\(\int (308+44 e^{4/3}+968 x+e^x (14+102 x+44 x^2+e^{4/3} (2+2 x)) \log (5)+e^{2 x} (2 x+2 x^2) \log ^2(5)) \, dx\) [87]

Optimal result

Integrand size = 58, antiderivative size = 19 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=\left (7+e^{4/3}+22 x+e^x x \log (5)\right )^2 \]

[Out]

(22*x+x*exp(x)*ln(5)+exp(4/3)+7)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(19)=38\).

Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.79, number of steps used = 19, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2227, 2225, 2207, 1607} \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=484 x^2+e^{2 x} x^2 \log ^2(5)+44 e^x x^2 \log (5)+44 \left (7+e^{4/3}\right ) x+14 e^x x \log (5)-2 e^{x+\frac {4}{3}} \log (5)+2 e^{x+\frac {4}{3}} (x+1) \log (5) \]

[In]

Int[308 + 44*E^(4/3) + 968*x + E^x*(14 + 102*x + 44*x^2 + E^(4/3)*(2 + 2*x))*Log[5] + E^(2*x)*(2*x + 2*x^2)*Lo
g[5]^2,x]

[Out]

44*(7 + E^(4/3))*x + 484*x^2 - 2*E^(4/3 + x)*Log[5] + 14*E^x*x*Log[5] + 44*E^x*x^2*Log[5] + 2*E^(4/3 + x)*(1 +
x)*Log[5] + E^(2*x)*x^2*Log[5]^2

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

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] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = 44 \left (7+e^{4/3}\right ) x+484 x^2+\log (5) \int e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \, dx+\log ^2(5) \int e^{2 x} \left (2 x+2 x^2\right ) \, dx \\ & = 44 \left (7+e^{4/3}\right ) x+484 x^2+\log (5) \int \left (14 e^x+102 e^x x+44 e^x x^2+2 e^{\frac {4}{3}+x} (1+x)\right ) \, dx+\log ^2(5) \int e^{2 x} x (2+2 x) \, dx \\ & = 44 \left (7+e^{4/3}\right ) x+484 x^2+(2 \log (5)) \int e^{\frac {4}{3}+x} (1+x) \, dx+(14 \log (5)) \int e^x \, dx+(44 \log (5)) \int e^x x^2 \, dx+(102 \log (5)) \int e^x x \, dx+\log ^2(5) \int \left (2 e^{2 x} x+2 e^{2 x} x^2\right ) \, dx \\ & = 44 \left (7+e^{4/3}\right ) x+484 x^2+14 e^x \log (5)+102 e^x x \log (5)+44 e^x x^2 \log (5)+2 e^{\frac {4}{3}+x} (1+x) \log (5)-(2 \log (5)) \int e^{\frac {4}{3}+x} \, dx-(88 \log (5)) \int e^x x \, dx-(102 \log (5)) \int e^x \, dx+\left (2 \log ^2(5)\right ) \int e^{2 x} x \, dx+\left (2 \log ^2(5)\right ) \int e^{2 x} x^2 \, dx \\ & = 44 \left (7+e^{4/3}\right ) x+484 x^2-88 e^x \log (5)-2 e^{\frac {4}{3}+x} \log (5)+14 e^x x \log (5)+44 e^x x^2 \log (5)+2 e^{\frac {4}{3}+x} (1+x) \log (5)+e^{2 x} x \log ^2(5)+e^{2 x} x^2 \log ^2(5)+(88 \log (5)) \int e^x \, dx-\log ^2(5) \int e^{2 x} \, dx-\left (2 \log ^2(5)\right ) \int e^{2 x} x \, dx \\ & = 44 \left (7+e^{4/3}\right ) x+484 x^2-2 e^{\frac {4}{3}+x} \log (5)+14 e^x x \log (5)+44 e^x x^2 \log (5)+2 e^{\frac {4}{3}+x} (1+x) \log (5)-\frac {1}{2} e^{2 x} \log ^2(5)+e^{2 x} x^2 \log ^2(5)+\log ^2(5) \int e^{2 x} \, dx \\ & = 44 \left (7+e^{4/3}\right ) x+484 x^2-2 e^{\frac {4}{3}+x} \log (5)+14 e^x x \log (5)+44 e^x x^2 \log (5)+2 e^{\frac {4}{3}+x} (1+x) \log (5)+e^{2 x} x^2 \log ^2(5) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=\left (7+e^{4/3}+22 x+e^x x \log (5)\right )^2 \]

