3.73.6 \(\int \frac {1}{3} (-6-640 x-336 x^2+16 x^3+15 x^4+e^8 (96 x+72 x^2+12 x^3)) \, dx\)

Optimal. Leaf size=25 \[ -2 x+16 \left (-\frac {20}{3}+e^8+x\right ) \left (x+\frac {x^2}{4}\right )^2 \]

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Rubi [B]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 3, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {12} \begin {gather*} x^5+e^8 x^4+\frac {4 x^4}{3}+8 e^8 x^3-\frac {112 x^3}{3}+16 e^8 x^2-\frac {320 x^2}{3}-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 640*x - 336*x^2 + 16*x^3 + 15*x^4 + E^8*(96*x + 72*x^2 + 12*x^3))/3,x]

[Out]

-2*x - (320*x^2)/3 + 16*E^8*x^2 - (112*x^3)/3 + 8*E^8*x^3 + (4*x^4)/3 + E^8*x^4 + x^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 {1}{3} \int \left (-6-640 x-336 x^2+16 x^3+15 x^4+e^8 \left (96 x+72 x^2+12 x^3\right )\right ) \, dx\\ &=-2 x-\frac {320 x^2}{3}-\frac {112 x^3}{3}+\frac {4 x^4}{3}+x^5+\frac {1}{3} e^8 \int \left (96 x+72 x^2+12 x^3\right ) \, dx\\ &=-2 x-\frac {320 x^2}{3}+16 e^8 x^2-\frac {112 x^3}{3}+8 e^8 x^3+\frac {4 x^4}{3}+e^8 x^4+x^5\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.96 \begin {gather*} -2 x+\frac {16}{3} \left (-20+3 e^8\right ) x^2+\frac {8}{3} \left (-14+3 e^8\right ) x^3+\frac {1}{3} \left (4+3 e^8\right ) x^4+x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 640*x - 336*x^2 + 16*x^3 + 15*x^4 + E^8*(96*x + 72*x^2 + 12*x^3))/3,x]

[Out]

-2*x + (16*(-20 + 3*E^8)*x^2)/3 + (8*(-14 + 3*E^8)*x^3)/3 + ((4 + 3*E^8)*x^4)/3 + x^5

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fricas [A]  time = 1.24, size = 39, normalized size = 1.56 \begin {gather*} x^{5} + \frac {4}{3} \, x^{4} - \frac {112}{3} \, x^{3} - \frac {320}{3} \, x^{2} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{8} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(12*x^3+72*x^2+96*x)*exp(8)+5*x^4+16/3*x^3-112*x^2-640/3*x-2,x, algorithm="fricas")

[Out]

x^5 + 4/3*x^4 - 112/3*x^3 - 320/3*x^2 + (x^4 + 8*x^3 + 16*x^2)*e^8 - 2*x

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giac [A]  time = 0.13, size = 39, normalized size = 1.56 \begin {gather*} x^{5} + \frac {4}{3} \, x^{4} - \frac {112}{3} \, x^{3} - \frac {320}{3} \, x^{2} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{8} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(12*x^3+72*x^2+96*x)*exp(8)+5*x^4+16/3*x^3-112*x^2-640/3*x-2,x, algorithm="giac")

[Out]

x^5 + 4/3*x^4 - 112/3*x^3 - 320/3*x^2 + (x^4 + 8*x^3 + 16*x^2)*e^8 - 2*x

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maple [A]  time = 0.02, size = 36, normalized size = 1.44

method	result	size
norman	\(x^{5}+\left ({\mathrm e}^{8}+\frac {4}{3}\right ) x^{4}+\left (8 \,{\mathrm e}^{8}-\frac {112}{3}\right ) x^{3}+\left (16 \,{\mathrm e}^{8}-\frac {320}{3}\right ) x^{2}-2 x\)	\(36\)
gosper	\(\frac {x \left (3 x^{3} {\mathrm e}^{8}+3 x^{4}+24 x^{2} {\mathrm e}^{8}+4 x^{3}+48 x \,{\mathrm e}^{8}-112 x^{2}-320 x -6\right )}{3}\)	\(43\)
default	\(\frac {{\mathrm e}^{8} \left (3 x^{4}+24 x^{3}+48 x^{2}\right )}{3}+x^{5}+\frac {4 x^{4}}{3}-\frac {112 x^{3}}{3}-\frac {320 x^{2}}{3}-2 x\)	\(43\)
risch	\(x^{4} {\mathrm e}^{8}+8 x^{3} {\mathrm e}^{8}+16 x^{2} {\mathrm e}^{8}+x^{5}+\frac {4 x^{4}}{3}-\frac {112 x^{3}}{3}-\frac {320 x^{2}}{3}-2 x\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(12*x^3+72*x^2+96*x)*exp(8)+5*x^4+16/3*x^3-112*x^2-640/3*x-2,x,method=_RETURNVERBOSE)

[Out]

x^5+(exp(8)+4/3)*x^4+(8*exp(8)-112/3)*x^3+(16*exp(8)-320/3)*x^2-2*x

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maxima [A]  time = 0.37, size = 39, normalized size = 1.56 \begin {gather*} x^{5} + \frac {4}{3} \, x^{4} - \frac {112}{3} \, x^{3} - \frac {320}{3} \, x^{2} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{8} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(12*x^3+72*x^2+96*x)*exp(8)+5*x^4+16/3*x^3-112*x^2-640/3*x-2,x, algorithm="maxima")

[Out]

x^5 + 4/3*x^4 - 112/3*x^3 - 320/3*x^2 + (x^4 + 8*x^3 + 16*x^2)*e^8 - 2*x

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mupad [B]  time = 0.03, size = 35, normalized size = 1.40 \begin {gather*} x^5+\left ({\mathrm {e}}^8+\frac {4}{3}\right )\,x^4+\left (8\,{\mathrm {e}}^8-\frac {112}{3}\right )\,x^3+\left (16\,{\mathrm {e}}^8-\frac {320}{3}\right )\,x^2-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8)*(96*x + 72*x^2 + 12*x^3))/3 - (640*x)/3 - 112*x^2 + (16*x^3)/3 + 5*x^4 - 2,x)

[Out]

x^3*(8*exp(8) - 112/3) - 2*x + x^2*(16*exp(8) - 320/3) + x^5 + x^4*(exp(8) + 4/3)

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sympy [B]  time = 0.09, size = 39, normalized size = 1.56 \begin {gather*} x^{5} + x^{4} \left (\frac {4}{3} + e^{8}\right ) + x^{3} \left (- \frac {112}{3} + 8 e^{8}\right ) + x^{2} \left (- \frac {320}{3} + 16 e^{8}\right ) - 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(12*x**3+72*x**2+96*x)*exp(8)+5*x**4+16/3*x**3-112*x**2-640/3*x-2,x)

[Out]

x**5 + x**4*(4/3 + exp(8)) + x**3*(-112/3 + 8*exp(8)) + x**2*(-320/3 + 16*exp(8)) - 2*x

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