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

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

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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 {gather*} \frac {9 e^{x+256} \left (9 x^5+19 x^2\right )}{\left (9 x^3+19\right )^2} \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=81 \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{\left (171+81 x^3\right )^2} \, dx\\ &=\frac {9 e^{256+x} \left (19 x^2+9 x^5\right )}{\left (19+9 x^3\right )^2}\\ \end {aligned} \end {gather*}

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

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 {gather*} \frac {9 \, x^{2} e^{\left (x + 256\right )}}{9 \, x^{3} + 19} \end {gather*}

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 {gather*} \frac {9 \, x^{2} e^{\left (x + 256\right )}}{9 \, x^{3} + 19} \end {gather*}

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	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
norman	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
risch	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
derivativedivides	\(-\frac {29802774360576 \,{\mathrm e}^{256+x} x}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}-\frac {9934258120192 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (\textit {\_R1} -258\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{171}+\frac {581632495218 \,{\mathrm e}^{256+x} \left (256+x \right ) x}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}+\frac {64625832802 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-257 \textit {\_R1} -256\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{57}-\frac {1513488365 \,{\mathrm e}^{256+x} \left (4608 \left (256+x \right )^{2}-301989907-1769472 x \right )}{57 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}-\frac {1513488365 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (4608 \textit {\_R1}^{2}-1774080 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{1539}+\frac {1969152 \,{\mathrm e}^{256+x} \left (1769472 \left (256+x \right )^{2}-115964131584-754974739 x \right )}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}+\frac {656384 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (1769472 \textit {\_R1}^{2}-756744211 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{171}-\frac {3843 \,{\mathrm e}^{256+x} \left (603979757 \left (256+x \right )^{2}-39582424825856-270582949376 x \right )}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}-\frac {427 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (603979757 \textit {\_R1}^{2}-271186929190 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{57}+\frac {{\mathrm e}^{256+x} \left (1739461536000 \left (256+x \right )^{2}-113997379912334999-801543976648704 x \right )}{513 \left (256+x \right )^{3}-393984 \left (256+x \right )^{2}[phone]+100859904 x}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (1739461535487 \textit {\_R1}^{2}-803283438578688 \textit {\_R1} +91465059401662825\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{1539}\)	\(520\)
default	\(-\frac {29802774360576 \,{\mathrm e}^{256+x} x}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}-\frac {9934258120192 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (\textit {\_R1} -258\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{171}+\frac {581632495218 \,{\mathrm e}^{256+x} \left (256+x \right ) x}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}+\frac {64625832802 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-257 \textit {\_R1} -256\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{57}-\frac {1513488365 \,{\mathrm e}^{256+x} \left (4608 \left (256+x \right )^{2}-301989907-1769472 x \right )}{57 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}-\frac {1513488365 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (4608 \textit {\_R1}^{2}-1774080 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{1539}+\frac {1969152 \,{\mathrm e}^{256+x} \left (1769472 \left (256+x \right )^{2}-115964131584-754974739 x \right )}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}+\frac {656384 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (1769472 \textit {\_R1}^{2}-756744211 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{171}-\frac {3843 \,{\mathrm e}^{256+x} \left (603979757 \left (256+x \right )^{2}-39582424825856-270582949376 x \right )}{19 \left (9 \left (256+x \right )^{3}-6912 \left (256+x \right )^{2}[phone]+1769472 x \right )}-\frac {427 \left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (603979757 \textit {\_R1}^{2}-271186929190 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{57}+\frac {{\mathrm e}^{256+x} \left (1739461536000 \left (256+x \right )^{2}-113997379912334999-801543976648704 x \right )}{513 \left (256+x \right )^{3}-393984 \left (256+x \right )^{2}[phone]+100859904 x}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (9 \textit {\_Z}^{3}-6912 \textit {\_Z}^{2}+1769472 \textit {\_Z} -150994925\right )}{\sum }\frac {\left (1739461535487 \textit {\_R1}^{2}-803283438578688 \textit {\_R1} +91465059401662825\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -256-x +\textit {\_R1} \right )}{\textit {\_R1}^{2}-512 \textit {\_R1} +65536}\right )}{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 {gather*} \frac {9 \, x^{2} e^{\left (x + 256\right )}}{9 \, x^{3} + 19} \end {gather*}

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 {gather*} \frac {9\,x^2\,{\mathrm {e}}^{256}\,{\mathrm {e}}^x}{9\,x^3+19} \end {gather*}

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 {gather*} \frac {9 x^{2} e^{x + 256}}{9 x^{3} + 19} \end {gather*}

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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