3.13.97 \(\int \frac {-8 x-6 x^2+(-5-12 x-6 x^2) \log ^2(3)}{2 \log ^2(3)} \, dx\)

Optimal. Leaf size=25 \[ \frac {3 x}{2}-(2+x) \left (2+x+x^2+\frac {x^2}{\log ^2(3)}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {12} \begin {gather*} -x^3-\frac {x^3}{\log ^2(3)}-3 x^2-\frac {2 x^2}{\log ^2(3)}-\frac {5 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8*x - 6*x^2 + (-5 - 12*x - 6*x^2)*Log[3]^2)/(2*Log[3]^2),x]

[Out]

(-5*x)/2 - 3*x^2 - x^3 - (2*x^2)/Log[3]^2 - x^3/Log[3]^2

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} &=\frac {\int \left (-8 x-6 x^2+\left (-5-12 x-6 x^2\right ) \log ^2(3)\right ) \, dx}{2 \log ^2(3)}\\ &=-\frac {2 x^2}{\log ^2(3)}-\frac {x^3}{\log ^2(3)}+\frac {1}{2} \int \left (-5-12 x-6 x^2\right ) \, dx\\ &=-\frac {5 x}{2}-3 x^2-x^3-\frac {2 x^2}{\log ^2(3)}-\frac {x^3}{\log ^2(3)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.36 \begin {gather*} -\frac {5 x}{2}-3 x^2-x^3-\frac {2 x^2}{\log ^2(3)}-\frac {x^3}{\log ^2(3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x - 6*x^2 + (-5 - 12*x - 6*x^2)*Log[3]^2)/(2*Log[3]^2),x]

[Out]

(-5*x)/2 - 3*x^2 - x^3 - (2*x^2)/Log[3]^2 - x^3/Log[3]^2

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fricas [A]  time = 0.87, size = 36, normalized size = 1.44 \begin {gather*} -\frac {2 \, x^{3} + {\left (2 \, x^{3} + 6 \, x^{2} + 5 \, x\right )} \log \relax (3)^{2} + 4 \, x^{2}}{2 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-6*x^2-12*x-5)*log(3)^2-6*x^2-8*x)/log(3)^2,x, algorithm="fricas")

[Out]

-1/2*(2*x^3 + (2*x^3 + 6*x^2 + 5*x)*log(3)^2 + 4*x^2)/log(3)^2

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giac [A]  time = 0.55, size = 36, normalized size = 1.44 \begin {gather*} -\frac {2 \, x^{3} + {\left (2 \, x^{3} + 6 \, x^{2} + 5 \, x\right )} \log \relax (3)^{2} + 4 \, x^{2}}{2 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-6*x^2-12*x-5)*log(3)^2-6*x^2-8*x)/log(3)^2,x, algorithm="giac")

[Out]

-1/2*(2*x^3 + (2*x^3 + 6*x^2 + 5*x)*log(3)^2 + 4*x^2)/log(3)^2

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maple [A]  time = 0.03, size = 33, normalized size = 1.32

method	result	size
risch	\(-x^{3}-3 x^{2}-\frac {5 x}{2}-\frac {x^{3}}{\ln \relax (3)^{2}}-\frac {2 x^{2}}{\ln \relax (3)^{2}}\)	\(33\)
default	\(\frac {\ln \relax (3)^{2} \left (-2 x^{3}-6 x^{2}-5 x \right )-2 x^{3}-4 x^{2}}{2 \ln \relax (3)^{2}}\)	\(37\)
gosper	\(-\frac {x \left (2 x^{2} \ln \relax (3)^{2}+6 x \ln \relax (3)^{2}+5 \ln \relax (3)^{2}+2 x^{2}+4 x \right )}{2 \ln \relax (3)^{2}}\)	\(39\)
norman	\(\frac {-\frac {5 x \ln \relax (3)}{2}-\frac {\left (\ln \relax (3)^{2}+1\right ) x^{3}}{\ln \relax (3)}-\frac {\left (3 \ln \relax (3)^{2}+2\right ) x^{2}}{\ln \relax (3)}}{\ln \relax (3)}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-6*x^2-12*x-5)*ln(3)^2-6*x^2-8*x)/ln(3)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.34, size = 36, normalized size = 1.44 \begin {gather*} -\frac {2 \, x^{3} + {\left (2 \, x^{3} + 6 \, x^{2} + 5 \, x\right )} \log \relax (3)^{2} + 4 \, x^{2}}{2 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-6*x^2-12*x-5)*log(3)^2-6*x^2-8*x)/log(3)^2,x, algorithm="maxima")

[Out]

-1/2*(2*x^3 + (2*x^3 + 6*x^2 + 5*x)*log(3)^2 + 4*x^2)/log(3)^2

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mupad [B]  time = 0.92, size = 38, normalized size = 1.52 \begin {gather*} -\frac {\left (6\,{\ln \relax (3)}^2+6\right )\,x^3}{6\,{\ln \relax (3)}^2}-\frac {\left (12\,{\ln \relax (3)}^2+8\right )\,x^2}{4\,{\ln \relax (3)}^2}-\frac {5\,x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + (log(3)^2*(12*x + 6*x^2 + 5))/2 + 3*x^2)/log(3)^2,x)

[Out]

- (5*x)/2 - (x^3*(6*log(3)^2 + 6))/(6*log(3)^2) - (x^2*(12*log(3)^2 + 8))/(4*log(3)^2)

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sympy [A]  time = 0.06, size = 39, normalized size = 1.56 \begin {gather*} \frac {x^{3} \left (- \log {\relax (3 )}^{2} - 1\right )}{\log {\relax (3 )}^{2}} + \frac {x^{2} \left (- 3 \log {\relax (3 )}^{2} - 2\right )}{\log {\relax (3 )}^{2}} - \frac {5 x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-6*x**2-12*x-5)*ln(3)**2-6*x**2-8*x)/ln(3)**2,x)

[Out]

x**3*(-log(3)**2 - 1)/log(3)**2 + x**2*(-3*log(3)**2 - 2)/log(3)**2 - 5*x/2

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