3.1.67 \(\int \frac {-78 x^3+36 x^4-3 x^5+e^{\frac {2 (5+3 x^2)}{3 x}} (-45 x+30 x^2-3 x^3)+e^{\frac {5+3 x^2}{3 x}} (-50-30 x-90 x^2+66 x^3-6 x^4)}{768 x^3-672 x^4+243 x^5-42 x^6+3 x^7+e^{\frac {2 (5+3 x^2)}{3 x}} (675 x-540 x^2+198 x^3-36 x^4+3 x^5)+e^{\frac {5+3 x^2}{3 x}} (1440 x^2-1206 x^3+438 x^4-78 x^5+6 x^6)} \, dx\)

Optimal. Leaf size=37 \[ \frac {x}{-x+x^2+\frac {10 x}{-5+x-\frac {x}{e^{\frac {5}{3 x}+x}+x}}} \]

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Rubi [F]  time = 13.00, 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 {-78 x^3+36 x^4-3 x^5+e^{\frac {2 \left (5+3 x^2\right )}{3 x}} \left (-45 x+30 x^2-3 x^3\right )+e^{\frac {5+3 x^2}{3 x}} \left (-50-30 x-90 x^2+66 x^3-6 x^4\right )}{768 x^3-672 x^4+243 x^5-42 x^6+3 x^7+e^{\frac {2 \left (5+3 x^2\right )}{3 x}} \left (675 x-540 x^2+198 x^3-36 x^4+3 x^5\right )+e^{\frac {5+3 x^2}{3 x}} \left (1440 x^2-1206 x^3+438 x^4-78 x^5+6 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-78*x^3 + 36*x^4 - 3*x^5 + E^((2*(5 + 3*x^2))/(3*x))*(-45*x + 30*x^2 - 3*x^3) + E^((5 + 3*x^2)/(3*x))*(-5
0 - 30*x - 90*x^2 + 66*x^3 - 6*x^4))/(768*x^3 - 672*x^4 + 243*x^5 - 42*x^6 + 3*x^7 + E^((2*(5 + 3*x^2))/(3*x))
*(675*x - 540*x^2 + 198*x^3 - 36*x^4 + 3*x^5) + E^((5 + 3*x^2)/(3*x))*(1440*x^2 - 1206*x^3 + 438*x^4 - 78*x^5
+ 6*x^6)),x]

[Out]

