3.32.19 \(\int (2560 x+768 x^2-2560 x^3-640 x^4+480 x^5+112 x^6+e^8 (10 x+3 x^2)+e^4 (-320 x-96 x^2+160 x^3+40 x^4)) \, dx\)

Optimal. Leaf size=22 \[ x^2 (5+x) \left (e^4-4 \left (4-x^2\right )\right )^2 \]

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Rubi [B]  time = 0.03, antiderivative size = 78, normalized size of antiderivative = 3.55, number of steps used = 3, number of rules used = 0, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} 16 x^7+80 x^6+8 e^4 x^5-128 x^5+40 e^4 x^4-640 x^4+e^8 x^3-32 e^4 x^3+256 x^3+5 e^8 x^2-160 e^4 x^2+1280 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2560*x + 768*x^2 - 2560*x^3 - 640*x^4 + 480*x^5 + 112*x^6 + E^8*(10*x + 3*x^2) + E^4*(-320*x - 96*x^2 + 16
0*x^3 + 40*x^4),x]

[Out]

1280*x^2 - 160*E^4*x^2 + 5*E^8*x^2 + 256*x^3 - 32*E^4*x^3 + E^8*x^3 - 640*x^4 + 40*E^4*x^4 - 128*x^5 + 8*E^4*x
^5 + 80*x^6 + 16*x^7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=1280 x^2+256 x^3-640 x^4-128 x^5+80 x^6+16 x^7+e^4 \int \left (-320 x-96 x^2+160 x^3+40 x^4\right ) \, dx+e^8 \int \left (10 x+3 x^2\right ) \, dx\\ &=1280 x^2-160 e^4 x^2+5 e^8 x^2+256 x^3-32 e^4 x^3+e^8 x^3-640 x^4+40 e^4 x^4-128 x^5+8 e^4 x^5+80 x^6+16 x^7\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.91 \begin {gather*} x^2 (5+x) \left (e^4+4 \left (-4+x^2\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2560*x + 768*x^2 - 2560*x^3 - 640*x^4 + 480*x^5 + 112*x^6 + E^8*(10*x + 3*x^2) + E^4*(-320*x - 96*x^
2 + 160*x^3 + 40*x^4),x]

[Out]

x^2*(5 + x)*(E^4 + 4*(-4 + x^2))^2

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fricas [B]  time = 0.63, size = 66, normalized size = 3.00 \begin {gather*} 16 \, x^{7} + 80 \, x^{6} - 128 \, x^{5} - 640 \, x^{4} + 256 \, x^{3} + 1280 \, x^{2} + {\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 8 \, {\left (x^{5} + 5 \, x^{4} - 4 \, x^{3} - 20 \, x^{2}\right )} e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+10*x)*exp(2)^4+(40*x^4+160*x^3-96*x^2-320*x)*exp(2)^2+112*x^6+480*x^5-640*x^4-2560*x^3+768*x^
2+2560*x,x, algorithm="fricas")

[Out]

16*x^7 + 80*x^6 - 128*x^5 - 640*x^4 + 256*x^3 + 1280*x^2 + (x^3 + 5*x^2)*e^8 + 8*(x^5 + 5*x^4 - 4*x^3 - 20*x^2
)*e^4

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giac [B]  time = 0.22, size = 66, normalized size = 3.00 \begin {gather*} 16 \, x^{7} + 80 \, x^{6} - 128 \, x^{5} - 640 \, x^{4} + 256 \, x^{3} + 1280 \, x^{2} + {\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 8 \, {\left (x^{5} + 5 \, x^{4} - 4 \, x^{3} - 20 \, x^{2}\right )} e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+10*x)*exp(2)^4+(40*x^4+160*x^3-96*x^2-320*x)*exp(2)^2+112*x^6+480*x^5-640*x^4-2560*x^3+768*x^
2+2560*x,x, algorithm="giac")

[Out]

16*x^7 + 80*x^6 - 128*x^5 - 640*x^4 + 256*x^3 + 1280*x^2 + (x^3 + 5*x^2)*e^8 + 8*(x^5 + 5*x^4 - 4*x^3 - 20*x^2
)*e^4

