3.63.80 \(\int (-48 x^2+128 x^3-120 x^4+48 x^5-7 x^6+(96 x^2-192 x^3+120 x^4-24 x^5) \log (9)+(-72 x^2+96 x^3-30 x^4) \log ^2(9)+(24 x^2-16 x^3) \log ^3(9)-3 x^2 \log ^4(9)) \, dx\)

Optimal. Leaf size=18 \[ 8-x^3 (2-x-\log (9))^4 \]

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Rubi [B]  time = 0.04, antiderivative size = 105, normalized size of antiderivative = 5.83, number of steps used = 5, number of rules used = 1, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {6} \begin {gather*} -x^7+8 x^6-4 x^6 \log (9)-24 x^5-6 x^5 \log ^2(9)+24 x^5 \log (9)+32 x^4-4 x^4 \log ^3(9)+24 x^4 \log ^2(9)-48 x^4 \log (9)-x^3 \left (16+\log ^4(9)\right )+8 x^3 \log ^3(9)-24 x^3 \log ^2(9)+32 x^3 \log (9) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-48*x^2 + 128*x^3 - 120*x^4 + 48*x^5 - 7*x^6 + (96*x^2 - 192*x^3 + 120*x^4 - 24*x^5)*Log[9] + (-72*x^2 + 9
6*x^3 - 30*x^4)*Log[9]^2 + (24*x^2 - 16*x^3)*Log[9]^3 - 3*x^2*Log[9]^4,x]

[Out]

32*x^4 - 24*x^5 + 8*x^6 - x^7 + 32*x^3*Log[9] - 48*x^4*Log[9] + 24*x^5*Log[9] - 4*x^6*Log[9] - 24*x^3*Log[9]^2
+ 24*x^4*Log[9]^2 - 6*x^5*Log[9]^2 + 8*x^3*Log[9]^3 - 4*x^4*Log[9]^3 - x^3*(16 + Log[9]^4)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (128 x^3-120 x^4+48 x^5-7 x^6+\left (96 x^2-192 x^3+120 x^4-24 x^5\right ) \log (9)+\left (-72 x^2+96 x^3-30 x^4\right ) \log ^2(9)+\left (24 x^2-16 x^3\right ) \log ^3(9)+x^2 \left (-48-3 \log ^4(9)\right )\right ) \, dx\\ &=32 x^4-24 x^5+8 x^6-x^7-x^3 \left (16+\log ^4(9)\right )+\log (9) \int \left (96 x^2-192 x^3+120 x^4-24 x^5\right ) \, dx+\log ^2(9) \int \left (-72 x^2+96 x^3-30 x^4\right ) \, dx+\log ^3(9) \int \left (24 x^2-16 x^3\right ) \, dx\\ &=32 x^4-24 x^5+8 x^6-x^7+32 x^3 \log (9)-48 x^4 \log (9)+24 x^5 \log (9)-4 x^6 \log (9)-24 x^3 \log ^2(9)+24 x^4 \log ^2(9)-6 x^5 \log ^2(9)+8 x^3 \log ^3(9)-4 x^4 \log ^3(9)-x^3 \left (16+\log ^4(9)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 74, normalized size = 4.11 \begin {gather*} -x^7-\frac {1}{3} x^3 (-2+\log (9))^3 (-6+\log (729))-\frac {3}{5} x^5 (-2+\log (9)) (-20+7 \log (9)+\log (729))-\frac {1}{4} x^4 (-2+\log (9))^2 (-32+7 \log (9)+3 \log (729))-x^6 (-8+\log (6561)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-48*x^2 + 128*x^3 - 120*x^4 + 48*x^5 - 7*x^6 + (96*x^2 - 192*x^3 + 120*x^4 - 24*x^5)*Log[9] + (-72*x
^2 + 96*x^3 - 30*x^4)*Log[9]^2 + (24*x^2 - 16*x^3)*Log[9]^3 - 3*x^2*Log[9]^4,x]

[Out]

-x^7 - (x^3*(-2 + Log[9])^3*(-6 + Log[729]))/3 - (3*x^5*(-2 + Log[9])*(-20 + 7*Log[9] + Log[729]))/5 - (x^4*(-
2 + Log[9])^2*(-32 + 7*Log[9] + 3*Log[729]))/4 - x^6*(-8 + Log[6561])

