3.23.65 \(\int (12+4 x-6 x^2+(24 x^2-16 x^3) \log ^2(8)+(16 x^3-10 x^4) \log ^4(8)) \, dx\)

Optimal. Leaf size=23 \[ 2+(4-2 x) \left (-4+x+\left (x+x^2 \log ^2(8)\right )^2\right ) \]

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Rubi [B]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 2.17, number of steps used = 3, number of rules used = 0, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} -2 x^5 \log ^4(8)+4 x^4 \log ^4(8)-4 x^4 \log ^2(8)-2 x^3+8 x^3 \log ^2(8)+2 x^2+12 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[12 + 4*x - 6*x^2 + (24*x^2 - 16*x^3)*Log[8]^2 + (16*x^3 - 10*x^4)*Log[8]^4,x]

[Out]

12*x + 2*x^2 - 2*x^3 + 8*x^3*Log[8]^2 - 4*x^4*Log[8]^2 + 4*x^4*Log[8]^4 - 2*x^5*Log[8]^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=12 x+2 x^2-2 x^3+\log ^2(8) \int \left (24 x^2-16 x^3\right ) \, dx+\log ^4(8) \int \left (16 x^3-10 x^4\right ) \, dx\\ &=12 x+2 x^2-2 x^3+8 x^3 \log ^2(8)-4 x^4 \log ^2(8)+4 x^4 \log ^4(8)-2 x^5 \log ^4(8)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 50, normalized size = 2.17 \begin {gather*} 12 x+2 x^2-2 x^3+8 x^3 \log ^2(8)-4 x^4 \log ^2(8)+4 x^4 \log ^4(8)-2 x^5 \log ^4(8) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[12 + 4*x - 6*x^2 + (24*x^2 - 16*x^3)*Log[8]^2 + (16*x^3 - 10*x^4)*Log[8]^4,x]

[Out]

12*x + 2*x^2 - 2*x^3 + 8*x^3*Log[8]^2 - 4*x^4*Log[8]^2 + 4*x^4*Log[8]^4 - 2*x^5*Log[8]^4

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fricas [A]  time = 0.79, size = 44, normalized size = 1.91 \begin {gather*} -162 \, {\left (x^{5} - 2 \, x^{4}\right )} \log \relax (2)^{4} - 2 \, x^{3} - 36 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (2)^{2} + 2 \, x^{2} + 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(81*(-10*x^4+16*x^3)*log(2)^4+9*(-16*x^3+24*x^2)*log(2)^2-6*x^2+4*x+12,x, algorithm="fricas")

[Out]

-162*(x^5 - 2*x^4)*log(2)^4 - 2*x^3 - 36*(x^4 - 2*x^3)*log(2)^2 + 2*x^2 + 12*x

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giac [A]  time = 0.41, size = 44, normalized size = 1.91 \begin {gather*} -162 \, {\left (x^{5} - 2 \, x^{4}\right )} \log \relax (2)^{4} - 2 \, x^{3} - 36 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (2)^{2} + 2 \, x^{2} + 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(81*(-10*x^4+16*x^3)*log(2)^4+9*(-16*x^3+24*x^2)*log(2)^2-6*x^2+4*x+12,x, algorithm="giac")

[Out]

-162*(x^5 - 2*x^4)*log(2)^4 - 2*x^3 - 36*(x^4 - 2*x^3)*log(2)^2 + 2*x^2 + 12*x

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maple [A]  time = 0.03, size = 48, normalized size = 2.09

method	result	size
gosper	\(-2 x \left (81 \ln \relax (2)^{4} x^{4}-162 x^{3} \ln \relax (2)^{4}+18 x^{3} \ln \relax (2)^{2}-36 x^{2} \ln \relax (2)^{2}+x^{2}-x -6\right )\)	\(48\)
norman	\(\left (72 \ln \relax (2)^{2}-2\right ) x^{3}+\left (324 \ln \relax (2)^{4}-36 \ln \relax (2)^{2}\right ) x^{4}+12 x +2 x^{2}-162 x^{5} \ln \relax (2)^{4}\)	\(48\)
default	\(81 \ln \relax (2)^{4} \left (-2 x^{5}+4 x^{4}\right )+9 \ln \relax (2)^{2} \left (-4 x^{4}+8 x^{3}\right )-2 x^{3}+2 x^{2}+12 x\)	\(49\)
risch	\(-162 x^{5} \ln \relax (2)^{4}+324 \ln \relax (2)^{4} x^{4}-36 x^{4} \ln \relax (2)^{2}+72 x^{3} \ln \relax (2)^{2}-2 x^{3}+2 x^{2}+12 x\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(81*(-10*x^4+16*x^3)*ln(2)^4+9*(-16*x^3+24*x^2)*ln(2)^2-6*x^2+4*x+12,x,method=_RETURNVERBOSE)

[Out]

-2*x*(81*ln(2)^4*x^4-162*x^3*ln(2)^4+18*x^3*ln(2)^2-36*x^2*ln(2)^2+x^2-x-6)

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maxima [A]  time = 0.36, size = 44, normalized size = 1.91 \begin {gather*} -162 \, {\left (x^{5} - 2 \, x^{4}\right )} \log \relax (2)^{4} - 2 \, x^{3} - 36 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (2)^{2} + 2 \, x^{2} + 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(81*(-10*x^4+16*x^3)*log(2)^4+9*(-16*x^3+24*x^2)*log(2)^2-6*x^2+4*x+12,x, algorithm="maxima")

[Out]

-162*(x^5 - 2*x^4)*log(2)^4 - 2*x^3 - 36*(x^4 - 2*x^3)*log(2)^2 + 2*x^2 + 12*x

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mupad [B]  time = 0.03, size = 48, normalized size = 2.09 \begin {gather*} -162\,{\ln \relax (2)}^4\,x^5+\left (324\,{\ln \relax (2)}^4-36\,{\ln \relax (2)}^2\right )\,x^4+\left (72\,{\ln \relax (2)}^2-2\right )\,x^3+2\,x^2+12\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x - 6*x^2 + 81*log(2)^4*(16*x^3 - 10*x^4) + 9*log(2)^2*(24*x^2 - 16*x^3) + 12,x)

[Out]

12*x - 162*x^5*log(2)^4 + x^3*(72*log(2)^2 - 2) + 2*x^2 - x^4*(36*log(2)^2 - 324*log(2)^4)

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sympy [B]  time = 0.07, size = 46, normalized size = 2.00 \begin {gather*} - 162 x^{5} \log {\relax (2 )}^{4} + x^{4} \left (- 36 \log {\relax (2 )}^{2} + 324 \log {\relax (2 )}^{4}\right ) + x^{3} \left (-2 + 72 \log {\relax (2 )}^{2}\right ) + 2 x^{2} + 12 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(81*(-10*x**4+16*x**3)*ln(2)**4+9*(-16*x**3+24*x**2)*ln(2)**2-6*x**2+4*x+12,x)

[Out]

-162*x**5*log(2)**4 + x**4*(-36*log(2)**2 + 324*log(2)**4) + x**3*(-2 + 72*log(2)**2) + 2*x**2 + 12*x

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