3.56.24 \(\int \frac {1}{4} e^x (9+57 x+90 x^2+54 x^3+13 x^4+x^5+(18+78 x+72 x^2+22 x^3+2 x^4) \log (3)+(9+21 x+9 x^2+x^3) \log ^2(3)) \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{4} e^x x (3+x)^2 (1+x+\log (3))^2 \]

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Rubi [B]  time = 0.39, antiderivative size = 131, normalized size of antiderivative = 6.55, number of steps used = 53, number of rules used = 4, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 2196, 2194, 2176} \begin {gather*} \frac {e^x x^5}{4}+2 e^x x^4+\frac {1}{2} e^x x^4 \log (3)+\frac {11 e^x x^3}{2}+\frac {1}{4} e^x x^3 \log ^2(3)+\frac {7}{2} e^x x^3 \log (3)+6 e^x x^2+\frac {3}{2} e^x x^2 \log ^2(3)+\frac {15}{2} e^x x^2 \log (3)+\frac {9 e^x x}{4}+\frac {9}{4} e^x x \log ^2(3)+\frac {9}{2} e^x x \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(9 + 57*x + 90*x^2 + 54*x^3 + 13*x^4 + x^5 + (18 + 78*x + 72*x^2 + 22*x^3 + 2*x^4)*Log[3] + (9 + 21*x
+ 9*x^2 + x^3)*Log[3]^2))/4,x]

[Out]