[In]

Integrate[308 + 44*E^(4/3) + 968*x + E^x*(14 + 102*x + 44*x^2 + E^(4/3)*(2 + 2*x))*Log[5] + E^(2*x)*(2*x + 2*x
^2)*Log[5]^2,x]

[Out]

(7 + E^(4/3) + 22*x + E^x*x*Log[5])^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(15)=30\).

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42

method	result	size
risch	\({\mathrm e}^{2 x} \ln \left (5\right )^{2} x^{2}+\ln \left (5\right ) \left (2 x \,{\mathrm e}^{\frac {4}{3}}+44 x^{2}+14 x \right ) {\mathrm e}^{x}+44 x \,{\mathrm e}^{\frac {4}{3}}+484 x^{2}+308 x\)	\(46\)
norman	\(\left (44 \,{\mathrm e}^{\frac {4}{3}}+308\right ) x +\left (2 \ln \left (5\right ) {\mathrm e}^{\frac {4}{3}}+14 \ln \left (5\right )\right ) x \,{\mathrm e}^{x}+{\mathrm e}^{2 x} \ln \left (5\right )^{2} x^{2}+484 x^{2}+44 x^{2} \ln \left (5\right ) {\mathrm e}^{x}\)	\(51\)
default	\({\mathrm e}^{2 x} \ln \left (5\right )^{2} x^{2}+2 \,{\mathrm e}^{x} \ln \left (5\right ) {\mathrm e}^{\frac {4}{3}} x +44 x^{2} \ln \left (5\right ) {\mathrm e}^{x}+14 x \,{\mathrm e}^{x} \ln \left (5\right )+44 x \,{\mathrm e}^{\frac {4}{3}}+484 x^{2}+308 x\)	\(52\)
parallelrisch	\({\mathrm e}^{2 x} \ln \left (5\right )^{2} x^{2}+2 \,{\mathrm e}^{x} \ln \left (5\right ) {\mathrm e}^{\frac {4}{3}} x +44 x^{2} \ln \left (5\right ) {\mathrm e}^{x}+14 x \,{\mathrm e}^{x} \ln \left (5\right )+484 x^{2}+\left (44 \,{\mathrm e}^{\frac {4}{3}}+308\right ) x\)	\(52\)
parts	\({\mathrm e}^{2 x} \ln \left (5\right )^{2} x^{2}+2 \,{\mathrm e}^{x} \ln \left (5\right ) {\mathrm e}^{\frac {4}{3}} x +44 x^{2} \ln \left (5\right ) {\mathrm e}^{x}+14 x \,{\mathrm e}^{x} \ln \left (5\right )+44 x \,{\mathrm e}^{\frac {4}{3}}+484 x^{2}+308 x\)	\(52\)

[In]

int((2*x^2+2*x)*ln(5)^2*exp(x)^2+((2+2*x)*exp(4/3)+44*x^2+102*x+14)*ln(5)*exp(x)+44*exp(4/3)+968*x+308,x,metho
d=_RETURNVERBOSE)

[Out]

exp(x)^2*ln(5)^2*x^2+ln(5)*(2*x*exp(4/3)+44*x^2+14*x)*exp(x)+44*x*exp(4/3)+484*x^2+308*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (15) = 30\).

Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.37 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 2 \, {\left (22 \, x^{2} + x e^{\frac {4}{3}} + 7 \, x\right )} e^{x} \log \left (5\right ) + 484 \, x^{2} + 44 \, x e^{\frac {4}{3}} + 308 \, x \]

[In]

integrate((2*x^2+2*x)*log(5)^2*exp(x)^2+((2+2*x)*exp(4/3)+44*x^2+102*x+14)*log(5)*exp(x)+44*exp(4/3)+968*x+308
,x, algorithm="fricas")

[Out]

x^2*e^(2*x)*log(5)^2 + 2*(22*x^2 + x*e^(4/3) + 7*x)*e^x*log(5) + 484*x^2 + 44*x*e^(4/3) + 308*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (19) = 38\).

Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.05 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=x^{2} e^{2 x} \log {\left (5 \right )}^{2} + 484 x^{2} + x \left (44 e^{\frac {4}{3}} + 308\right ) + \left (44 x^{2} \log {\left (5 \right )} + 2 x e^{\frac {4}{3}} \log {\left (5 \right )} + 14 x \log {\left (5 \right )}\right ) e^{x} \]

[In]

integrate((2*x**2+2*x)*ln(5)**2*exp(x)**2+((2+2*x)*exp(4/3)+44*x**2+102*x+14)*ln(5)*exp(x)+44*exp(4/3)+968*x+3
08,x)

[Out]

x**2*exp(2*x)*log(5)**2 + 484*x**2 + x*(44*exp(4/3) + 308) + (44*x**2*log(5) + 2*x*exp(4/3)*log(5) + 14*x*log(
5))*exp(x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.32 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 2 \, {\left (22 \, x^{2} + x {\left (e^{\frac {4}{3}} + 7\right )}\right )} e^{x} \log \left (5\right ) + 484 \, x^{2} + 44 \, x e^{\frac {4}{3}} + 308 \, x \]

[In]

integrate((2*x^2+2*x)*log(5)^2*exp(x)^2+((2+2*x)*exp(4/3)+44*x^2+102*x+14)*log(5)*exp(x)+44*exp(4/3)+968*x+308
,x, algorithm="maxima")

[Out]

x^2*e^(2*x)*log(5)^2 + 2*(22*x^2 + x*(e^(4/3) + 7))*e^x*log(5) + 484*x^2 + 44*x*e^(4/3) + 308*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 484 \, x^{2} + 44 \, x e^{\frac {4}{3}} + 2 \, {\left (x e^{\left (x + \frac {4}{3}\right )} + {\left (22 \, x^{2} + 7 \, x\right )} e^{x}\right )} \log \left (5\right ) + 308 \, x \]

[In]

integrate((2*x^2+2*x)*log(5)^2*exp(x)^2+((2+2*x)*exp(4/3)+44*x^2+102*x+14)*log(5)*exp(x)+44*exp(4/3)+968*x+308
,x, algorithm="giac")

[Out]

x^2*e^(2*x)*log(5)^2 + 484*x^2 + 44*x*e^(4/3) + 2*(x*e^(x + 4/3) + (22*x^2 + 7*x)*e^x)*log(5) + 308*x

Mupad [B] (verification not implemented)

Time = 7.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \left (308+44 e^{4/3}+968 x+e^x \left (14+102 x+44 x^2+e^{4/3} (2+2 x)\right ) \log (5)+e^{2 x} \left (2 x+2 x^2\right ) \log ^2(5)\right ) \, dx=x\,\left ({\mathrm {e}}^x\,\ln \left (5\right )+22\right )\,\left (22\,x+2\,{\mathrm {e}}^{4/3}+x\,{\mathrm {e}}^x\,\ln \left (5\right )+14\right ) \]

[In]

int(968*x + 44*exp(4/3) + exp(x)*log(5)*(102*x + 44*x^2 + exp(4/3)*(2*x + 2) + 14) + exp(2*x)*log(5)^2*(2*x +
2*x^2) + 308,x)

[Out]

x*(exp(x)*log(5) + 22)*(22*x + 2*exp(4/3) + x*exp(x)*log(5) + 14)