrt[29] + 2*x)/2])/29 - (E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/(49*Sqrt[29]) - ((29 -

181*Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/2842 - ((29 - 15*Sqrt[29])*E^(-1/2

- Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/2842 + (9*(29 - 13*Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIn

tegralEi[(1 + Sqrt[29] + 2*x)/2])/2842 + ((87 - 4*Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29]

+ 2*x)/2])/1421 - ((13 - Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/98 - ((1 + Sqr

t[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/29 - (E^x*Log[x])/(7*x) + (30*E^x*Log[x])/

(203*(1 - Sqrt[29] + 2*x)) - ((1 - Sqrt[29])*E^x*Log[x])/(203*(1 - Sqrt[29] + 2*x)) + (30*E^x*Log[x])/(203*(1

+ Sqrt[29] + 2*x)) - ((1 + Sqrt[29])*E^x*Log[x])/(203*(1 + Sqrt[29] + 2*x)) - (15*E^((-1 + Sqrt[29])/2)*ExpInt

egralEi[(1 - Sqrt[29] + 2*x)/2]*Log[x])/203 + (E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2]*Log

[x])/(7*Sqrt[29]) + (2*(29 - 9*Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2]*Log[x])/1

421 + ((1 - Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2]*Log[x])/406 + ((3 + Sqrt[29]

)*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2]*Log[x])/98 - (15*E^(-1/2 - Sqrt[29]/2)*ExpIntegr

alEi[(1 + Sqrt[29] + 2*x)/2]*Log[x])/203 - (E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2]*Log[x]

)/(7*Sqrt[29]) + ((3 - Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2]*Log[x])/98 + ((1

+ Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2]*Log[x])/406 + (2*(29 + 9*Sqrt[29])*E^(

-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2]*Log[x])/1421

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 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

Rule 2268

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr

and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]

)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6475

Int[ExpIntegralE[1, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)], x

] + (-Simp[EulerGamma*Log[x], x] - Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]

Rule 6483

Int[ExpIntegralEi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -(b*x)]), x

] - Int[ExpIntegralE[1, -(b*x)]/x, x] /; FreeQ[b, x]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{x^2 \left (7-x-x^2\right )^2} \, dx\\ &=\int \left (-\frac {e^x \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{49 x^2}-\frac {2 e^x \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{343 x}-\frac {e^x (8+x) \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{49 \left (-7+x+x^2\right )^2}+\frac {e^x (9+2 x) \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{343 \left (-7+x+x^2\right )}\right ) \, dx\\ &=\frac {1}{343} \int \frac {e^x (9+2 x) \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{-7+x+x^2} \, dx-\frac {2}{343} \int \frac {e^x \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{x} \, dx-\frac {1}{49} \int \frac {e^x \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{x^2} \, dx-\frac {1}{49} \int \frac {e^x (8+x) \left (7-x-9 x^2-x^3+x^4-7 \log (x)+9 x \log (x)+2 x^2 \log (x)-x^3 \log (x)\right )}{\left (-7+x+x^2\right )^2} \, dx\\ &=\frac {1}{343} \int \left (\frac {e^x \left (63+5 x-83 x^2-27 x^3+7 x^4+2 x^5\right )}{-7+x+x^2}-\frac {e^x \left (63-67 x-36 x^2+5 x^3+2 x^4\right ) \log (x)}{-7+x+x^2}\right ) \, dx-\frac {2}{343} \int \left (\frac {e^x \left (7-x-9 x^2-x^3+x^4\right )}{x}-\frac {e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{x}\right ) \, dx-\frac {1}{49} \int \left (\frac {e^x \left (7-x-9 x^2-x^3+x^4\right )}{x^2}-\frac {e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{x^2}\right ) \, dx-\frac {1}{49} \int \left (\frac {7 e^x (8+x)}{\left (-7+x+x^2\right )^2}-\frac {e^x x (8+x)}{\left (-7+x+x^2\right )^2}-\frac {9 e^x x^2 (8+x)}{\left (-7+x+x^2\right )^2}-\frac {e^x x^3 (8+x)}{\left (-7+x+x^2\right )^2}+\frac {e^x x^4 (8+x)}{\left (-7+x+x^2\right )^2}-\frac {e^x \left (56-65 x-25 x^2+6 x^3+x^4\right ) \log (x)}{\left (-7+x+x^2\right )^2}\right ) \, dx\\ &=\frac {1}{343} \int \frac {e^x \left (63+5 x-83 x^2-27 x^3+7 x^4+2 x^5\right )}{-7+x+x^2} \, dx-\frac {1}{343} \int \frac {e^x \left (63-67 x-36 x^2+5 x^3+2 x^4\right ) \log (x)}{-7+x+x^2} \, dx-\frac {2}{343} \int \frac {e^x \left (7-x-9 x^2-x^3+x^4\right )}{x} \, dx+\frac {2}{343} \int \frac {e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{x} \, dx+\frac {1}{49} \int \frac {e^x x (8+x)}{\left (-7+x+x^2\right )^2} \, dx+\frac {1}{49} \int \frac {e^x x^3 (8+x)}{\left (-7+x+x^2\right )^2} \, dx-\frac {1}{49} \int \frac {e^x x^4 (8+x)}{\left (-7+x+x^2\right )^2} \, dx-\frac {1}{49} \int \frac {e^x \left (7-x-9 x^2-x^3+x^4\right )}{x^2} \, dx+\frac {1}{49} \int \frac {e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{x^2} \, dx+\frac {1}{49} \int \frac {e^x \left (56-65 x-25 x^2+6 x^3+x^4\right ) \log (x)}{\left (-7+x+x^2\right )^2} \, dx-\frac {1}{7} \int \frac {e^x (8+x)}{\left (-7+x+x^2\right )^2} \, dx+\frac {9}{49} \int \frac {e^x x^2 (8+x)}{\left (-7+x+x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.61, size = 21, normalized size = 0.84 \begin {gather*} \frac {e^x (-x+\log (x))}{x \left (-7+x+x^2\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^x*(-7 + x + 9*x^2 + x^3 - x^4) + E^x*(7 - 9*x - 2*x^2 + x^3)*Log[x])/(49*x^2 - 14*x^3 - 13*x^4 +