(9*E^x*x)/4 + 6*E^x*x^2 + (11*E^x*x^3)/2 + 2*E^x*x^4 + (E^x*x^5)/4 + (9*E^x*x*Log[3])/2 + (15*E^x*x^2*Log[3])/
2 + (7*E^x*x^3*Log[3])/2 + (E^x*x^4*Log[3])/2 + (9*E^x*x*Log[3]^2)/4 + (3*E^x*x^2*Log[3]^2)/2 + (E^x*x^3*Log[3
]^2)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\frac {1}{4} \int e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx\\ &=\frac {1}{4} \int \left (9 e^x+57 e^x x+90 e^x x^2+54 e^x x^3+13 e^x x^4+e^x x^5+2 e^x \left (9+39 x+36 x^2+11 x^3+x^4\right ) \log (3)+e^x \left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx\\ &=\frac {1}{4} \int e^x x^5 \, dx+\frac {9 \int e^x \, dx}{4}+\frac {13}{4} \int e^x x^4 \, dx+\frac {27}{2} \int e^x x^3 \, dx+\frac {57}{4} \int e^x x \, dx+\frac {45}{2} \int e^x x^2 \, dx+\frac {1}{2} \log (3) \int e^x \left (9+39 x+36 x^2+11 x^3+x^4\right ) \, dx+\frac {1}{4} \log ^2(3) \int e^x \left (9+21 x+9 x^2+x^3\right ) \, dx\\ &=\frac {9 e^x}{4}+\frac {57 e^x x}{4}+\frac {45 e^x x^2}{2}+\frac {27 e^x x^3}{2}+\frac {13 e^x x^4}{4}+\frac {e^x x^5}{4}-\frac {5}{4} \int e^x x^4 \, dx-13 \int e^x x^3 \, dx-\frac {57 \int e^x \, dx}{4}-\frac {81}{2} \int e^x x^2 \, dx-45 \int e^x x \, dx+\frac {1}{2} \log (3) \int \left (9 e^x+39 e^x x+36 e^x x^2+11 e^x x^3+e^x x^4\right ) \, dx+\frac {1}{4} \log ^2(3) \int \left (9 e^x+21 e^x x+9 e^x x^2+e^x x^3\right ) \, dx\\ &=-12 e^x-\frac {123 e^x x}{4}-18 e^x x^2+\frac {e^x x^3}{2}+2 e^x x^4+\frac {e^x x^5}{4}+5 \int e^x x^3 \, dx+39 \int e^x x^2 \, dx+45 \int e^x \, dx+81 \int e^x x \, dx+\frac {1}{2} \log (3) \int e^x x^4 \, dx+\frac {1}{2} (9 \log (3)) \int e^x \, dx+\frac {1}{2} (11 \log (3)) \int e^x x^3 \, dx+(18 \log (3)) \int e^x x^2 \, dx+\frac {1}{2} (39 \log (3)) \int e^x x \, dx+\frac {1}{4} \log ^2(3) \int e^x x^3 \, dx+\frac {1}{4} \left (9 \log ^2(3)\right ) \int e^x \, dx+\frac {1}{4} \left (9 \log ^2(3)\right ) \int e^x x^2 \, dx+\frac {1}{4} \left (21 \log ^2(3)\right ) \int e^x x \, dx\\ &=33 e^x+\frac {201 e^x x}{4}+21 e^x x^2+\frac {11 e^x x^3}{2}+2 e^x x^4+\frac {e^x x^5}{4}+\frac {9}{2} e^x \log (3)+\frac {39}{2} e^x x \log (3)+18 e^x x^2 \log (3)+\frac {11}{2} e^x x^3 \log (3)+\frac {1}{2} e^x x^4 \log (3)+\frac {9}{4} e^x \log ^2(3)+\frac {21}{4} e^x x \log ^2(3)+\frac {9}{4} e^x x^2 \log ^2(3)+\frac {1}{4} e^x x^3 \log ^2(3)-15 \int e^x x^2 \, dx-78 \int e^x x \, dx-81 \int e^x \, dx-(2 \log (3)) \int e^x x^3 \, dx-\frac {1}{2} (33 \log (3)) \int e^x x^2 \, dx-\frac {1}{2} (39 \log (3)) \int e^x \, dx-(36 \log (3)) \int e^x x \, dx-\frac {1}{4} \left (3 \log ^2(3)\right ) \int e^x x^2 \, dx-\frac {1}{2} \left (9 \log ^2(3)\right ) \int e^x x \, dx-\frac {1}{4} \left (21 \log ^2(3)\right ) \int e^x \, dx\\ &=-48 e^x-\frac {111 e^x x}{4}+6 e^x x^2+\frac {11 e^x x^3}{2}+2 e^x x^4+\frac {e^x x^5}{4}-15 e^x \log (3)-\frac {33}{2} e^x x \log (3)+\frac {3}{2} e^x x^2 \log (3)+\frac {7}{2} e^x x^3 \log (3)+\frac {1}{2} e^x x^4 \log (3)-3 e^x \log ^2(3)+\frac {3}{4} e^x x \log ^2(3)+\frac {3}{2} e^x x^2 \log ^2(3)+\frac {1}{4} e^x x^3 \log ^2(3)+30 \int e^x x \, dx+78 \int e^x \, dx+(6 \log (3)) \int e^x x^2 \, dx+(33 \log (3)) \int e^x x \, dx+(36 \log (3)) \int e^x \, dx+\frac {1}{2} \left (3 \log ^2(3)\right ) \int e^x x \, dx+\frac {1}{2} \left (9 \log ^2(3)\right ) \int e^x \, dx\\ &=30 e^x+\frac {9 e^x x}{4}+6 e^x x^2+\frac {11 e^x x^3}{2}+2 e^x x^4+\frac {e^x x^5}{4}+21 e^x \log (3)+\frac {33}{2} e^x x \log (3)+\frac {15}{2} e^x x^2 \log (3)+\frac {7}{2} e^x x^3 \log (3)+\frac {1}{2} e^x x^4 \log (3)+\frac {3}{2} e^x \log ^2(3)+\frac {9}{4} e^x x \log ^2(3)+\frac {3}{2} e^x x^2 \log ^2(3)+\frac {1}{4} e^x x^3 \log ^2(3)-30 \int e^x \, dx-(12 \log (3)) \int e^x x \, dx-(33 \log (3)) \int e^x \, dx-\frac {1}{2} \left (3 \log ^2(3)\right ) \int e^x \, dx\\ &=\frac {9 e^x x}{4}+6 e^x x^2+\frac {11 e^x x^3}{2}+2 e^x x^4+\frac {e^x x^5}{4}-12 e^x \log (3)+\frac {9}{2} e^x x \log (3)+\frac {15}{2} e^x x^2 \log (3)+\frac {7}{2} e^x x^3 \log (3)+\frac {1}{2} e^x x^4 \log (3)+\frac {9}{4} e^x x \log ^2(3)+\frac {3}{2} e^x x^2 \log ^2(3)+\frac {1}{4} e^x x^3 \log ^2(3)+(12 \log (3)) \int e^x \, dx\\ &=\frac {9 e^x x}{4}+6 e^x x^2+\frac {11 e^x x^3}{2}+2 e^x x^4+\frac {e^x x^5}{4}+\frac {9}{2} e^x x \log (3)+\frac {15}{2} e^x x^2 \log (3)+\frac {7}{2} e^x x^3 \log (3)+\frac {1}{2} e^x x^4 \log (3)+\frac {9}{4} e^x x \log ^2(3)+\frac {3}{2} e^x x^2 \log ^2(3)+\frac {1}{4} e^x x^3 \log ^2(3)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.32, size = 54, normalized size = 2.70 \begin {gather*} \frac {1}{4} e^x x \left (9+x^4+9 \log ^2(3)+x^3 (8+\log (9))+3 x \left (8+2 \log ^2(3)+\log (59049)\right )+x^2 \left (22+\log ^2(3)+\log (4782969)\right )+\log (387420489)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(9 + 57*x + 90*x^2 + 54*x^3 + 13*x^4 + x^5 + (18 + 78*x + 72*x^2 + 22*x^3 + 2*x^4)*Log[3] + (9
+ 21*x + 9*x^2 + x^3)*Log[3]^2))/4,x]