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maple [B]  time = 0.04, size = 70, normalized size = 3.18

method	result	size
norman	\(16 x^{7}+80 x^{6}+\left (8 \,{\mathrm e}^{4}-128\right ) x^{5}+\left (40 \,{\mathrm e}^{4}-640\right ) x^{4}+\left ({\mathrm e}^{8}-32 \,{\mathrm e}^{4}+256\right ) x^{3}+\left (5 \,{\mathrm e}^{8}-160 \,{\mathrm e}^{4}+1280\right ) x^{2}\)	\(70\)
default	\({\mathrm e}^{8} \left (x^{3}+5 x^{2}\right )+{\mathrm e}^{4} \left (8 x^{5}+40 x^{4}-32 x^{3}-160 x^{2}\right )+16 x^{7}+80 x^{6}-128 x^{5}-640 x^{4}+256 x^{3}+1280 x^{2}\)	\(72\)
gosper	\(x^{2} \left (x \,{\mathrm e}^{8}+8 x^{3} {\mathrm e}^{4}+16 x^{5}+5 \,{\mathrm e}^{8}+40 x^{2} {\mathrm e}^{4}+80 x^{4}-32 x \,{\mathrm e}^{4}-128 x^{3}-160 \,{\mathrm e}^{4}-640 x^{2}+256 x +1280\right )\)	\(73\)
risch	\({\mathrm e}^{8} x^{3}+5 x^{2} {\mathrm e}^{8}+8 x^{5} {\mathrm e}^{4}+40 x^{4} {\mathrm e}^{4}-32 x^{3} {\mathrm e}^{4}-160 x^{2} {\mathrm e}^{4}+16 x^{7}+80 x^{6}-128 x^{5}-640 x^{4}+256 x^{3}+1280 x^{2}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+10*x)*exp(2)^4+(40*x^4+160*x^3-96*x^2-320*x)*exp(2)^2+112*x^6+480*x^5-640*x^4-2560*x^3+768*x^2+2560
*x,x,method=_RETURNVERBOSE)

[Out]

16*x^7+80*x^6+(8*exp(2)^2-128)*x^5+(40*exp(2)^2-640)*x^4+(exp(2)^4-32*exp(2)^2+256)*x^3+(5*exp(2)^4-160*exp(2)
^2+1280)*x^2

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maxima [B]  time = 0.49, size = 66, normalized size = 3.00 \begin {gather*} 16 \, x^{7} + 80 \, x^{6} - 128 \, x^{5} - 640 \, x^{4} + 256 \, x^{3} + 1280 \, x^{2} + {\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 8 \, {\left (x^{5} + 5 \, x^{4} - 4 \, x^{3} - 20 \, x^{2}\right )} e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+10*x)*exp(2)^4+(40*x^4+160*x^3-96*x^2-320*x)*exp(2)^2+112*x^6+480*x^5-640*x^4-2560*x^3+768*x^
2+2560*x,x, algorithm="maxima")

[Out]

16*x^7 + 80*x^6 - 128*x^5 - 640*x^4 + 256*x^3 + 1280*x^2 + (x^3 + 5*x^2)*e^8 + 8*(x^5 + 5*x^4 - 4*x^3 - 20*x^2
)*e^4

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mupad [B]  time = 1.76, size = 57, normalized size = 2.59 \begin {gather*} 16\,x^7+80\,x^6+\left (8\,{\mathrm {e}}^4-128\right )\,x^5+\left (40\,{\mathrm {e}}^4-640\right )\,x^4+\left ({\mathrm {e}}^8-32\,{\mathrm {e}}^4+256\right )\,x^3+\left (5\,{\mathrm {e}}^8-160\,{\mathrm {e}}^4+1280\right )\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2560*x + exp(8)*(10*x + 3*x^2) - exp(4)*(320*x + 96*x^2 - 160*x^3 - 40*x^4) + 768*x^2 - 2560*x^3 - 640*x^4
+ 480*x^5 + 112*x^6,x)

[Out]

x^2*(5*exp(8) - 160*exp(4) + 1280) + x^5*(8*exp(4) - 128) + x^4*(40*exp(4) - 640) + x^3*(exp(8) - 32*exp(4) +
256) + 80*x^6 + 16*x^7

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sympy [B]  time = 0.06, size = 58, normalized size = 2.64 \begin {gather*} 16 x^{7} + 80 x^{6} + x^{5} \left (-128 + 8 e^{4}\right ) + x^{4} \left (-640 + 40 e^{4}\right ) + x^{3} \left (- 32 e^{4} + 256 + e^{8}\right ) + x^{2} \left (- 160 e^{4} + 1280 + 5 e^{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+10*x)*exp(2)**4+(40*x**4+160*x**3-96*x**2-320*x)*exp(2)**2+112*x**6+480*x**5-640*x**4-2560*x
**3+768*x**2+2560*x,x)

[Out]

16*x**7 + 80*x**6 + x**5*(-128 + 8*exp(4)) + x**4*(-640 + 40*exp(4)) + x**3*(-32*exp(4) + 256 + exp(8)) + x**2
*(-160*exp(4) + 1280 + 5*exp(8))

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