3.42.82 $\int \frac{e^{256+x}(342x+171x^2-81x^4+81x^5)}{361+342x^3+81x^6}dx$

Optimal. Leaf size=17 $$\frac{e^{256+x}}{\frac{19}{9x^2}+x}$$

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Rubi [A]  time = 0.08, antiderivative size = 27, normalized size of antiderivative = 1.59, number of steps used = 2, number of rules used = 2, integrand size = 39, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.051, Rules used = {28, 2288} $$\begin{array}{c}\frac{9e^{x+256}(9x^5+19x^2)}{{(9x^3+19)}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(256 + x)*(342*x + 171*x^2 - 81*x^4 + 81*x^5))/(361 + 342*x^3 + 81*x^6),x]

[Out]

(9*E^(256 + x)*(19*x^2 + 9*x^5))/(19 + 9*x^3)^2

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =81\int \frac{e^{256+x}(342x+171x^2-81x^4+81x^5)}{{(171+81x^3)}^2}dx\\ & =\frac{9e^{256+x}(19x^2+9x^5)}{{(19+9x^3)}^2}\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.12 $$\begin{array}{c}\frac{9e^{256+x}{x}^2}{19+9x^3}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(256 + x)*(342*x + 171*x^2 - 81*x^4 + 81*x^5))/(361 + 342*x^3 + 81*x^6),x]

[Out]

(9*E^(256 + x)*x^2)/(19 + 9*x^3)

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fricas [A]  time = 0.68, size = 18, normalized size = 1.06 $$\begin{array}{c}\frac{9x^2{e}^{(x+256)}}{9x^3+19}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x, algorithm="fricas")

[Out]

9*x^2*e^(x + 256)/(9*x^3 + 19)

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giac [A]  time = 0.18, size = 18, normalized size = 1.06 $$\begin{array}{c}\frac{9x^2{e}^{(x+256)}}{9x^3+19}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x, algorithm="giac")

[Out]

9*x^2*e^(x + 256)/(9*x^3 + 19)

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maple [A]  time = 0.08, size = 19, normalized size = 1.12

method	result	size
gosper	$\frac{9x^2{\mathrm{e}}^{256+x}}{9x^3+19}$	$19$
norman	$\frac{9x^2{\mathrm{e}}^{256+x}}{9x^3+19}$	$19$
risch	$\frac{9x^2{\mathrm{e}}^{256+x}}{9x^3+19}$	$19$
derivativedivides	$-\frac{29802774360576{\mathrm{e}}^{256+x}x}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}-\frac{9934258120192(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(\mathit{\text{\_R1}}-258){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{171}+\frac{581632495218{\mathrm{e}}^{256+x}(256+x)x}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}[phone]\frac{\mathrm{}(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{({\mathit{\text{\_R1}}}^2-257\mathit{\text{\_R1}}-256){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{57}-\frac{1513488365{\mathrm{e}}^{256+x}(4608{(256+x)}^2-301989907-1769472x)}{57(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}-\frac{1513488365(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(4608{\mathit{\text{\_R1}}}^2-1774080\mathit{\text{\_R1}}+150994925){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{1539}+\frac{1969152{\mathrm{e}}^{256+x}(1769472{(256+x)}^2-115964131584-754974739x)}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}+\frac{656384(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(1769472{\mathit{\text{\_R1}}}^2-756744211\mathit{\text{\_R1}}+77460396525){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{171}-\frac{3843{\mathrm{e}}^{256+x}(603979757{(256+x)}^2-39582424825856-270582949376x)}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}-\frac{427(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(603979757{\mathit{\text{\_R1}}}^2-271186929190\mathit{\text{\_R1}}+29764119616000){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{57}+\frac{{\mathrm{e}}^{256+x}(1739461536000{(256+x)}^2-113997379912334999-801543976648704x)}{513{(256+x)}^3-393984{(256+x)}^2+17213424699+100859904x}+\frac{(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(1739461535487{\mathit{\text{\_R1}}}^2-803283438578688\mathit{\text{\_R1}}+91465059401662825){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{1539}$	$520$
default	$-\frac{29802774360576{\mathrm{e}}^{256+x}x}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}-\frac{9934258120192(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(\mathit{\text{\_R1}}-258){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{171}+\frac{581632495218{\mathrm{e}}^{256+x}(256+x)x}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}[phone]\frac{\mathrm{}(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{({\mathit{\text{\_R1}}}^2-257\mathit{\text{\_R1}}-256){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{57}-\frac{1513488365{\mathrm{e}}^{256+x}(4608{(256+x)}^2-301989907-1769472x)}{57(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}-\frac{1513488365(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(4608{\mathit{\text{\_R1}}}^2-1774080\mathit{\text{\_R1}}+150994925){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{1539}+\frac{1969152{\mathrm{e}}^{256+x}(1769472{(256+x)}^2-115964131584-754974739x)}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}+\frac{656384(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(1769472{\mathit{\text{\_R1}}}^2-756744211\mathit{\text{\_R1}}+77460396525){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{171}-\frac{3843{\mathrm{e}}^{256+x}(603979757{(256+x)}^2-39582424825856-270582949376x)}{19(9{(256+x)}^3-6912{(256+x)}^2+301989907+1769472x)}-\frac{427(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(603979757{\mathit{\text{\_R1}}}^2-271186929190\mathit{\text{\_R1}}+29764119616000){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{57}+\frac{{\mathrm{e}}^{256+x}(1739461536000{(256+x)}^2-113997379912334999-801543976648704x)}{513{(256+x)}^3-393984{(256+x)}^2+17213424699+100859904x}+\frac{(\sum _{\mathit{\text{\_R1}}=\mathrm{RootOf}(9{\mathit{\text{\_Z}}}^3-6912{\mathit{\text{\_Z}}}^2+1769472\mathit{\text{\_Z}}-150994925)}\frac{(1739461535487{\mathit{\text{\_R1}}}^2-803283438578688\mathit{\text{\_R1}}+91465059401662825){\mathrm{e}}^{\mathit{\text{\_R1}}}\text{expIntegralEi}(1,-256-x+\mathit{\text{\_R1}})}{{\mathit{\text{\_R1}}}^2-512\mathit{\text{\_R1}}+65536})}{1539}$	$520$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x,method=_RETURNVERBOSE)

[Out]

9*x^2*exp(256+x)/(9*x^3+19)

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maxima [A]  time = 0.50, size = 18, normalized size = 1.06 $$\begin{array}{c}\frac{9x^2{e}^{(x+256)}}{9x^3+19}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x, algorithm="maxima")

[Out]

9*x^2*e^(x + 256)/(9*x^3 + 19)

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mupad [B]  time = 3.03, size = 18, normalized size = 1.06 $$\begin{array}{c}\frac{9x^2{\mathrm{e}}^{256}{\mathrm{e}}^x}{9x^3+19}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 256)*(342*x + 171*x^2 - 81*x^4 + 81*x^5))/(342*x^3 + 81*x^6 + 361),x)

[Out]

(9*x^2*exp(256)*exp(x))/(9*x^3 + 19)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.88 $$\begin{array}{c}\frac{9x^2{e}^{x+256}}{9x^3+19}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x**5-81*x**4+171*x**2+342*x)*exp(256+x)/(81*x**6+342*x**3+361),x)

[Out]

9*x**2*exp(x + 256)/(9*x**3 + 19)

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