[Out]

(E^x*x*(9 + x^4 + 9*Log[3]^2 + x^3*(8 + Log[9]) + 3*x*(8 + 2*Log[3]^2 + Log[59049]) + x^2*(22 + Log[3]^2 + Log
[4782969]) + Log[387420489]))/4

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fricas [B]  time = 0.59, size = 64, normalized size = 3.20 \begin {gather*} \frac {1}{4} \, {\left (x^{5} + 8 \, x^{4} + 22 \, x^{3} + {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \relax (3)^{2} + 24 \, x^{2} + 2 \, {\left (x^{4} + 7 \, x^{3} + 15 \, x^{2} + 9 \, x\right )} \log \relax (3) + 9 \, x\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*log(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+
9)*exp(x),x, algorithm="fricas")

[Out]

1/4*(x^5 + 8*x^4 + 22*x^3 + (x^3 + 6*x^2 + 9*x)*log(3)^2 + 24*x^2 + 2*(x^4 + 7*x^3 + 15*x^2 + 9*x)*log(3) + 9*
x)*e^x

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giac [B]  time = 0.14, size = 76, normalized size = 3.80 \begin {gather*} \frac {1}{4} \, {\left (x^{5} + 2 \, x^{4} \log \relax (3) + x^{3} \log \relax (3)^{2} + 8 \, x^{4} + 14 \, x^{3} \log \relax (3) + 6 \, x^{2} \log \relax (3)^{2} + 22 \, x^{3} + 30 \, x^{2} \log \relax (3) + 9 \, x \log \relax (3)^{2} + 24 \, x^{2} + 18 \, x \log \relax (3) + 9 \, x\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*log(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+
9)*exp(x),x, algorithm="giac")

[Out]

1/4*(x^5 + 2*x^4*log(3) + x^3*log(3)^2 + 8*x^4 + 14*x^3*log(3) + 6*x^2*log(3)^2 + 22*x^3 + 30*x^2*log(3) + 9*x
*log(3)^2 + 24*x^2 + 18*x*log(3) + 9*x)*e^x