2*x^5 + x^6),x]

[Out]

(E^x*(-x + Log[x]))/(x*(-7 + x + x^2))

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fricas [A] time = 0.58, size = 25, normalized size = 1.00 \begin {gather*} -\frac {x e^{x} - e^{x} \log \relax (x)}{x^{3} + x^{2} - 7 \, x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-2*x^2-9*x+7)*exp(x)*log(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x^6+2*x^5-13*x^4-14*x^3+49*x^2),x, al

gorithm="fricas")

[Out]

-(x*e^x - e^x*log(x))/(x^3 + x^2 - 7*x)

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giac [A] time = 0.13, size = 25, normalized size = 1.00 \begin {gather*} -\frac {x e^{x} - e^{x} \log \relax (x)}{x^{3} + x^{2} - 7 \, x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-2*x^2-9*x+7)*exp(x)*log(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x^6+2*x^5-13*x^4-14*x^3+49*x^2),x, al

gorithm="giac")

[Out]

-(x*e^x - e^x*log(x))/(x^3 + x^2 - 7*x)

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maple [A] time = 0.03, size = 30, normalized size = 1.20


method	result	size

risch	$\frac {{\mathrm e}^{x} \ln \relax (x )}{\left (x^{2}+x -7\right ) x}-\frac {{\mathrm e}^{x}}{x^{2}+x -7}$	$30$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-2*x^2-9*x+7)*exp(x)*ln(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x^6+2*x^5-13*x^4-14*x^3+49*x^2),x,method=_RE

TURNVERBOSE)

[Out]

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

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maxima [A] time = 0.40, size = 22, normalized size = 0.88 \begin {gather*} -\frac {{\left (x - \log \relax (x)\right )} e^{x}}{x^{3} + x^{2} - 7 \, x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-2*x^2-9*x+7)*exp(x)*log(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x^6+2*x^5-13*x^4-14*x^3+49*x^2),x, al

gorithm="maxima")

[Out]

-(x - log(x))*e^x/(x^3 + x^2 - 7*x)

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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (-x^4+x^3+9\,x^2+x-7\right )-{\mathrm {e}}^x\,\ln \relax (x)\,\left (-x^3+2\,x^2+9\,x-7\right )}{x^6+2\,x^5-13\,x^4-14\,x^3+49\,x^2} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(x + 9*x^2 + x^3 - x^4 - 7) - exp(x)*log(x)*(9*x + 2*x^2 - x^3 - 7))/(49*x^2 - 14*x^3 - 13*x^4 + 2

*x^5 + x^6),x)

[Out]

int((exp(x)*(x + 9*x^2 + x^3 - x^4 - 7) - exp(x)*log(x)*(9*x + 2*x^2 - x^3 - 7))/(49*x^2 - 14*x^3 - 13*x^4 + 2

*x^5 + x^6), x)

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sympy [A] time = 0.32, size = 17, normalized size = 0.68 \begin {gather*} \frac {\left (- x + \log {\relax (x )}\right ) e^{x}}{x^{3} + x^{2} - 7 x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-2*x**2-9*x+7)*exp(x)*ln(x)+(-x**4+x**3+9*x**2+x-7)*exp(x))/(x**6+2*x**5-13*x**4-14*x**3+49*x*

*2),x)

[Out]

(-x + log(x))*exp(x)/(x**3 + x**2 - 7*x)

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3.50.55 \(\int \frac {e^x (-7+x+9 x^2+x^3-x^4)+e^x (7-9 x-2 x^2+x^3) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx\)