3.1.73 \(\int \frac {e (-12+72 x-144 x^2+240 x^3+144 x^4)+e (-16+84 x-192 x^2+128 x^3+96 x^4) \log (2)+e (-5+28 x-48 x^2+16 x^3+16 x^4) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx\)

Optimal. Leaf size=26 \[ e \left (x+\frac {(-1+2 x)^2 (2+\log (2))}{4 (1+x)}\right )^2 \]

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Rubi [B]  time = 0.13, antiderivative size = 60, normalized size of antiderivative = 2.31, number of steps used = 2, number of rules used = 1, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2074} \begin {gather*} e x^2 (3+\log (2))^2-4 e x \left (6+\log ^2(2)+\log (32)\right )-\frac {9 e \left (14+3 \log ^2(2)+\log (8192)\right )}{2 (x+1)}+\frac {81 e (2+\log (2))^2}{16 (x+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E*(-12 + 72*x - 144*x^2 + 240*x^3 + 144*x^4) + E*(-16 + 84*x - 192*x^2 + 128*x^3 + 96*x^4)*Log[2] + E*(-5
+ 28*x - 48*x^2 + 16*x^3 + 16*x^4)*Log[2]^2)/(8 + 24*x + 24*x^2 + 8*x^3),x]

[Out]

(81*E*(2 + Log[2])^2)/(16*(1 + x)^2) + E*x^2*(3 + Log[2])^2 - 4*E*x*(6 + Log[2]^2 + Log[32]) - (9*E*(14 + 3*Lo
g[2]^2 + Log[8192]))/(2*(1 + x))

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {81 e (2+\log (2))^2}{8 (1+x)^3}+2 e x (3+\log (2))^2-4 e \left (6+\log ^2(2)+\log (32)\right )+\frac {9 e \left (14+3 \log ^2(2)+\log (8192)\right )}{2 (1+x)^2}\right ) \, dx\\ &=\frac {81 e (2+\log (2))^2}{16 (1+x)^2}+e x^2 (3+\log (2))^2-4 e x \left (6+\log ^2(2)+\log (32)\right )-\frac {9 e \left (14+3 \log ^2(2)+\log (8192)\right )}{2 (1+x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 81, normalized size = 3.12 \begin {gather*} \frac {e \left (324+81 \log ^2(2)+16 (1+x)^4 (3+\log (2))^2+49 \log (16)+16 \log (256)-24 (1+x) \left (42-25 \log (2)+9 \log ^2(2)+8 \log (16)+4 \log (256)\right )-32 (1+x)^3 \left (21+3 \log ^2(2)+\log (65536)\right )\right )}{16 (1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(-12 + 72*x - 144*x^2 + 240*x^3 + 144*x^4) + E*(-16 + 84*x - 192*x^2 + 128*x^3 + 96*x^4)*Log[2] +
E*(-5 + 28*x - 48*x^2 + 16*x^3 + 16*x^4)*Log[2]^2)/(8 + 24*x + 24*x^2 + 8*x^3),x]

[Out]

(E*(324 + 81*Log[2]^2 + 16*(1 + x)^4*(3 + Log[2])^2 + 49*Log[16] + 16*Log[256] - 24*(1 + x)*(42 - 25*Log[2] +
9*Log[2]^2 + 8*Log[16] + 4*Log[256]) - 32*(1 + x)^3*(21 + 3*Log[2]^2 + Log[65536])))/(16*(1 + x)^2)

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fricas [B]  time = 0.65, size = 90, normalized size = 3.46 \begin {gather*} \frac {{\left (16 \, x^{4} - 32 \, x^{3} - 112 \, x^{2} - 280 \, x - 135\right )} e \log \relax (2)^{2} + 4 \, {\left (24 \, x^{4} - 32 \, x^{3} - 136 \, x^{2} - 314 \, x - 153\right )} e \log \relax (2) + 12 \, {\left (12 \, x^{4} - 8 \, x^{3} - 52 \, x^{2} - 116 \, x - 57\right )} e}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-192*x^2+84*x-16)*exp(1)*log(2)+(144*x
^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8*x^3+24*x^2+24*x+8),x, algorithm="fricas")

[Out]

1/16*((16*x^4 - 32*x^3 - 112*x^2 - 280*x - 135)*e*log(2)^2 + 4*(24*x^4 - 32*x^3 - 136*x^2 - 314*x - 153)*e*log
(2) + 12*(12*x^4 - 8*x^3 - 52*x^2 - 116*x - 57)*e)/(x^2 + 2*x + 1)

