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

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

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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} $$\begin{array}{c}-x^3+x^2+6x-\frac{x}{\log (5)}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int (-1+(6+2x-3x^2)\log (5))dx}{\log (5)}\\ & =-\frac{x}{\log (5)}+\int (6+2x-3x^2)dx\\ & =6x+x^2-x^3-\frac{x}{\log (5)}\end{array}\end{array}$$

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

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 $$\begin{array}{c}-\frac{(x^3-x^2-6x)\log (5)+x}{\log (5)}\end{array}$$

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 $$\begin{array}{c}-\frac{(x^3-x^2-6x)\log (5)+x}{\log (5)}\end{array}$$

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	$-x^3+x^2+6x-\frac{x}{\ln (5)}$	$20$
norman	$x^2+\frac{(6\ln (5)-1)x}{\ln (5)}-x^3$	$22$
gosper	$-\frac{x(x^2\ln (5)-x\ln (5)-6\ln (5)+1)}{\ln (5)}$	$25$
default	$\frac{\ln (5)(-x^3+x^2+6x)-x}{\ln (5)}$	$25$

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 $$\begin{array}{c}-\frac{(x^3-x^2-6x)\log (5)+x}{\log (5)}\end{array}$$

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 $$\begin{array}{c}-x^3+x^2+\frac{(6\ln (5)-1)x}{\ln (5)}\end{array}$$

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 $$\begin{array}{c}-x^3+x^2+\frac{x(-1+6\log (5))}{\log (5)}\end{array}$$

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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