3.59.31 \(\int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x (-5000 x^3-5000 x^4)+(150 x+1250 x^2+1250 x^3) \log (x)}{625 x^3} \, dx\)

Optimal. Leaf size=31 \[ -4+x+\left (-e^x+4 x\right )^2+\left (\frac {3}{25 x}-x-\log (x)\right )^2 \]

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Rubi [A]  time = 0.11, antiderivative size = 50, normalized size of antiderivative = 1.61, number of steps used = 15, number of rules used = 8, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 14, 2194, 2176, 2357, 2295, 2304, 2301} \begin {gather*} 17 x^2+\frac {9}{625 x^2}+x+8 e^x+e^{2 x}-8 e^x (x+1)+\log ^2(x)+2 x \log (x)-\frac {6 \log (x)}{25 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 - 150*x + 1875*x^3 + 1250*E^(2*x)*x^3 + 21250*x^4 + E^x*(-5000*x^3 - 5000*x^4) + (150*x + 1250*x^2 +
1250*x^3)*Log[x])/(625*x^3),x]

[Out]

8*E^x + E^(2*x) + 9/(625*x^2) + x + 17*x^2 - 8*E^x*(1 + x) - (6*Log[x])/(25*x) + 2*x*Log[x] + Log[x]^2

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 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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \frac {-18-150 x+1875 x^3+1250 e^{2 x} x^3+21250 x^4+e^x \left (-5000 x^3-5000 x^4\right )+\left (150 x+1250 x^2+1250 x^3\right ) \log (x)}{x^3} \, dx\\ &=\frac {1}{625} \int \left (1250 e^{2 x}-5000 e^x (1+x)+\frac {-18-150 x+1875 x^3+21250 x^4+150 x \log (x)+1250 x^2 \log (x)+1250 x^3 \log (x)}{x^3}\right ) \, dx\\ &=\frac {1}{625} \int \frac {-18-150 x+1875 x^3+21250 x^4+150 x \log (x)+1250 x^2 \log (x)+1250 x^3 \log (x)}{x^3} \, dx+2 \int e^{2 x} \, dx-8 \int e^x (1+x) \, dx\\ &=e^{2 x}-8 e^x (1+x)+\frac {1}{625} \int \left (\frac {-18-150 x+1875 x^3+21250 x^4}{x^3}+\frac {50 \left (3+25 x+25 x^2\right ) \log (x)}{x^2}\right ) \, dx+8 \int e^x \, dx\\ &=8 e^x+e^{2 x}-8 e^x (1+x)+\frac {1}{625} \int \frac {-18-150 x+1875 x^3+21250 x^4}{x^3} \, dx+\frac {2}{25} \int \frac {\left (3+25 x+25 x^2\right ) \log (x)}{x^2} \, dx\\ &=8 e^x+e^{2 x}-8 e^x (1+x)+\frac {1}{625} \int \left (1875-\frac {18}{x^3}-\frac {150}{x^2}+21250 x\right ) \, dx+\frac {2}{25} \int \left (25 \log (x)+\frac {3 \log (x)}{x^2}+\frac {25 \log (x)}{x}\right ) \, dx\\ &=8 e^x+e^{2 x}+\frac {9}{625 x^2}+\frac {6}{25 x}+3 x+17 x^2-8 e^x (1+x)+\frac {6}{25} \int \frac {\log (x)}{x^2} \, dx+2 \int \log (x) \, dx+2 \int \frac {\log (x)}{x} \, dx\\ &=8 e^x+e^{2 x}+\frac {9}{625 x^2}+x+17 x^2-8 e^x (1+x)-\frac {6 \log (x)}{25 x}+2 x \log (x)+\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 43, normalized size = 1.39 \begin {gather*} e^{2 x}+\frac {9}{625 x^2}+x-8 e^x x+17 x^2-\frac {6 \log (x)}{25 x}+2 x \log (x)+\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 - 150*x + 1875*x^3 + 1250*E^(2*x)*x^3 + 21250*x^4 + E^x*(-5000*x^3 - 5000*x^4) + (150*x + 1250*
x^2 + 1250*x^3)*Log[x])/(625*x^3),x]

[Out]

E^(2*x) + 9/(625*x^2) + x - 8*E^x*x + 17*x^2 - (6*Log[x])/(25*x) + 2*x*Log[x] + Log[x]^2

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fricas [A]  time = 0.65, size = 55, normalized size = 1.77 \begin {gather*} \frac {10625 \, x^{4} - 5000 \, x^{3} e^{x} + 625 \, x^{2} \log \relax (x)^{2} + 625 \, x^{3} + 625 \, x^{2} e^{\left (2 \, x\right )} + 50 \, {\left (25 \, x^{3} - 3 \, x\right )} \log \relax (x) + 9}{625 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000*x^4-5000*x^3)*exp(x)+21250*x^4+1875
*x^3-150*x-18)/x^3,x, algorithm="fricas")