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fricas [B]  time = 0.61, size = 93, normalized size = 5.17 \begin {gather*} -x^{7} - 16 \, x^{3} \log \relax (3)^{4} + 8 \, x^{6} - 24 \, x^{5} + 32 \, x^{4} - 32 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (3)^{3} - 16 \, x^{3} - 24 \, {\left (x^{5} - 4 \, x^{4} + 4 \, x^{3}\right )} \log \relax (3)^{2} - 8 \, {\left (x^{6} - 6 \, x^{5} + 12 \, x^{4} - 8 \, x^{3}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-48*x^2*log(3)^4+8*(-16*x^3+24*x^2)*log(3)^3+4*(-30*x^4+96*x^3-72*x^2)*log(3)^2+2*(-24*x^5+120*x^4-1
92*x^3+96*x^2)*log(3)-7*x^6+48*x^5-120*x^4+128*x^3-48*x^2,x, algorithm="fricas")

[Out]

-x^7 - 16*x^3*log(3)^4 + 8*x^6 - 24*x^5 + 32*x^4 - 32*(x^4 - 2*x^3)*log(3)^3 - 16*x^3 - 24*(x^5 - 4*x^4 + 4*x^
3)*log(3)^2 - 8*(x^6 - 6*x^5 + 12*x^4 - 8*x^3)*log(3)

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giac [B]  time = 0.14, size = 93, normalized size = 5.17 \begin {gather*} -x^{7} - 16 \, x^{3} \log \relax (3)^{4} + 8 \, x^{6} - 24 \, x^{5} + 32 \, x^{4} - 32 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (3)^{3} - 16 \, x^{3} - 24 \, {\left (x^{5} - 4 \, x^{4} + 4 \, x^{3}\right )} \log \relax (3)^{2} - 8 \, {\left (x^{6} - 6 \, x^{5} + 12 \, x^{4} - 8 \, x^{3}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-48*x^2*log(3)^4+8*(-16*x^3+24*x^2)*log(3)^3+4*(-30*x^4+96*x^3-72*x^2)*log(3)^2+2*(-24*x^5+120*x^4-1
92*x^3+96*x^2)*log(3)-7*x^6+48*x^5-120*x^4+128*x^3-48*x^2,x, algorithm="giac")

[Out]

-x^7 - 16*x^3*log(3)^4 + 8*x^6 - 24*x^5 + 32*x^4 - 32*(x^4 - 2*x^3)*log(3)^3 - 16*x^3 - 24*(x^5 - 4*x^4 + 4*x^
3)*log(3)^2 - 8*(x^6 - 6*x^5 + 12*x^4 - 8*x^3)*log(3)

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maple [B]  time = 0.05, size = 83, normalized size = 4.61

method	result	size
norman	\(\left (-8 \ln \relax (3)+8\right ) x^{6}+\left (-24 \ln \relax (3)^{2}+48 \ln \relax (3)-24\right ) x^{5}+\left (-32 \ln \relax (3)^{3}+96 \ln \relax (3)^{2}-96 \ln \relax (3)+32\right ) x^{4}+\left (-16 \ln \relax (3)^{4}+64 \ln \relax (3)^{3}-96 \ln \relax (3)^{2}+64 \ln \relax (3)-16\right ) x^{3}-x^{7}\)	\(83\)
gosper	\(-x^{3} \left (16 \ln \relax (3)^{4}+32 x \ln \relax (3)^{3}+24 x^{2} \ln \relax (3)^{2}+8 x^{3} \ln \relax (3)+x^{4}-64 \ln \relax (3)^{3}-96 x \ln \relax (3)^{2}-48 x^{2} \ln \relax (3)-8 x^{3}+96 \ln \relax (3)^{2}+96 x \ln \relax (3)+24 x^{2}-64 \ln \relax (3)-32 x +16\right )\)	\(88\)
default	\(-16 \ln \relax (3)^{4} x^{3}+8 \ln \relax (3)^{3} \left (-4 x^{4}+8 x^{3}\right )+4 \ln \relax (3)^{2} \left (-6 x^{5}+24 x^{4}-24 x^{3}\right )+2 \ln \relax (3) \left (-4 x^{6}+24 x^{5}-48 x^{4}+32 x^{3}\right )-x^{7}+8 x^{6}-24 x^{5}+32 x^{4}-16 x^{3}\)	\(100\)
risch	\(-16 \ln \relax (3)^{4} x^{3}-32 x^{4} \ln \relax (3)^{3}+64 x^{3} \ln \relax (3)^{3}-24 x^{5} \ln \relax (3)^{2}+96 x^{4} \ln \relax (3)^{2}-96 x^{3} \ln \relax (3)^{2}-8 x^{6} \ln \relax (3)+48 x^{5} \ln \relax (3)-96 x^{4} \ln \relax (3)+64 x^{3} \ln \relax (3)-x^{7}+8 x^{6}-24 x^{5}+32 x^{4}-16 x^{3}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-48*x^2*ln(3)^4+8*(-16*x^3+24*x^2)*ln(3)^3+4*(-30*x^4+96*x^3-72*x^2)*ln(3)^2+2*(-24*x^5+120*x^4-192*x^3+96
*x^2)*ln(3)-7*x^6+48*x^5-120*x^4+128*x^3-48*x^2,x,method=_RETURNVERBOSE)

