3.78.39 \(\int \frac {-1+(6+2 x-3 x^2) \log (5)}{\log (5)} \, dx\) [7739]

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

[Out]

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

________________________________________________________________________________________

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*}

________________________________________________________________________________________

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]

________________________________________________________________________________________

Maple [A]
time = 0.55, size = 28, normalized size = 1.27

method	result	size
risch	\(-x^{3}+x^{2}+6 x -\frac {x}{\ln \left (5\right )}\)	\(20\)
norman	\(x^{2}+\frac {\left (6 \ln \left (5\right )-1\right ) x}{\ln \left (5\right )}-x^{3}\)	\(22\)
gosper	\(-\frac {x \left (x^{2} \ln \left (5\right )-x \ln \left (5\right )-6 \ln \left (5\right )+1\right )}{\ln \left (5\right )}\)	\(25\)
default	\(\frac {-x^{3} \ln \left (5\right )+x^{2} \ln \left (5\right )+6 x \ln \left (5\right )-x}{\ln \left (5\right )}\)	\(28\)

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]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 23, normalized size = 1.05 \begin {gather*} -\frac {{\left (x^{3} - x^{2} - 6 \, x\right )} \log \left (5\right ) + x}{\log \left (5\right )} \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)

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 23, normalized size = 1.05 \begin {gather*} -\frac {{\left (x^{3} - x^{2} - 6 \, x\right )} \log \left (5\right ) + x}{\log \left (5\right )} \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)

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 17, normalized size = 0.77 \begin {gather*} - x^{3} + x^{2} + \frac {x \left (-1 + 6 \log {\left (5 \right )}\right )}{\log {\left (5 \right )}} \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)

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 23, normalized size = 1.05 \begin {gather*} -\frac {{\left (x^{3} - x^{2} - 6 \, x\right )} \log \left (5\right ) + x}{\log \left (5\right )} \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)

________________________________________________________________________________________

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

________________________________________________________________________________________