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maple [A]  time = 0.07, size = 24, normalized size = 1.20

method	result	size
gosper	\(\frac {{\mathrm e}^{x} \left (x \ln \relax (3)+x^{2}+3 \ln \relax (3)+4 x +3\right )^{2} x}{4}\)	\(24\)
norman	\(\left (\frac {\ln \relax (3)}{2}+2\right ) x^{4} {\mathrm e}^{x}+\left (\frac {\ln \relax (3)^{2}}{4}+\frac {7 \ln \relax (3)}{2}+\frac {11}{2}\right ) x^{3} {\mathrm e}^{x}+\left (\frac {3 \ln \relax (3)^{2}}{2}+\frac {15 \ln \relax (3)}{2}+6\right ) x^{2} {\mathrm e}^{x}+\left (\frac {9 \ln \relax (3)^{2}}{4}+\frac {9 \ln \relax (3)}{2}+\frac {9}{4}\right ) x \,{\mathrm e}^{x}+\frac {x^{5} {\mathrm e}^{x}}{4}\)	\(73\)
risch	\(\frac {\left (x^{3} \ln \relax (3)^{2}+2 x^{4} \ln \relax (3)+x^{5}+6 x^{2} \ln \relax (3)^{2}+14 x^{3} \ln \relax (3)+8 x^{4}+9 x \ln \relax (3)^{2}+30 x^{2} \ln \relax (3)+22 x^{3}+18 x \ln \relax (3)+24 x^{2}+9 x \right ) {\mathrm e}^{x}}{4}\)	\(77\)
default	\(\frac {x^{5} {\mathrm e}^{x}}{4}+2 \,{\mathrm e}^{x} x^{4}+\frac {11 \,{\mathrm e}^{x} x^{3}}{2}+6 \,{\mathrm e}^{x} x^{2}+\frac {9 \,{\mathrm e}^{x} x}{4}+\frac {{\mathrm e}^{x} x^{3} \ln \relax (3)^{2}}{4}+\frac {3 \,{\mathrm e}^{x} x^{2} \ln \relax (3)^{2}}{2}+\frac {9 x \ln \relax (3)^{2} {\mathrm e}^{x}}{4}+\frac {9 x \ln \relax (3) {\mathrm e}^{x}}{2}+\frac {15 x^{2} \ln \relax (3) {\mathrm e}^{x}}{2}+\frac {7 \,{\mathrm e}^{x} x^{3} \ln \relax (3)}{2}+\frac {{\mathrm e}^{x} x^{4} \ln \relax (3)}{2}\)	\(100\)
meijerg	\(\frac {111}{4}+\frac {9 \,{\mathrm e}^{x}}{4}-\left (-\frac {\ln \relax (3)^{2}}{4}-\frac {11 \ln \relax (3)}{2}-\frac {27}{2}\right ) \left (6-\frac {\left (-4 x^{3}+12 x^{2}-24 x +24\right ) {\mathrm e}^{x}}{4}\right )-\left (\frac {9 \ln \relax (3)^{2}}{4}+18 \ln \relax (3)+\frac {45}{2}\right ) \left (2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\right )-\left (-\frac {21 \ln \relax (3)^{2}}{4}-\frac {39 \ln \relax (3)}{2}-\frac {57}{4}\right ) \left (1-\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}\right )-\frac {\left (-6 x^{5}+30 x^{4}-120 x^{3}+360 x^{2}-720 x +720\right ) {\mathrm e}^{x}}{24}-\left (\frac {\ln \relax (3)}{2}+\frac {13}{4}\right ) \left (24-\frac {\left (5 x^{4}-20 x^{3}+60 x^{2}-120 x +120\right ) {\mathrm e}^{x}}{5}\right )-\frac {9 \ln \relax (3)^{2} \left (1-{\mathrm e}^{x}\right )}{4}-\frac {9 \ln \relax (3) \left (1-{\mathrm e}^{x}\right )}{2}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((x^3+9*x^2+21*x+9)*ln(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*ln(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+9)*exp(x
),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(x)*(x*ln(3)+x^2+3*ln(3)+4*x+3)^2*x

