3.45.63 \(\int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6)}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx\)

Optimal. Leaf size=18 \[ 4+e^x+x+\frac {x}{-1+x+(1+x)^4} \]

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Rubi [F]  time = 180.01, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(19 + 52*x + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + E^x*(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 +
x^6))/(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.44, size = 20, normalized size = 1.11 \begin {gather*} e^x+x+\frac {1}{5+6 x+4 x^2+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(19 + 52*x + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + E^x*(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*
x^5 + x^6))/(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6),x]

[Out]

E^x + x + (5 + 6*x + 4*x^2 + x^3)^(-1)

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fricas [B]  time = 0.53, size = 50, normalized size = 2.78 \begin {gather*} \frac {x^{4} + 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 6 \, x + 5\right )} e^{x} + 5 \, x + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+
28*x^4+58*x^3+76*x^2+60*x+25),x, algorithm="fricas")

[Out]

(x^4 + 4*x^3 + 6*x^2 + (x^3 + 4*x^2 + 6*x + 5)*e^x + 5*x + 1)/(x^3 + 4*x^2 + 6*x + 5)

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giac [B]  time = 0.22, size = 56, normalized size = 3.11 \begin {gather*} \frac {x^{4} + x^{3} e^{x} + 4 \, x^{3} + 4 \, x^{2} e^{x} + 6 \, x^{2} + 6 \, x e^{x} + 5 \, x + 5 \, e^{x} + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+
28*x^4+58*x^3+76*x^2+60*x+25),x, algorithm="giac")

[Out]

(x^4 + x^3*e^x + 4*x^3 + 4*x^2*e^x + 6*x^2 + 6*x*e^x + 5*x + 5*e^x + 1)/(x^3 + 4*x^2 + 6*x + 5)

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maple [A]  time = 0.12, size = 20, normalized size = 1.11

method	result	size
risch	\(x +\frac {1}{x^{3}+4 x^{2}+6 x +5}+{\mathrm e}^{x}\)	\(20\)
norman	\(\frac {x^{4}-19 x -10 x^{2}+{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{x}-19}{x^{3}+4 x^{2}+6 x +5}\)	\(52\)
default	\(x +{\mathrm e}^{x}-\frac {\frac {936}{83} x^{2}+\frac {1403}{83} x +\frac {1840}{83}}{x^{3}+4 x^{2}+6 x +5}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (272 \textit {\_R1}^{2}+799 \textit {\_R1} +620\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}+\frac {60 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (21 \textit {\_R1}^{2}+15 \textit {\_R1} +32\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}-\frac {28 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (161 \textit {\_R1}^{2}+198 \textit {\_R1} +190\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}-\frac {76 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (48 \textit {\_R1}^{2}+58 \textit {\_R1} +85\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}+\frac {40 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (57 \textit {\_R1}^{2}+100 \textit {\_R1} +75\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}+\frac {58 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (86 \textit {\_R1}^{2}+97 \textit {\_R1} +135\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}+\frac {168 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (-13 \textit {\_R} -40\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}-\frac {25 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (4 \textit {\_R1}^{2}-9 \textit {\_R1} +14\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, \textit {\_R1} -x \right )}{3 \textit {\_R1}^{2}+8 \textit {\_R1} +6}\right )}{83}+\frac {58 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (86 \textit {\_R} +105\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}+\frac {292 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (-12 \textit {\_R} -5\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (166 \textit {\_R}^{2}+151 \textit {\_R} +305\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}+\frac {38 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (-2 \textit {\_R} +13\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}+\frac {156 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (7 \textit {\_R} -4\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}+\frac {8 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+6 \textit {\_Z} +5\right )}{\sum }\frac {\left (83 \textit {\_R}^{2}+36 \textit {\_R} +430\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +6}\right )}{83}-\frac {76 \,{\mathrm e}^{x} \left (48 x^{2}+106 x +105\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}-\frac {25 \,{\mathrm e}^{x} \left (4 x^{2}-5 x -12\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}+\frac {-\frac {76}{83} x^{2}+\frac {95}{83} x +\frac {228}{83}}{x^{3}+4 x^{2}+6 x +5}+\frac {\frac {2944}{83} x^{2}+\frac {4288}{83} x +\frac {6440}{83}}{x^{3}+4 x^{2}+6 x +5}+\frac {\frac {1092}{83} x^{2}+\frac {1872}{83} x +\frac {1040}{83}}{x^{3}+4 x^{2}+6 x +5}+\frac {58 \,{\mathrm e}^{x} \left (86 x^{2}+183 x +240\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}+\frac {8 \,{\mathrm e}^{x} \left (368 x^{2}+536 x +805\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}-\frac {{\mathrm e}^{x} \left (936 x^{2}+1403 x +1840\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}-\frac {28 \,{\mathrm e}^{x} \left (161 x^{2}+276 x +430\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}+\frac {60 \,{\mathrm e}^{x} \left (21 x^{2}+36 x +20\right )}{83 \left (x^{3}+4 x^{2}+6 x +5\right )}+\frac {-\frac {3504}{83} x^{2}-\frac {7738}{83} x -\frac {7665}{83}}{x^{3}+4 x^{2}+6 x +5}+\frac {\frac {4988}{83} x^{2}+\frac {10614}{83} x +\frac {13920}{83}}{x^{3}+4 x^{2}+6 x +5}+\frac {-\frac {4508}{83} x^{2}-\frac {7728}{83} x -\frac {12040}{83}}{x^{3}+4 x^{2}+6 x +5}\)	\(1065\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4
+58*x^3+76*x^2+60*x+25),x,method=_RETURNVERBOSE)

[Out]

x+1/(x^3+4*x^2+6*x+5)+exp(x)

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maxima [B]  time = 0.36, size = 50, normalized size = 2.78 \begin {gather*} \frac {x^{4} + 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 6 \, x + 5\right )} e^{x} + 5 \, x + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+
28*x^4+58*x^3+76*x^2+60*x+25),x, algorithm="maxima")

[Out]

(x^4 + 4*x^3 + 6*x^2 + (x^3 + 4*x^2 + 6*x + 5)*e^x + 5*x + 1)/(x^3 + 4*x^2 + 6*x + 5)

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mupad [B]  time = 0.16, size = 19, normalized size = 1.06 \begin {gather*} x+{\mathrm {e}}^x+\frac {1}{x^3+4\,x^2+6\,x+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((52*x + exp(x)*(60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + 25) + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x
^6 + 19)/(60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + 25),x)

[Out]

x + exp(x) + 1/(6*x + 4*x^2 + x^3 + 5)

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sympy [A]  time = 0.17, size = 19, normalized size = 1.06 \begin {gather*} x + e^{x} + \frac {1}{x^{3} + 4 x^{2} + 6 x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**6+8*x**5+28*x**4+58*x**3+76*x**2+60*x+25)*exp(x)+x**6+8*x**5+28*x**4+58*x**3+73*x**2+52*x+19)/(
x**6+8*x**5+28*x**4+58*x**3+76*x**2+60*x+25),x)

[Out]

x + exp(x) + 1/(x**3 + 4*x**2 + 6*x + 5)

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