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giac [B]  time = 0.33, size = 95, normalized size = 3.65 \begin {gather*} x^{2} e \log \relax (2)^{2} + 6 \, x^{2} e \log \relax (2) - 4 \, x e \log \relax (2)^{2} + 9 \, x^{2} e - 20 \, x e \log \relax (2) - 24 \, x e - \frac {9 \, {\left (24 \, x e \log \relax (2)^{2} + 104 \, x e \log \relax (2) + 15 \, e \log \relax (2)^{2} + 112 \, x e + 68 \, e \log \relax (2) + 76 \, e\right )}}{16 \, {\left (x + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-192*x^2+84*x-16)*exp(1)*log(2)+(144*x
^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8*x^3+24*x^2+24*x+8),x, algorithm="giac")

[Out]

x^2*e*log(2)^2 + 6*x^2*e*log(2) - 4*x*e*log(2)^2 + 9*x^2*e - 20*x*e*log(2) - 24*x*e - 9/16*(24*x*e*log(2)^2 +
104*x*e*log(2) + 15*e*log(2)^2 + 112*x*e + 68*e*log(2) + 76*e)/(x + 1)^2

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maple [B]  time = 0.07, size = 80, normalized size = 3.08

method	result	size
default	\(\frac {{\mathrm e} \left (8 x^{2} \ln \relax (2)^{2}-32 x \ln \relax (2)^{2}+48 x^{2} \ln \relax (2)-160 x \ln \relax (2)+72 x^{2}-192 x -\frac {-81 \ln \relax (2)^{2}-324 \ln \relax (2)-324}{2 \left (x +1\right )^{2}}-\frac {108 \ln \relax (2)^{2}+468 \ln \relax (2)+504}{x +1}\right )}{8}\)	\(80\)
gosper	\(\frac {\left (16 x^{4} \ln \relax (2)^{2}-32 x^{3} \ln \relax (2)^{2}+96 x^{4} \ln \relax (2)-128 x^{3} \ln \relax (2)+144 x^{4}-56 x \ln \relax (2)^{2}-96 x^{3}-23 \ln \relax (2)^{2}-168 x \ln \relax (2)-68 \ln \relax (2)-144 x -60\right ) {\mathrm e}}{16 x^{2}+32 x +16}\)	\(84\)
norman	\(\frac {\left (-2 \,{\mathrm e} \ln \relax (2)^{2}-8 \,{\mathrm e} \ln \relax (2)-6 \,{\mathrm e}\right ) x^{3}+\left ({\mathrm e} \ln \relax (2)^{2}+6 \,{\mathrm e} \ln \relax (2)+9 \,{\mathrm e}\right ) x^{4}+\left (-\frac {7 \,{\mathrm e} \ln \relax (2)^{2}}{2}-\frac {21 \,{\mathrm e} \ln \relax (2)}{2}-9 \,{\mathrm e}\right ) x -\frac {23 \,{\mathrm e} \ln \relax (2)^{2}}{16}-\frac {17 \,{\mathrm e} \ln \relax (2)}{4}-\frac {15 \,{\mathrm e}}{4}}{\left (x +1\right )^{2}}\)	\(92\)
risch	\(x^{2} {\mathrm e} \ln \relax (2)^{2}-4 \ln \relax (2)^{2} {\mathrm e} x +6 \ln \relax (2) {\mathrm e} x^{2}-20 x \,{\mathrm e} \ln \relax (2)+9 x^{2} {\mathrm e}-24 x \,{\mathrm e}+\frac {\left (-\frac {27 \,{\mathrm e} \ln \relax (2)^{2}}{2}-\frac {117 \,{\mathrm e} \ln \relax (2)}{2}-63 \,{\mathrm e}\right ) x -\frac {135 \,{\mathrm e} \ln \relax (2)^{2}}{16}-\frac {153 \,{\mathrm e} \ln \relax (2)}{4}-\frac {171 \,{\mathrm e}}{4}}{x^{2}+2 x +1}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*ln(2)^2+(96*x^4+128*x^3-192*x^2+84*x-16)*exp(1)*ln(2)+(144*x^4+240*x
^3-144*x^2+72*x-12)*exp(1))/(8*x^3+24*x^2+24*x+8),x,method=_RETURNVERBOSE)

[Out]