[Out]

1/625*(10625*x^4 - 5000*x^3*e^x + 625*x^2*log(x)^2 + 625*x^3 + 625*x^2*e^(2*x) + 50*(25*x^3 - 3*x)*log(x) + 9)
/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {21250 \, x^{4} + 1250 \, x^{3} e^{\left (2 \, x\right )} + 1875 \, x^{3} - 5000 \, {\left (x^{4} + x^{3}\right )} e^{x} + 50 \, {\left (25 \, x^{3} + 25 \, x^{2} + 3 \, x\right )} \log \relax (x) - 150 \, x - 18}{625 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000*x^4-5000*x^3)*exp(x)+21250*x^4+1875
*x^3-150*x-18)/x^3,x, algorithm="giac")

[Out]

integrate(1/625*(21250*x^4 + 1250*x^3*e^(2*x) + 1875*x^3 - 5000*(x^4 + x^3)*e^x + 50*(25*x^3 + 25*x^2 + 3*x)*l
og(x) - 150*x - 18)/x^3, x)

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maple [A]  time = 0.03, size = 38, normalized size = 1.23

method	result	size
default	\(-8 \,{\mathrm e}^{x} x +17 x^{2}+x +\frac {9}{625 x^{2}}+{\mathrm e}^{2 x}+2 x \ln \relax (x )+\ln \relax (x )^{2}-\frac {6 \ln \relax (x )}{25 x}\)	\(38\)
risch	\(\ln \relax (x )^{2}+\frac {2 \left (25 x^{2}-3\right ) \ln \relax (x )}{25 x}+\frac {10625 x^{4}-5000 \,{\mathrm e}^{x} x^{3}+625 \,{\mathrm e}^{2 x} x^{2}+625 x^{3}+9}{625 x^{2}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/625*((1250*x^3+1250*x^2+150*x)*ln(x)+1250*exp(x)^2*x^3+(-5000*x^4-5000*x^3)*exp(x)+21250*x^4+1875*x^3-15
0*x-18)/x^3,x,method=_RETURNVERBOSE)

[Out]

-8*exp(x)*x+17*x^2+x+9/625/x^2+exp(x)^2+2*x*ln(x)+ln(x)^2-6/25*ln(x)/x

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maxima [A]  time = 0.36, size = 43, normalized size = 1.39 \begin {gather*} 17 \, x^{2} - 8 \, {\left (x - 1\right )} e^{x} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + x - \frac {6 \, \log \relax (x)}{25 \, x} + \frac {9}{625 \, x^{2}} + e^{\left (2 \, x\right )} - 8 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((1250*x^3+1250*x^2+150*x)*log(x)+1250*exp(x)^2*x^3+(-5000*x^4-5000*x^3)*exp(x)+21250*x^4+1875
*x^3-150*x-18)/x^3,x, algorithm="maxima")

[Out]

17*x^2 - 8*(x - 1)*e^x + 2*x*log(x) + log(x)^2 + x - 6/25*log(x)/x + 9/625/x^2 + e^(2*x) - 8*e^x

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mupad [B]  time = 4.08, size = 38, normalized size = 1.23 \begin {gather*} {\mathrm {e}}^{2\,x}+{\ln \relax (x)}^2-\frac {\frac {6\,x\,\ln \relax (x)}{25}-\frac {9}{625}}{x^2}+x\,\left (2\,\ln \relax (x)-8\,{\mathrm {e}}^x+1\right )+17\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3*exp(2*x) - (exp(x)*(5000*x^3 + 5000*x^4))/625 - (6*x)/25 + 3*x^3 + 34*x^4 + (log(x)*(150*x + 1250*x
^2 + 1250*x^3))/625 - 18/625)/x^3,x)

[Out]

exp(2*x) + log(x)^2 - ((6*x*log(x))/25 - 9/625)/x^2 + x*(2*log(x) - 8*exp(x) + 1) + 17*x^2

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sympy [A]  time = 0.35, size = 42, normalized size = 1.35 \begin {gather*} 17 x^{2} - 8 x e^{x} + x + e^{2 x} + \log {\relax (x )}^{2} + \frac {\left (50 x^{2} - 6\right ) \log {\relax (x )}}{25 x} + \frac {9}{625 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((1250*x**3+1250*x**2+150*x)*ln(x)+1250*exp(x)**2*x**3+(-5000*x**4-5000*x**3)*exp(x)+21250*x**
4+1875*x**3-150*x-18)/x**3,x)

[Out]

17*x**2 - 8*x*exp(x) + x + exp(2*x) + log(x)**2 + (50*x**2 - 6)*log(x)/(25*x) + 9/(625*x**2)

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