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maxima [B]  time = 0.38, size = 218, normalized size = 10.90 \begin {gather*} \frac {1}{4} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \log \relax (3)^{2} + \frac {9}{4} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \log \relax (3)^{2} + \frac {21}{4} \, {\left (x - 1\right )} e^{x} \log \relax (3)^{2} + \frac {1}{2} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} \log \relax (3) + \frac {11}{2} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \log \relax (3) + 18 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \log \relax (3) + \frac {39}{2} \, {\left (x - 1\right )} e^{x} \log \relax (3) + \frac {9}{4} \, e^{x} \log \relax (3)^{2} + \frac {1}{4} \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + \frac {13}{4} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + \frac {27}{2} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + \frac {45}{2} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac {57}{4} \, {\left (x - 1\right )} e^{x} + \frac {9}{2} \, e^{x} \log \relax (3) + \frac {9}{4} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*log(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+
9)*exp(x),x, algorithm="maxima")

[Out]

1/4*(x^3 - 3*x^2 + 6*x - 6)*e^x*log(3)^2 + 9/4*(x^2 - 2*x + 2)*e^x*log(3)^2 + 21/4*(x - 1)*e^x*log(3)^2 + 1/2*
(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x*log(3) + 11/2*(x^3 - 3*x^2 + 6*x - 6)*e^x*log(3) + 18*(x^2 - 2*x + 2)*e
^x*log(3) + 39/2*(x - 1)*e^x*log(3) + 9/4*e^x*log(3)^2 + 1/4*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x
+ 13/4*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x + 27/2*(x^3 - 3*x^2 + 6*x - 6)*e^x + 45/2*(x^2 - 2*x + 2)*e^x +
57/4*(x - 1)*e^x + 9/2*e^x*log(3) + 9/4*e^x

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mupad [B]  time = 0.09, size = 62, normalized size = 3.10 \begin {gather*} \frac {x^5\,{\mathrm {e}}^x}{4}+\frac {x^3\,{\mathrm {e}}^x\,\left (14\,\ln \relax (3)+{\ln \relax (3)}^2+22\right )}{4}+\frac {9\,x\,{\mathrm {e}}^x\,{\left (\ln \relax (3)+1\right )}^2}{4}+\frac {x^4\,{\mathrm {e}}^x\,\left (\ln \relax (9)+8\right )}{4}+\frac {3\,x^2\,{\mathrm {e}}^x\,\left (\ln \relax (3)+1\right )\,\left (\ln \relax (3)+4\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(57*x + log(3)^2*(21*x + 9*x^2 + x^3 + 9) + log(3)*(78*x + 72*x^2 + 22*x^3 + 2*x^4 + 18) + 90*x^2
+ 54*x^3 + 13*x^4 + x^5 + 9))/4,x)

[Out]

(x^5*exp(x))/4 + (x^3*exp(x)*(14*log(3) + log(3)^2 + 22))/4 + (9*x*exp(x)*(log(3) + 1)^2)/4 + (x^4*exp(x)*(log
(9) + 8))/4 + (3*x^2*exp(x)*(log(3) + 1)*(log(3) + 4))/2

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sympy [B]  time = 0.20, size = 85, normalized size = 4.25 \begin {gather*} \frac {\left (x^{5} + 2 x^{4} \log {\relax (3 )} + 8 x^{4} + x^{3} \log {\relax (3 )}^{2} + 14 x^{3} \log {\relax (3 )} + 22 x^{3} + 6 x^{2} \log {\relax (3 )}^{2} + 24 x^{2} + 30 x^{2} \log {\relax (3 )} + 9 x + 9 x \log {\relax (3 )}^{2} + 18 x \log {\relax (3 )}\right ) e^{x}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x**3+9*x**2+21*x+9)*ln(3)**2+(2*x**4+22*x**3+72*x**2+78*x+18)*ln(3)+x**5+13*x**4+54*x**3+90*x*
*2+57*x+9)*exp(x),x)

[Out]

(x**5 + 2*x**4*log(3) + 8*x**4 + x**3*log(3)**2 + 14*x**3*log(3) + 22*x**3 + 6*x**2*log(3)**2 + 24*x**2 + 30*x
**2*log(3) + 9*x + 9*x*log(3)**2 + 18*x*log(3))*exp(x)/4

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