1/8*exp(1)*(8*x^2*ln(2)^2-32*x*ln(2)^2+48*x^2*ln(2)-160*x*ln(2)+72*x^2-192*x-1/2*(-81*ln(2)^2-324*ln(2)-324)/(
x+1)^2-(108*ln(2)^2+468*ln(2)+504)/(x+1))

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maxima [B]  time = 0.41, size = 97, normalized size = 3.73 \begin {gather*} {\left (e \log \relax (2)^{2} + 6 \, e \log \relax (2) + 9 \, e\right )} x^{2} - 4 \, {\left (e \log \relax (2)^{2} + 5 \, e \log \relax (2) + 6 \, e\right )} x - \frac {9 \, {\left (15 \, e \log \relax (2)^{2} + 8 \, {\left (3 \, e \log \relax (2)^{2} + 13 \, e \log \relax (2) + 14 \, e\right )} x + 68 \, e \log \relax (2) + 76 \, e\right )}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-192*x^2+84*x-16)*exp(1)*log(2)+(144*x
^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8*x^3+24*x^2+24*x+8),x, algorithm="maxima")

[Out]

(e*log(2)^2 + 6*e*log(2) + 9*e)*x^2 - 4*(e*log(2)^2 + 5*e*log(2) + 6*e)*x - 9/16*(15*e*log(2)^2 + 8*(3*e*log(2
)^2 + 13*e*log(2) + 14*e)*x + 68*e*log(2) + 76*e)/(x^2 + 2*x + 1)

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mupad [B]  time = 0.20, size = 99, normalized size = 3.81 \begin {gather*} x^2\,\left (9\,\mathrm {e}+6\,\mathrm {e}\,\ln \relax (2)+\mathrm {e}\,{\ln \relax (2)}^2\right )-x\,\left (24\,\mathrm {e}+20\,\mathrm {e}\,\ln \relax (2)+4\,\mathrm {e}\,{\ln \relax (2)}^2\right )-\frac {342\,\mathrm {e}+306\,\mathrm {e}\,\ln \relax (2)+\frac {135\,\mathrm {e}\,{\ln \relax (2)}^2}{2}+x\,\left (504\,\mathrm {e}+468\,\mathrm {e}\,\ln \relax (2)+108\,\mathrm {e}\,{\ln \relax (2)}^2\right )}{8\,x^2+16\,x+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)*(72*x - 144*x^2 + 240*x^3 + 144*x^4 - 12) + exp(1)*log(2)*(84*x - 192*x^2 + 128*x^3 + 96*x^4 - 16)
+ exp(1)*log(2)^2*(28*x - 48*x^2 + 16*x^3 + 16*x^4 - 5))/(24*x + 24*x^2 + 8*x^3 + 8),x)

[Out]

x^2*(9*exp(1) + 6*exp(1)*log(2) + exp(1)*log(2)^2) - x*(24*exp(1) + 20*exp(1)*log(2) + 4*exp(1)*log(2)^2) - (3
42*exp(1) + 306*exp(1)*log(2) + (135*exp(1)*log(2)^2)/2 + x*(504*exp(1) + 468*exp(1)*log(2) + 108*exp(1)*log(2
)^2))/(16*x + 8*x^2 + 8)

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sympy [B]  time = 0.50, size = 112, normalized size = 4.31 \begin {gather*} x^{2} \left (e \log {\relax (2 )}^{2} + 6 e \log {\relax (2 )} + 9 e\right ) + x \left (- 24 e - 20 e \log {\relax (2 )} - 4 e \log {\relax (2 )}^{2}\right ) + \frac {x \left (- 1008 e - 936 e \log {\relax (2 )} - 216 e \log {\relax (2 )}^{2}\right ) - 684 e - 612 e \log {\relax (2 )} - 135 e \log {\relax (2 )}^{2}}{16 x^{2} + 32 x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**4+16*x**3-48*x**2+28*x-5)*exp(1)*ln(2)**2+(96*x**4+128*x**3-192*x**2+84*x-16)*exp(1)*ln(2)+(
144*x**4+240*x**3-144*x**2+72*x-12)*exp(1))/(8*x**3+24*x**2+24*x+8),x)

[Out]

x**2*(E*log(2)**2 + 6*E*log(2) + 9*E) + x*(-24*E - 20*E*log(2) - 4*E*log(2)**2) + (x*(-1008*E - 936*E*log(2) -
216*E*log(2)**2) - 684*E - 612*E*log(2) - 135*E*log(2)**2)/(16*x**2 + 32*x + 16)

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