3.29.35 \(\int \frac {27-18 x-24 x^2+12 x^3+8 x^4-2 x^5-x^6+e^{\frac {4 x+4 x^2}{-3+x+x^2}} (12 x^2+24 x^3)}{9 x^2-6 x^3-5 x^4+2 x^5+x^6} \, dx\)

Optimal. Leaf size=32 \[ -e^{\frac {4 x}{x-\frac {3}{1+x}}}+\frac {-3+(2-x) x}{x} \]

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Rubi [F]  time = 2.16, 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 {27-18 x-24 x^2+12 x^3+8 x^4-2 x^5-x^6+e^{\frac {4 x+4 x^2}{-3+x+x^2}} \left (12 x^2+24 x^3\right )}{9 x^2-6 x^3-5 x^4+2 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(27 - 18*x - 24*x^2 + 12*x^3 + 8*x^4 - 2*x^5 - x^6 + E^((4*x + 4*x^2)/(-3 + x + x^2))*(12*x^2 + 24*x^3))/(
9*x^2 - 6*x^3 - 5*x^4 + 2*x^5 + x^6),x]

[Out]

-60/(13*x) - (18*x)/13 + x^2/13 + (12*(6 - x))/(13*(3 - x - x^2)) + (8*(6 - x)*x)/(13*(3 - x - x^2)) - (2*(6 -
x)*x^2)/(13*(3 - x - x^2)) - ((6 - x)*x^3)/(13*(3 - x - x^2)) - (6*(7 + x))/(13*(3 - x - x^2)) + (9*(7 + x))/
(13*x*(3 - x - x^2)) - (24*(1 + 2*x))/(13*(3 - x - x^2)) - (216*ArcTanh[(1 + 2*x)/Sqrt[13]])/(13*Sqrt[13]) + (
(169 - 73*Sqrt[13])*Log[1 - Sqrt[13] + 2*x])/169 - ((169 - 19*Sqrt[13])*Log[1 - Sqrt[13] + 2*x])/169 + ((169 +
19*Sqrt[13])*Log[1 - Sqrt[13] + 2*x])/169 - ((169 + 73*Sqrt[13])*Log[1 - Sqrt[13] + 2*x])/169 - ((169 - 73*Sq
rt[13])*Log[1 + Sqrt[13] + 2*x])/169 + ((169 - 19*Sqrt[13])*Log[1 + Sqrt[13] + 2*x])/169 - ((169 + 19*Sqrt[13]
)*Log[1 + Sqrt[13] + 2*x])/169 + ((169 + 73*Sqrt[13])*Log[1 + Sqrt[13] + 2*x])/169 + (48*Defer[Int][E^((4*x*(1
+ x))/(-3 + x + x^2))/(-1 + Sqrt[13] - 2*x)^2, x])/13 - (48*(1 - Sqrt[13])*Defer[Int][E^((4*x*(1 + x))/(-3 +
x + x^2))/(-1 + Sqrt[13] - 2*x)^2, x])/13 + (48*Defer[Int][E^((4*x*(1 + x))/(-3 + x + x^2))/(1 + Sqrt[13] + 2*
x)^2, x])/13 - (48*(1 + Sqrt[13])*Defer[Int][E^((4*x*(1 + x))/(-3 + x + x^2))/(1 + Sqrt[13] + 2*x)^2, x])/13

