3.35.33 \(\int \frac {1}{5} e^{\frac {1}{5} (-256 x^3-256 x^4-96 x^5-16 x^6-x^7)} (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} (256 x^3+256 x^4+96 x^5+16 x^6+x^7)} (50+20 x)) \, dx\)

Optimal. Leaf size=26 \[ -4+(5+x) \left (2 x+e^{-\frac {1}{5} x^3 (4+x)^4} x\right ) \]

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Rubi [F]  time = 17.79, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-256*x^3 - 256*x^4 - 96*x^5 - 16*x^6 - x^7)/5)*(25 + 10*x - 3840*x^3 - 5888*x^4 - 3424*x^5 - 960*x^6
- 131*x^7 - 7*x^8 + E^((256*x^3 + 256*x^4 + 96*x^5 + 16*x^6 + x^7)/5)*(50 + 20*x)))/5,x]

[Out]

(5 + 2*x)^2/2 + 5*Defer[Int][E^(-1/5*(x^3*(4 + x)^4)), x] + 2*Defer[Int][x/E^((x^3*(4 + x)^4)/5), x] - 768*Def
er[Int][x^3/E^((x^3*(4 + x)^4)/5), x] - (5888*Defer[Int][x^4/E^((x^3*(4 + x)^4)/5), x])/5 - (3424*Defer[Int][x
^5/E^((x^3*(4 + x)^4)/5), x])/5 - 192*Defer[Int][x^6/E^((x^3*(4 + x)^4)/5), x] - (131*Defer[Int][x^7/E^((x^3*(
4 + x)^4)/5), x])/5 - (7*Defer[Int][x^8/E^((x^3*(4 + x)^4)/5), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx\\ &=\frac {1}{5} \int e^{-\frac {1}{5} x^3 (4+x)^4} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+10 e^{\frac {1}{5} x^3 (4+x)^4} (5+2 x)\right ) \, dx\\ &=\frac {1}{5} \int \left (25 e^{-\frac {1}{5} x^3 (4+x)^4}+10 e^{-\frac {1}{5} x^3 (4+x)^4} x-3840 e^{-\frac {1}{5} x^3 (4+x)^4} x^3-5888 e^{-\frac {1}{5} x^3 (4+x)^4} x^4-3424 e^{-\frac {1}{5} x^3 (4+x)^4} x^5-960 e^{-\frac {1}{5} x^3 (4+x)^4} x^6-131 e^{-\frac {1}{5} x^3 (4+x)^4} x^7-7 e^{-\frac {1}{5} x^3 (4+x)^4} x^8+10 (5+2 x)\right ) \, dx\\ &=\frac {1}{2} (5+2 x)^2-\frac {7}{5} \int e^{-\frac {1}{5} x^3 (4+x)^4} x^8 \, dx+2 \int e^{-\frac {1}{5} x^3 (4+x)^4} x \, dx+5 \int e^{-\frac {1}{5} x^3 (4+x)^4} \, dx-\frac {131}{5} \int e^{-\frac {1}{5} x^3 (4+x)^4} x^7 \, dx-192 \int e^{-\frac {1}{5} x^3 (4+x)^4} x^6 \, dx-\frac {3424}{5} \int e^{-\frac {1}{5} x^3 (4+x)^4} x^5 \, dx-768 \int e^{-\frac {1}{5} x^3 (4+x)^4} x^3 \, dx-\frac {5888}{5} \int e^{-\frac {1}{5} x^3 (4+x)^4} x^4 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.06, size = 37, normalized size = 1.42 \begin {gather*} e^{-\frac {1}{5} x^3 (4+x)^4} \left (1+2 e^{\frac {1}{5} x^3 (4+x)^4}\right ) x (5+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-256*x^3 - 256*x^4 - 96*x^5 - 16*x^6 - x^7)/5)*(25 + 10*x - 3840*x^3 - 5888*x^4 - 3424*x^5 - 96
0*x^6 - 131*x^7 - 7*x^8 + E^((256*x^3 + 256*x^4 + 96*x^5 + 16*x^6 + x^7)/5)*(50 + 20*x)))/5,x]

[Out]

((1 + 2*E^((x^3*(4 + x)^4)/5))*x*(5 + x))/E^((x^3*(4 + x)^4)/5)

