3.49.71 $\int \frac{1}{4}(-8x-12x^2+e^4(-4x-3x^2)+e^4(-4-4x)\log (5))dx$

Optimal. Leaf size=26 $$2-x(x+x^2+\frac{1}{4}{e}^4(2+x)(x+2\log (5)))$$

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.73, number of steps used = 3, number of rules used = 1, integrand size = 37, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.027, Rules used = {12} $$\begin{array}{c}-\frac{1}{4}{e}^4{x}^3-x^3-\frac{e^4{x}^2}{2}-x^2-\frac{1}{2}{e}^4(x+1{)}^2\log (5)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-8*x - 12*x^2 + E^4*(-4*x - 3*x^2) + E^4*(-4 - 4*x)*Log[5])/4,x]

[Out]

-x^2 - (E^4*x^2)/2 - x^3 - (E^4*x^3)/4 - (E^4*(1 + x)^2*Log[5])/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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{4}\int (-8x-12x^2+e^4(-4x-3x^2)+e^4(-4-4x)\log (5))dx\\ & =-x^2-x^3-\frac{1}{2}{e}^4(1+x{)}^2\log (5)+\frac{1}{4}{e}^4\int (-4x-3x^2)dx\\ & =-x^2-\frac{e^4{x}^2}{2}-x^3-\frac{e^4{x}^3}{4}-\frac{1}{2}{e}^4(1+x{)}^2\log (5)\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.46 $$\begin{array}{c}-\frac{1}{4}(4+e^4)x^3-e^{4}x\log (5)-\frac{1}{2}{x}^2(2+e^4(1+\log (5)))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-8*x - 12*x^2 + E^4*(-4*x - 3*x^2) + E^4*(-4 - 4*x)*Log[5])/4,x]

[Out]

-1/4*((4 + E^4)*x^3) - E^4*x*Log[5] - (x^2*(2 + E^4*(1 + Log[5])))/2

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fricas [A]  time = 0.57, size = 37, normalized size = 1.42 $$\begin{array}{c}-x^3-\frac{1}{2}(x^2+2x)e^4\log (5)-x^2-\frac{1}{4}(x^3+2x^2)e^4\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x-4)*exp(4)*log(5)+1/4*(-3*x^2-4*x)*exp(4)-3*x^2-2*x,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.76, size = 37, normalized size = 1.42 $$\begin{array}{c}-x^3-\frac{1}{2}(x^2+2x)e^4\log (5)-x^2-\frac{1}{4}(x^3+2x^2)e^4\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x-4)*exp(4)*log(5)+1/4*(-3*x^2-4*x)*exp(4)-3*x^2-2*x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 35, normalized size = 1.35

method	result	size
norman	$(-\frac{{\mathrm{e}}^4}{4}-1)x^3+(-\frac{{\mathrm{e}}^4\ln (5)}{2}-\frac{{\mathrm{e}}^4}{2}-1)x^2-{\mathrm{e}}^4\ln (5)x$	$35$
gosper	$-\frac{x(2{\mathrm{e}}^4\ln (5)x+x^2{\mathrm{e}}^4+4{\mathrm{e}}^4\ln (5)+2x{\mathrm{e}}^4+4x^2+4x)}{4}$	$37$
default	$\frac{{\mathrm{e}}^4\ln (5)(-2x^2-4x)}{4}+\frac{{\mathrm{e}}^4(-x^3-2x^2)}{4}-x^3-x^2$	$42$
risch	$-\frac{{\mathrm{e}}^4\ln (5)x^2}{2}-{\mathrm{e}}^4\ln (5)x-\frac{x^3{\mathrm{e}}^4}{4}-\frac{x^2{\mathrm{e}}^4}{2}-x^3-x^2$	$42$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-4*x-4)*exp(4)*ln(5)+1/4*(-3*x^2-4*x)*exp(4)-3*x^2-2*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 37, normalized size = 1.42 $$\begin{array}{c}-x^3-\frac{1}{2}(x^2+2x)e^4\log (5)-x^2-\frac{1}{4}(x^3+2x^2)e^4\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x-4)*exp(4)*log(5)+1/4*(-3*x^2-4*x)*exp(4)-3*x^2-2*x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 36, normalized size = 1.38 $$\begin{array}{c}(-\frac{{\mathrm{e}}^4}{4}-1)x^3+(-\frac{{\mathrm{e}}^4}{2}-\frac{{\mathrm{e}}^4\ln (5)}{2}-1)x^2-{\mathrm{e}}^4\ln (5)x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 39, normalized size = 1.50 $$\begin{array}{c}{x}^3(-\frac{e^4}{4}-1)+x^2(-\frac{e^4\log (5)}{2}-\frac{e^4}{2}-1)-x{e}^4\log (5)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x-4)*exp(4)*ln(5)+1/4*(-3*x**2-4*x)*exp(4)-3*x**2-2*x,x)

[Out]

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

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