[Out]

(-8*ln(3)+8)*x^6+(-24*ln(3)^2+48*ln(3)-24)*x^5+(-32*ln(3)^3+96*ln(3)^2-96*ln(3)+32)*x^4+(-16*ln(3)^4+64*ln(3)^
3-96*ln(3)^2+64*ln(3)-16)*x^3-x^7

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maxima [B]  time = 0.36, size = 93, normalized size = 5.17 \begin {gather*} -x^{7} - 16 \, x^{3} \log \relax (3)^{4} + 8 \, x^{6} - 24 \, x^{5} + 32 \, x^{4} - 32 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (3)^{3} - 16 \, x^{3} - 24 \, {\left (x^{5} - 4 \, x^{4} + 4 \, x^{3}\right )} \log \relax (3)^{2} - 8 \, {\left (x^{6} - 6 \, x^{5} + 12 \, x^{4} - 8 \, x^{3}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-48*x^2*log(3)^4+8*(-16*x^3+24*x^2)*log(3)^3+4*(-30*x^4+96*x^3-72*x^2)*log(3)^2+2*(-24*x^5+120*x^4-1
92*x^3+96*x^2)*log(3)-7*x^6+48*x^5-120*x^4+128*x^3-48*x^2,x, algorithm="maxima")

[Out]

-x^7 - 16*x^3*log(3)^4 + 8*x^6 - 24*x^5 + 32*x^4 - 32*(x^4 - 2*x^3)*log(3)^3 - 16*x^3 - 24*(x^5 - 4*x^4 + 4*x^
3)*log(3)^2 - 8*(x^6 - 6*x^5 + 12*x^4 - 8*x^3)*log(3)

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mupad [B]  time = 4.07, size = 56, normalized size = 3.11 \begin {gather*} -x^7+\left (8-8\,\ln \relax (3)\right )\,x^6-24\,{\left (\ln \relax (3)-1\right )}^2\,x^5-\frac {\left (32\,\ln \relax (3)-32\right )\,{\left (\ln \relax (9)-2\right )}^2\,x^4}{4}-16\,{\left (\ln \relax (3)-1\right )}^4\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*log(3)*(96*x^2 - 192*x^3 + 120*x^4 - 24*x^5) - 48*x^2*log(3)^4 - 4*log(3)^2*(72*x^2 - 96*x^3 + 30*x^4) -
48*x^2 + 128*x^3 - 120*x^4 + 48*x^5 - 7*x^6 + 8*log(3)^3*(24*x^2 - 16*x^3),x)

[Out]

- x^6*(8*log(3) - 8) - 16*x^3*(log(3) - 1)^4 - 24*x^5*(log(3) - 1)^2 - x^7 - (x^4*(32*log(3) - 32)*(log(9) - 2
)^2)/4

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sympy [B]  time = 0.08, size = 83, normalized size = 4.61 \begin {gather*} - x^{7} + x^{6} \left (8 - 8 \log {\relax (3 )}\right ) + x^{5} \left (- 24 \log {\relax (3 )}^{2} - 24 + 48 \log {\relax (3 )}\right ) + x^{4} \left (- 96 \log {\relax (3 )} - 32 \log {\relax (3 )}^{3} + 32 + 96 \log {\relax (3 )}^{2}\right ) + x^{3} \left (- 96 \log {\relax (3 )}^{2} - 16 \log {\relax (3 )}^{4} - 16 + 64 \log {\relax (3 )} + 64 \log {\relax (3 )}^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-48*x**2*ln(3)**4+8*(-16*x**3+24*x**2)*ln(3)**3+4*(-30*x**4+96*x**3-72*x**2)*ln(3)**2+2*(-24*x**5+12
0*x**4-192*x**3+96*x**2)*ln(3)-7*x**6+48*x**5-120*x**4+128*x**3-48*x**2,x)

[Out]

-x**7 + x**6*(8 - 8*log(3)) + x**5*(-24*log(3)**2 - 24 + 48*log(3)) + x**4*(-96*log(3) - 32*log(3)**3 + 32 + 9
6*log(3)**2) + x**3*(-96*log(3)**2 - 16*log(3)**4 - 16 + 64*log(3) + 64*log(3)**3)

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