3.12.91 \(\int \frac {72 x+108 x^2+36 x^3+(-36 x^2-24 x^3) \log (2)+4 x^3 \log ^2(2)}{\log ^2(2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {5}{3}+x^4 \left (-1+\frac {3 (2+x)}{x \log (2)}\right )^2 \]

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Rubi [B]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 12} \begin {gather*} x^4 \left (1+\frac {9}{\log ^2(2)}\right )-\frac {6 x^4}{\log (2)}+\frac {36 x^3}{\log ^2(2)}-\frac {12 x^3}{\log (2)}+\frac {36 x^2}{\log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(72*x + 108*x^2 + 36*x^3 + (-36*x^2 - 24*x^3)*Log[2] + 4*x^3*Log[2]^2)/Log[2]^2,x]

[Out]

x^4*(1 + 9/Log[2]^2) + (36*x^2)/Log[2]^2 + (36*x^3)/Log[2]^2 - (12*x^3)/Log[2] - (6*x^4)/Log[2]

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]

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {72 x+108 x^2+\left (-36 x^2-24 x^3\right ) \log (2)+x^3 \left (36+4 \log ^2(2)\right )}{\log ^2(2)} \, dx\\ &=\frac {\int \left (72 x+108 x^2+\left (-36 x^2-24 x^3\right ) \log (2)+x^3 \left (36+4 \log ^2(2)\right )\right ) \, dx}{\log ^2(2)}\\ &=x^4 \left (1+\frac {9}{\log ^2(2)}\right )+\frac {36 x^2}{\log ^2(2)}+\frac {36 x^3}{\log ^2(2)}+\frac {\int \left (-36 x^2-24 x^3\right ) \, dx}{\log (2)}\\ &=x^4 \left (1+\frac {9}{\log ^2(2)}\right )+\frac {36 x^2}{\log ^2(2)}+\frac {36 x^3}{\log ^2(2)}-\frac {12 x^3}{\log (2)}-\frac {6 x^4}{\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.75 \begin {gather*} \frac {x^2 (-6+x (-3+\log (2)))^2}{\log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(72*x + 108*x^2 + 36*x^3 + (-36*x^2 - 24*x^3)*Log[2] + 4*x^3*Log[2]^2)/Log[2]^2,x]

[Out]

(x^2*(-6 + x*(-3 + Log[2]))^2)/Log[2]^2

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fricas [A]  time = 0.58, size = 42, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \relax (2)^{2} + 9 \, x^{4} + 36 \, x^{3} + 36 \, x^{2} - 6 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \relax (2)}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(2)^2+(-24*x^3-36*x^2)*log(2)+36*x^3+108*x^2+72*x)/log(2)^2,x, algorithm="fricas")

[Out]

(x^4*log(2)^2 + 9*x^4 + 36*x^3 + 36*x^2 - 6*(x^4 + 2*x^3)*log(2))/log(2)^2

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giac [A]  time = 0.40, size = 42, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \relax (2)^{2} + 9 \, x^{4} + 36 \, x^{3} + 36 \, x^{2} - 6 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \relax (2)}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(2)^2+(-24*x^3-36*x^2)*log(2)+36*x^3+108*x^2+72*x)/log(2)^2,x, algorithm="giac")

[Out]

(x^4*log(2)^2 + 9*x^4 + 36*x^3 + 36*x^2 - 6*(x^4 + 2*x^3)*log(2))/log(2)^2

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maple [A]  time = 0.02, size = 39, normalized size = 1.62

method	result	size
gosper	\(\frac {\left (x^{2} \ln \relax (2)^{2}-6 x^{2} \ln \relax (2)-12 x \ln \relax (2)+9 x^{2}+36 x +36\right ) x^{2}}{\ln \relax (2)^{2}}\)	\(39\)
default	\(\frac {x^{4} \ln \relax (2)^{2}+\ln \relax (2) \left (-6 x^{4}-12 x^{3}\right )+9 x^{4}+36 x^{3}+36 x^{2}}{\ln \relax (2)^{2}}\)	\(44\)
norman	\(\frac {\frac {\left (\ln \relax (2)^{2}-6 \ln \relax (2)+9\right ) x^{4}}{\ln \relax (2)}+\frac {36 x^{2}}{\ln \relax (2)}-\frac {12 \left (\ln \relax (2)-3\right ) x^{3}}{\ln \relax (2)}}{\ln \relax (2)}\)	\(47\)
risch	\(x^{4}-\frac {6 x^{4}}{\ln \relax (2)}+\frac {9 x^{4}}{\ln \relax (2)^{2}}-\frac {12 x^{3}}{\ln \relax (2)}+\frac {36 x^{3}}{\ln \relax (2)^{2}}+\frac {36 x^{2}}{\ln \relax (2)^{2}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*ln(2)^2+(-24*x^3-36*x^2)*ln(2)+36*x^3+108*x^2+72*x)/ln(2)^2,x,method=_RETURNVERBOSE)

[Out]

(x^2*ln(2)^2-6*x^2*ln(2)-12*x*ln(2)+9*x^2+36*x+36)*x^2/ln(2)^2

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maxima [A]  time = 0.41, size = 42, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \relax (2)^{2} + 9 \, x^{4} + 36 \, x^{3} + 36 \, x^{2} - 6 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \relax (2)}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(2)^2+(-24*x^3-36*x^2)*log(2)+36*x^3+108*x^2+72*x)/log(2)^2,x, algorithm="maxima")

[Out]

(x^4*log(2)^2 + 9*x^4 + 36*x^3 + 36*x^2 - 6*(x^4 + 2*x^3)*log(2))/log(2)^2

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mupad [B]  time = 0.08, size = 20, normalized size = 0.83 \begin {gather*} \frac {x^2\,{\left (3\,x-x\,\ln \relax (2)+6\right )}^2}{{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((72*x + 4*x^3*log(2)^2 - log(2)*(36*x^2 + 24*x^3) + 108*x^2 + 36*x^3)/log(2)^2,x)

[Out]

(x^2*(3*x - x*log(2) + 6)^2)/log(2)^2

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sympy [B]  time = 0.07, size = 44, normalized size = 1.83 \begin {gather*} \frac {x^{4} \left (- 6 \log {\relax (2 )} + \log {\relax (2 )}^{2} + 9\right )}{\log {\relax (2 )}^{2}} + \frac {x^{3} \left (36 - 12 \log {\relax (2 )}\right )}{\log {\relax (2 )}^{2}} + \frac {36 x^{2}}{\log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3*ln(2)**2+(-24*x**3-36*x**2)*ln(2)+36*x**3+108*x**2+72*x)/ln(2)**2,x)

[Out]

x**4*(-6*log(2) + log(2)**2 + 9)/log(2)**2 + x**3*(36 - 12*log(2))/log(2)**2 + 36*x**2/log(2)**2

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