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fricas [B]  time = 1.04, size = 71, normalized size = 2.73 \begin {gather*} {\left (x^{2} + 2 \, {\left (x^{2} + 5 \, x\right )} e^{\left (\frac {1}{5} \, x^{7} + \frac {16}{5} \, x^{6} + \frac {96}{5} \, x^{5} + \frac {256}{5} \, x^{4} + \frac {256}{5} \, x^{3}\right )} + 5 \, x\right )} e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3)-7*x^8-131*x^7-960*x^6-3424*x^5-588
8*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x, algorithm="fricas")

[Out]

(x^2 + 2*(x^2 + 5*x)*e^(1/5*x^7 + 16/5*x^6 + 96/5*x^5 + 256/5*x^4 + 256/5*x^3) + 5*x)*e^(-1/5*x^7 - 16/5*x^6 -
96/5*x^5 - 256/5*x^4 - 256/5*x^3)

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giac [B]  time = 0.27, size = 70, normalized size = 2.69 \begin {gather*} x^{2} e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} + 2 \, x^{2} + 5 \, x e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} + 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3)-7*x^8-131*x^7-960*x^6-3424*x^5-588
8*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x, algorithm="giac")

[Out]

x^2*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5 - 256/5*x^4 - 256/5*x^3) + 2*x^2 + 5*x*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5
- 256/5*x^4 - 256/5*x^3) + 10*x

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maple [A]  time = 0.08, size = 32, normalized size = 1.23

method	result	size
risch	\(2 x^{2}+10 x +\frac {\left (5 x^{2}+25 x \right ) {\mathrm e}^{-\frac {\left (4+x \right )^{4} x^{3}}{5}}}{5}\)	\(32\)
norman	\(\left (x^{2}+5 x +2 x^{2} {\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}+10 \,{\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}} x \right ) {\mathrm e}^{-\frac {1}{5} x^{7}-\frac {16}{5} x^{6}-\frac {96}{5} x^{5}-\frac {256}{5} x^{4}-\frac {256}{5} x^{3}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3)-7*x^8-131*x^7-960*x^6-3424*x^5-5888*x^4-
3840*x^3+10*x+25)/exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x,method=_RETURNVERBOSE)

[Out]

2*x^2+10*x+1/5*(5*x^2+25*x)*exp(-1/5*(4+x)^4*x^3)

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maxima [A]  time = 0.61, size = 44, normalized size = 1.69 \begin {gather*} 2 \, x^{2} + {\left (x^{2} + 5 \, x\right )} e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} + 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3)-7*x^8-131*x^7-960*x^6-3424*x^5-588
8*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x, algorithm="maxima")

[Out]

2*x^2 + (x^2 + 5*x)*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5 - 256/5*x^4 - 256/5*x^3) + 10*x

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mupad [B]  time = 2.15, size = 34, normalized size = 1.31 \begin {gather*} x\,\left ({\mathrm {e}}^{-\frac {x^7}{5}-\frac {16\,x^6}{5}-\frac {96\,x^5}{5}-\frac {256\,x^4}{5}-\frac {256\,x^3}{5}}+2\right )\,\left (x+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- (256*x^3)/5 - (256*x^4)/5 - (96*x^5)/5 - (16*x^6)/5 - x^7/5)*(768*x^3 - (exp((256*x^3)/5 + (256*x^4
)/5 + (96*x^5)/5 + (16*x^6)/5 + x^7/5)*(20*x + 50))/5 - 2*x + (5888*x^4)/5 + (3424*x^5)/5 + 192*x^6 + (131*x^7
)/5 + (7*x^8)/5 - 5),x)

[Out]

x*(exp(- (256*x^3)/5 - (256*x^4)/5 - (96*x^5)/5 - (16*x^6)/5 - x^7/5) + 2)*(x + 5)

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sympy [B]  time = 0.21, size = 49, normalized size = 1.88 \begin {gather*} 2 x^{2} + 10 x + \left (x^{2} + 5 x\right ) e^{- \frac {x^{7}}{5} - \frac {16 x^{6}}{5} - \frac {96 x^{5}}{5} - \frac {256 x^{4}}{5} - \frac {256 x^{3}}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((20*x+50)*exp(1/5*x**7+16/5*x**6+96/5*x**5+256/5*x**4+256/5*x**3)-7*x**8-131*x**7-960*x**6-3424
*x**5-5888*x**4-3840*x**3+10*x+25)/exp(1/5*x**7+16/5*x**6+96/5*x**5+256/5*x**4+256/5*x**3),x)

[Out]

2*x**2 + 10*x + (x**2 + 5*x)*exp(-x**7/5 - 16*x**6/5 - 96*x**5/5 - 256*x**4/5 - 256*x**3/5)

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