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27-18 x-24 x^2+12 x^3+8 x^4-2 x^5-x^6+e^{\frac {4 x+4 x^2}{-3+x+x^2}} \left (12 x^2+24 x^3\right )}{x^2 \left (3-x-x^2\right )^2} \, dx\\ &=\int \left (-\frac {24}{\left (-3+x+x^2\right )^2}+\frac {27}{x^2 \left (-3+x+x^2\right )^2}-\frac {18}{x \left (-3+x+x^2\right )^2}+\frac {12 x}{\left (-3+x+x^2\right )^2}+\frac {8 x^2}{\left (-3+x+x^2\right )^2}-\frac {2 x^3}{\left (-3+x+x^2\right )^2}-\frac {x^4}{\left (-3+x+x^2\right )^2}+\frac {12 e^{\frac {4 x (1+x)}{-3+x+x^2}} (1+2 x)}{\left (-3+x+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^3}{\left (-3+x+x^2\right )^2} \, dx\right )+8 \int \frac {x^2}{\left (-3+x+x^2\right )^2} \, dx+12 \int \frac {x}{\left (-3+x+x^2\right )^2} \, dx+12 \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}} (1+2 x)}{\left (-3+x+x^2\right )^2} \, dx-18 \int \frac {1}{x \left (-3+x+x^2\right )^2} \, dx-24 \int \frac {1}{\left (-3+x+x^2\right )^2} \, dx+27 \int \frac {1}{x^2 \left (-3+x+x^2\right )^2} \, dx-\int \frac {x^4}{\left (-3+x+x^2\right )^2} \, dx\\ &=\frac {12 (6-x)}{13 \left (3-x-x^2\right )}+\frac {8 (6-x) x}{13 \left (3-x-x^2\right )}-\frac {2 (6-x) x^2}{13 \left (3-x-x^2\right )}-\frac {(6-x) x^3}{13 \left (3-x-x^2\right )}-\frac {6 (7+x)}{13 \left (3-x-x^2\right )}+\frac {9 (7+x)}{13 x \left (3-x-x^2\right )}-\frac {24 (1+2 x)}{13 \left (3-x-x^2\right )}+\frac {1}{13} \int \frac {x^2 (-18+2 x)}{-3+x+x^2} \, dx+\frac {2}{13} \int \frac {(-12+x) x}{-3+x+x^2} \, dx-\frac {6}{13} \int \frac {-13-x}{x \left (-3+x+x^2\right )} \, dx+\frac {9}{13} \int \frac {-20-2 x}{x^2 \left (-3+x+x^2\right )} \, dx+\frac {12}{13} \int \frac {1}{-3+x+x^2} \, dx+2 \left (\frac {48}{13} \int \frac {1}{-3+x+x^2} \, dx\right )+12 \int \left (\frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (-3+x+x^2\right )^2}+\frac {2 e^{\frac {4 x (1+x)}{-3+x+x^2}} x}{\left (-3+x+x^2\right )^2}\right ) \, dx\\ &=\frac {2 x}{13}+\frac {12 (6-x)}{13 \left (3-x-x^2\right )}+\frac {8 (6-x) x}{13 \left (3-x-x^2\right )}-\frac {2 (6-x) x^2}{13 \left (3-x-x^2\right )}-\frac {(6-x) x^3}{13 \left (3-x-x^2\right )}-\frac {6 (7+x)}{13 \left (3-x-x^2\right )}+\frac {9 (7+x)}{13 x \left (3-x-x^2\right )}-\frac {24 (1+2 x)}{13 \left (3-x-x^2\right )}+\frac {1}{13} \int \left (-20+2 x-\frac {2 (30-13 x)}{-3+x+x^2}\right ) \, dx+\frac {2}{13} \int \frac {3-13 x}{-3+x+x^2} \, dx-\frac {6}{13} \int \left (\frac {13}{3 x}+\frac {-16-13 x}{3 \left (-3+x+x^2\right )}\right ) \, dx+\frac {9}{13} \int \left (\frac {20}{3 x^2}+\frac {26}{9 x}-\frac {2 (43+13 x)}{9 \left (-3+x+x^2\right )}\right ) \, dx-\frac {24}{13} \operatorname {Subst}\left (\int \frac {1}{13-x^2} \, dx,x,1+2 x\right )-2 \left (\frac {96}{13} \operatorname {Subst}\left (\int \frac {1}{13-x^2} \, dx,x,1+2 x\right )\right )+12 \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (-3+x+x^2\right )^2} \, dx+24 \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}} x}{\left (-3+x+x^2\right )^2} \, dx\\ &=-\frac {60}{13 x}-\frac {18 x}{13}+\frac {x^2}{13}+\frac {12 (6-x)}{13 \left (3-x-x^2\right )}+\frac {8 (6-x) x}{13 \left (3-x-x^2\right )}-\frac {2 (6-x) x^2}{13 \left (3-x-x^2\right )}-\frac {(6-x) x^3}{13 \left (3-x-x^2\right )}-\frac {6 (7+x)}{13 \left (3-x-x^2\right )}+\frac {9 (7+x)}{13 x \left (3-x-x^2\right )}-\frac {24 (1+2 x)}{13 \left (3-x-x^2\right )}-\frac {216 \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {13}}\right )}{13 \sqrt {13}}-\frac {2}{13} \int \frac {-16-13 x}{-3+x+x^2} \, dx-\frac {2}{13} \int \frac {30-13 x}{-3+x+x^2} \, dx-\frac {2}{13} \int \frac {43+13 x}{-3+x+x^2} \, dx+12 \int \left (\frac {4 e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \left (-1+\sqrt {13}-2 x\right )^2}+\frac {4 e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \sqrt {13} \left (-1+\sqrt {13}-2 x\right )}+\frac {4 e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \left (1+\sqrt {13}+2 x\right )^2}+\frac {4 e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \sqrt {13} \left (1+\sqrt {13}+2 x\right )}\right ) \, dx+24 \int \left (\frac {2 \left (-1+\sqrt {13}\right ) e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \left (-1+\sqrt {13}-2 x\right )^2}-\frac {2 e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \sqrt {13} \left (-1+\sqrt {13}-2 x\right )}+\frac {2 \left (-1-\sqrt {13}\right ) e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \left (1+\sqrt {13}+2 x\right )^2}-\frac {2 e^{\frac {4 x (1+x)}{-3+x+x^2}}}{13 \sqrt {13} \left (1+\sqrt {13}+2 x\right )}\right ) \, dx+\frac {1}{169} \left (-169+19 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{169} \left (169+19 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx\\ &=-\frac {60}{13 x}-\frac {18 x}{13}+\frac {x^2}{13}+\frac {12 (6-x)}{13 \left (3-x-x^2\right )}+\frac {8 (6-x) x}{13 \left (3-x-x^2\right )}-\frac {2 (6-x) x^2}{13 \left (3-x-x^2\right )}-\frac {(6-x) x^3}{13 \left (3-x-x^2\right )}-\frac {6 (7+x)}{13 \left (3-x-x^2\right )}+\frac {9 (7+x)}{13 x \left (3-x-x^2\right )}-\frac {24 (1+2 x)}{13 \left (3-x-x^2\right )}-\frac {216 \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {13}}\right )}{13 \sqrt {13}}-\frac {1}{169} \left (169-19 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{169} \left (169+19 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {48}{13} \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (-1+\sqrt {13}-2 x\right )^2} \, dx+\frac {48}{13} \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (1+\sqrt {13}+2 x\right )^2} \, dx-\frac {1}{169} \left (169-73 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{13} \left (48 \left (1-\sqrt {13}\right )\right ) \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (-1+\sqrt {13}-2 x\right )^2} \, dx-\frac {1}{13} \left (48 \left (1+\sqrt {13}\right )\right ) \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (1+\sqrt {13}+2 x\right )^2} \, dx-\frac {1}{169} \left (-169+19 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{169} \left (169+19 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{169} \left (-169+73 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{169} \left (169+73 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{169} \left (169+73 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx\\ &=-\frac {60}{13 x}-\frac {18 x}{13}+\frac {x^2}{13}+\frac {12 (6-x)}{13 \left (3-x-x^2\right )}+\frac {8 (6-x) x}{13 \left (3-x-x^2\right )}-\frac {2 (6-x) x^2}{13 \left (3-x-x^2\right )}-\frac {(6-x) x^3}{13 \left (3-x-x^2\right )}-\frac {6 (7+x)}{13 \left (3-x-x^2\right )}+\frac {9 (7+x)}{13 x \left (3-x-x^2\right )}-\frac {24 (1+2 x)}{13 \left (3-x-x^2\right )}-\frac {216 \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {13}}\right )}{13 \sqrt {13}}+\frac {1}{169} \left (169-73 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{169} \left (169-19 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )+\frac {1}{169} \left (169+19 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{169} \left (169+73 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{169} \left (169-73 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {1}{169} \left (169-19 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {1}{169} \left (169+19 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {1}{169} \left (169+73 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {48}{13} \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (-1+\sqrt {13}-2 x\right )^2} \, dx+\frac {48}{13} \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (1+\sqrt {13}+2 x\right )^2} \, dx-\frac {1}{13} \left (48 \left (1-\sqrt {13}\right )\right ) \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (-1+\sqrt {13}-2 x\right )^2} \, dx-\frac {1}{13} \left (48 \left (1+\sqrt {13}\right )\right ) \int \frac {e^{\frac {4 x (1+x)}{-3+x+x^2}}}{\left (1+\sqrt {13}+2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 0.78 \begin {gather*} -e^{4+\frac {12}{-3+x+x^2}}-\frac {3}{x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(27 - 18*x - 24*x^2 + 12*x^3 + 8*x^4 - 2*x^5 - x^6 + E^((4*x + 4*x^2)/(-3 + x + x^2))*(12*x^2 + 24*x
^3))/(9*x^2 - 6*x^3 - 5*x^4 + 2*x^5 + x^6),x]

