3.78.39 \(\int \frac {-1+(6+2 x-3 x^2) \log (5)}{\log (5)} \, dx\)

Optimal. Leaf size=22 \[ -1+2 x-x^3+(2+x)^2-\frac {x}{\log (5)} \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {12} \begin {gather*} -x^3+x^2+6 x-\frac {x}{\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + (6 + 2*x - 3*x^2)*Log[5])/Log[5],x]

[Out]

6*x + x^2 - x^3 - x/Log[5]

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

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.86 \begin {gather*} 6 x+x^2-x^3-\frac {x}{\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + (6 + 2*x - 3*x^2)*Log[5])/Log[5],x]

[Out]

6*x + x^2 - x^3 - x/Log[5]

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fricas [A]  time = 0.63, size = 23, normalized size = 1.05 \begin {gather*} -\frac {{\left (x^{3} - x^{2} - 6 \, x\right )} \log \relax (5) + x}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 23, normalized size = 1.05 \begin {gather*} -\frac {{\left (x^{3} - x^{2} - 6 \, x\right )} \log \relax (5) + x}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 20, normalized size = 0.91

method	result	size
risch	\(-x^{3}+x^{2}+6 x -\frac {x}{\ln \relax (5)}\)	\(20\)
norman	\(x^{2}+\frac {\left (6 \ln \relax (5)-1\right ) x}{\ln \relax (5)}-x^{3}\)	\(22\)
gosper	\(-\frac {x \left (x^{2} \ln \relax (5)-x \ln \relax (5)-6 \ln \relax (5)+1\right )}{\ln \relax (5)}\)	\(25\)
default	\(\frac {\ln \relax (5) \left (-x^{3}+x^{2}+6 x \right )-x}{\ln \relax (5)}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^3+x^2+6*x-x/ln(5)

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maxima [A]  time = 0.39, size = 23, normalized size = 1.05 \begin {gather*} -\frac {{\left (x^{3} - x^{2} - 6 \, x\right )} \log \relax (5) + x}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.89, size = 21, normalized size = 0.95 \begin {gather*} -x^3+x^2+\frac {\left (6\,\ln \relax (5)-1\right )\,x}{\ln \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - x^3 + (x*(6*log(5) - 1))/log(5)

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sympy [A]  time = 0.06, size = 17, normalized size = 0.77 \begin {gather*} - x^{3} + x^{2} + \frac {x \left (-1 + 6 \log {\relax (5 )}\right )}{\log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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