3.12.91 \(\int \frac {72 x+108 x^2+36 x^3+(-36 x^2-24 x^3) \log (2)+4 x^3 \log ^2(2)}{\log ^2(2)} \, dx\) [1191]

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

[Out]

(3*(2+x)/x/ln(2)-1)^2*x^4+5/3

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
time = 0.01, 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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Maple [A]
time = 0.07, size = 33, normalized size = 1.38

method	result	size
default	\(\frac {\left (\ln \left (2\right )-3\right )^{2} x^{4}+\frac {4 \left (-9 \ln \left (2\right )+27\right ) x^{3}}{3}+36 x^{2}}{\ln \left (2\right )^{2}}\)	\(33\)
gosper	\(\frac {\left (x^{2} \ln \left (2\right )^{2}-6 x^{2} \ln \left (2\right )-12 x \ln \left (2\right )+9 x^{2}+36 x +36\right ) x^{2}}{\ln \left (2\right )^{2}}\)	\(39\)
norman	\(\frac {\frac {\left (\ln \left (2\right )^{2}-6 \ln \left (2\right )+9\right ) x^{4}}{\ln \left (2\right )}+\frac {36 x^{2}}{\ln \left (2\right )}-\frac {12 \left (\ln \left (2\right )-3\right ) x^{3}}{\ln \left (2\right )}}{\ln \left (2\right )}\)	\(47\)
risch	\(x^{4}-\frac {6 x^{4}}{\ln \left (2\right )}+\frac {9 x^{4}}{\ln \left (2\right )^{2}}-\frac {12 x^{3}}{\ln \left (2\right )}+\frac {36 x^{3}}{\ln \left (2\right )^{2}}+\frac {36 x^{2}}{\ln \left (2\right )^{2}}\)	\(50\)

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]

1/ln(2)^2*((ln(2)-3)^2*x^4+4/3*(-9*ln(2)+27)*x^3+36*x^2)

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Maxima [A]
time = 0.25, size = 42, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \left (2\right )^{2} + 9 \, x^{4} + 36 \, x^{3} + 36 \, x^{2} - 6 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \left (2\right )}{\log \left (2\right )^{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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Fricas [A]
time = 0.34, size = 42, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \left (2\right )^{2} + 9 \, x^{4} + 36 \, x^{3} + 36 \, x^{2} - 6 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \left (2\right )}{\log \left (2\right )^{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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
time = 0.01, size = 44, normalized size = 1.83 \begin {gather*} \frac {x^{4} \left (- 6 \log {\left (2 \right )} + \log {\left (2 \right )}^{2} + 9\right )}{\log {\left (2 \right )}^{2}} + \frac {x^{3} \cdot \left (36 - 12 \log {\left (2 \right )}\right )}{\log {\left (2 \right )}^{2}} + \frac {36 x^{2}}{\log {\left (2 \right )}^{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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Giac [A]
time = 0.40, size = 42, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \left (2\right )^{2} + 9 \, x^{4} + 36 \, x^{3} + 36 \, x^{2} - 6 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \left (2\right )}{\log \left (2\right )^{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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Mupad [B]
time = 0.08, size = 20, normalized size = 0.83 \begin {gather*} \frac {x^2\,{\left (3\,x-x\,\ln \left (2\right )+6\right )}^2}{{\ln \left (2\right )}^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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