[Out]

-E^(4 + 12/(-3 + x + x^2)) - 3/x - x

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fricas [A]  time = 0.83, size = 28, normalized size = 0.88 \begin {gather*} -\frac {x^{2} + x e^{\left (\frac {4 \, {\left (x^{2} + x\right )}}{x^{2} + x - 3}\right )} + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^3+12*x^2)*exp((4*x^2+4*x)/(x^2+x-3))-x^6-2*x^5+8*x^4+12*x^3-24*x^2-18*x+27)/(x^6+2*x^5-5*x^4-
6*x^3+9*x^2),x, algorithm="fricas")

[Out]

-(x^2 + x*e^(4*(x^2 + x)/(x^2 + x - 3)) + 3)/x

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giac [A]  time = 0.27, size = 37, normalized size = 1.16 \begin {gather*} -x - \frac {3}{x} - e^{\left (\frac {4 \, x^{2}}{x^{2} + x - 3} + \frac {4 \, x}{x^{2} + x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^3+12*x^2)*exp((4*x^2+4*x)/(x^2+x-3))-x^6-2*x^5+8*x^4+12*x^3-24*x^2-18*x+27)/(x^6+2*x^5-5*x^4-
6*x^3+9*x^2),x, algorithm="giac")

[Out]

-x - 3/x - e^(4*x^2/(x^2 + x - 3) + 4*x/(x^2 + x - 3))

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maple [A]  time = 0.16, size = 27, normalized size = 0.84

method	result	size
risch	\(-x -\frac {3}{x}-{\mathrm e}^{\frac {4 \left (x +1\right ) x}{x^{2}+x -3}}\)	\(27\)
norman	\(\frac {9-6 x +x^{2}-x^{4}+3 x \,{\mathrm e}^{\frac {4 x^{2}+4 x}{x^{2}+x -3}}-x^{2} {\mathrm e}^{\frac {4 x^{2}+4 x}{x^{2}+x -3}}-x^{3} {\mathrm e}^{\frac {4 x^{2}+4 x}{x^{2}+x -3}}}{x \left (x^{2}+x -3\right )}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^3+12*x^2)*exp((4*x^2+4*x)/(x^2+x-3))-x^6-2*x^5+8*x^4+12*x^3-24*x^2-18*x+27)/(x^6+2*x^5-5*x^4-6*x^3+
9*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.99, size = 129, normalized size = 4.03 \begin {gather*} -x - \frac {3 \, {\left (20 \, x^{2} + 23 \, x - 39\right )}}{13 \, {\left (x^{3} + x^{2} - 3 \, x\right )}} + \frac {31 \, x - 30}{13 \, {\left (x^{2} + x - 3\right )}} - \frac {2 \, {\left (10 \, x - 21\right )}}{13 \, {\left (x^{2} + x - 3\right )}} - \frac {8 \, {\left (7 \, x - 3\right )}}{13 \, {\left (x^{2} + x - 3\right )}} + \frac {24 \, {\left (2 \, x + 1\right )}}{13 \, {\left (x^{2} + x - 3\right )}} + \frac {6 \, {\left (x + 7\right )}}{13 \, {\left (x^{2} + x - 3\right )}} + \frac {12 \, {\left (x - 6\right )}}{13 \, {\left (x^{2} + x - 3\right )}} - e^{\left (\frac {12}{x^{2} + x - 3} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^3+12*x^2)*exp((4*x^2+4*x)/(x^2+x-3))-x^6-2*x^5+8*x^4+12*x^3-24*x^2-18*x+27)/(x^6+2*x^5-5*x^4-
6*x^3+9*x^2),x, algorithm="maxima")

[Out]

-x - 3/13*(20*x^2 + 23*x - 39)/(x^3 + x^2 - 3*x) + 1/13*(31*x - 30)/(x^2 + x - 3) - 2/13*(10*x - 21)/(x^2 + x
- 3) - 8/13*(7*x - 3)/(x^2 + x - 3) + 24/13*(2*x + 1)/(x^2 + x - 3) + 6/13*(x + 7)/(x^2 + x - 3) + 12/13*(x -
6)/(x^2 + x - 3) - e^(12/(x^2 + x - 3) + 4)

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mupad [B]  time = 1.85, size = 37, normalized size = 1.16 \begin {gather*} -x-{\mathrm {e}}^{\frac {4\,x}{x^2+x-3}}\,{\mathrm {e}}^{\frac {4\,x^2}{x^2+x-3}}-\frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x - exp((4*x + 4*x^2)/(x + x^2 - 3))*(12*x^2 + 24*x^3) + 24*x^2 - 12*x^3 - 8*x^4 + 2*x^5 + x^6 - 27)/
(9*x^2 - 6*x^3 - 5*x^4 + 2*x^5 + x^6),x)

[Out]

- x - exp((4*x)/(x + x^2 - 3))*exp((4*x^2)/(x + x^2 - 3)) - 3/x

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sympy [A]  time = 0.22, size = 22, normalized size = 0.69 \begin {gather*} - x - e^{\frac {4 x^{2} + 4 x}{x^{2} + x - 3}} - \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**3+12*x**2)*exp((4*x**2+4*x)/(x**2+x-3))-x**6-2*x**5+8*x**4+12*x**3-24*x**2-18*x+27)/(x**6+2*
x**5-5*x**4-6*x**3+9*x**2),x)

[Out]

-x - exp((4*x**2 + 4*x)/(x**2 + x - 3)) - 3/x

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