-((5 - x)/(15 - 6*x + x^2)) + (200*Defer[Int][(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))^(-2), x]
)/3 - ((1100*I)/3)*Sqrt[2/3]*Defer[Int][1/((6 + (2*I)*Sqrt[6] - 2*x)*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15
- 6*x + x^2))^2), x] + 20*Defer[Int][x/(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))^2, x] - 10*Def
er[Int][x^2/(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))^2, x] + (275*(2 - I*Sqrt[6])*Defer[Int][1/
((-6 - (2*I)*Sqrt[6] + 2*x)*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))^2), x])/3 - ((1100*I)/3)*S
qrt[2/3]*Defer[Int][1/((-6 + (2*I)*Sqrt[6] + 2*x)*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))^2),
x] + (275*(2 + I*Sqrt[6])*Defer[Int][1/((-6 + (2*I)*Sqrt[6] + 2*x)*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 -
6*x + x^2))^2), x])/3 - 600*Defer[Int][x/((15 - 6*x + x^2)^2*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x
+ x^2))^2), x] + ((35*I)/3)*Sqrt[2/3]*Defer[Int][1/((6 + (2*I)*Sqrt[6] - 2*x)*(x*(16 - 7*x + x^2) + E^(5/(3*x)
+ x)*(15 - 6*x + x^2))), x] - (10*Defer[Int][1/(x*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))), x
])/9 + (50*(2 - I*Sqrt[6])*Defer[Int][1/((-6 - (2*I)*Sqrt[6] + 2*x)*(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15
- 6*x + x^2))), x])/9 + ((35*I)/3)*Sqrt[2/3]*Defer[Int][1/((-6 + (2*I)*Sqrt[6] + 2*x)*(x*(16 - 7*x + x^2) + E^
(5/(3*x) + x)*(15 - 6*x + x^2))), x] + (50*(2 + I*Sqrt[6])*Defer[Int][1/((-6 + (2*I)*Sqrt[6] + 2*x)*(x*(16 - 7
*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))), x])/9 - 600*Defer[Int][1/((15 - 6*x + x^2)^2*(x*(16 - 7*x + x^
2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))), x] + 120*Defer[Int][x/((15 - 6*x + x^2)^2*(x*(16 - 7*x + x^2) + E^(5/
(3*x) + x)*(15 - 6*x + x^2))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 x^3 \left (26-12 x+x^2\right )-3 e^{\frac {10}{3 x}+2 x} x \left (15-10 x+x^2\right )-e^{\frac {5}{3 x}+x} \left (50+30 x+90 x^2-66 x^3+6 x^4\right )}{3 x \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {-3 x^3 \left (26-12 x+x^2\right )-3 e^{\frac {10}{3 x}+2 x} x \left (15-10 x+x^2\right )-e^{\frac {5}{3 x}+x} \left (50+30 x+90 x^2-66 x^3+6 x^4\right )}{x \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {3 \left (15-10 x+x^2\right )}{\left (15-6 x+x^2\right )^2}+\frac {10 \left (-75-15 x+22 x^2-9 x^3+3 x^4\right )}{x \left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}-\frac {10 \left (-1200+285 x+985 x^2-751 x^3+250 x^4-42 x^5+3 x^6\right )}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}\right ) \, dx\\ &=\frac {10}{3} \int \frac {-75-15 x+22 x^2-9 x^3+3 x^4}{x \left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx-\frac {10}{3} \int \frac {-1200+285 x+985 x^2-751 x^3+250 x^4-42 x^5+3 x^6}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx-\int \frac {15-10 x+x^2}{\left (15-6 x+x^2\right )^2} \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {\int 0 \, dx}{24}-\frac {10}{3} \int \frac {-1200+285 x+985 x^2-751 x^3+250 x^4-42 x^5+3 x^6}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {10}{3} \int \frac {-75-15 x+22 x^2-9 x^3+3 x^4}{x \left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {10}{3} \int \left (-\frac {20}{\left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}-\frac {6 x}{\left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}+\frac {3 x^2}{\left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}+\frac {180 x}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}-\frac {55 (-4+x)}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}\right ) \, dx+\frac {10}{3} \int \left (-\frac {1}{3 x \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}+\frac {36 (-5+x)}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}+\frac {21+10 x}{3 \left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}\right ) \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {10}{9} \int \frac {1}{x \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx+\frac {10}{9} \int \frac {21+10 x}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx-10 \int \frac {x^2}{\left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx+20 \int \frac {x}{\left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx+\frac {200}{3} \int \frac {1}{\left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx+120 \int \frac {-5+x}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx+\frac {550}{3} \int \frac {-4+x}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx-600 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {10}{9} \int \frac {1}{x \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx+\frac {10}{9} \int \frac {21+10 x}{\left (15-6 x+x^2\right ) \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx-10 \int \frac {x^2}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+20 \int \frac {x}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {200}{3} \int \frac {1}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+120 \int \frac {-5+x}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx+\frac {550}{3} \int \frac {-4+x}{\left (15-6 x+x^2\right ) \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx-600 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {10}{9} \int \frac {1}{x \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx+\frac {10}{9} \int \left (\frac {21}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}+\frac {10 x}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}\right ) \, dx-10 \int \frac {x^2}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+20 \int \frac {x}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {200}{3} \int \frac {1}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+120 \int \left (-\frac {5}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}+\frac {x}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )}\right ) \, dx+\frac {550}{3} \int \left (-\frac {4}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}+\frac {x}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2}\right ) \, dx-600 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {10}{9} \int \frac {1}{x \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx-10 \int \frac {x^2}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {100}{9} \int \frac {x}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx+20 \int \frac {x}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {70}{3} \int \frac {1}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx+\frac {200}{3} \int \frac {1}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+120 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx+\frac {550}{3} \int \frac {x}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx-600 \int \frac {1}{\left (15-6 x+x^2\right )^2 \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )} \, dx-600 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx-\frac {2200}{3} \int \frac {1}{\left (15-6 x+x^2\right ) \left (15 e^{\frac {5}{3 x}+x}+16 x-6 e^{\frac {5}{3 x}+x} x-7 x^2+e^{\frac {5}{3 x}+x} x^2+x^3\right )^2} \, dx\\ &=-\frac {5-x}{15-6 x+x^2}-\frac {10}{9} \int \frac {1}{x \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx-10 \int \frac {x^2}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {100}{9} \int \frac {x}{\left (15-6 x+x^2\right ) \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx+20 \int \frac {x}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+\frac {70}{3} \int \frac {1}{\left (15-6 x+x^2\right ) \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx+\frac {200}{3} \int \frac {1}{\left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx+120 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx+\frac {550}{3} \int \frac {x}{\left (15-6 x+x^2\right ) \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx-600 \int \frac {x}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx-600 \int \frac {1}{\left (15-6 x+x^2\right )^2 \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )} \, dx-\frac {2200}{3} \int \frac {1}{\left (15-6 x+x^2\right ) \left (x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 55, normalized size = 1.49 \begin {gather*} \frac {e^{\frac {5}{3 x}+x} (-5+x)+(-6+x) x}{x \left (16-7 x+x^2\right )+e^{\frac {5}{3 x}+x} \left (15-6 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-78*x^3 + 36*x^4 - 3*x^5 + E^((2*(5 + 3*x^2))/(3*x))*(-45*x + 30*x^2 - 3*x^3) + E^((5 + 3*x^2)/(3*x
))*(-50 - 30*x - 90*x^2 + 66*x^3 - 6*x^4))/(768*x^3 - 672*x^4 + 243*x^5 - 42*x^6 + 3*x^7 + E^((2*(5 + 3*x^2))/
(3*x))*(675*x - 540*x^2 + 198*x^3 - 36*x^4 + 3*x^5) + E^((5 + 3*x^2)/(3*x))*(1440*x^2 - 1206*x^3 + 438*x^4 - 7
8*x^5 + 6*x^6)),x]

[Out]

(E^(5/(3*x) + x)*(-5 + x) + (-6 + x)*x)/(x*(16 - 7*x + x^2) + E^(5/(3*x) + x)*(15 - 6*x + x^2))

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fricas [A]  time = 0.79, size = 61, normalized size = 1.65 \begin {gather*} \frac {x^{2} + {\left (x - 5\right )} e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )} - 6 \, x}{x^{3} - 7 \, x^{2} + {\left (x^{2} - 6 \, x + 15\right )} e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )} + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^3+30*x^2-45*x)*exp(1/3*(3*x^2+5)/x)^2+(-6*x^4+66*x^3-90*x^2-30*x-50)*exp(1/3*(3*x^2+5)/x)-3*x
^5+36*x^4-78*x^3)/((3*x^5-36*x^4+198*x^3-540*x^2+675*x)*exp(1/3*(3*x^2+5)/x)^2+(6*x^6-78*x^5+438*x^4-1206*x^3+
1440*x^2)*exp(1/3*(3*x^2+5)/x)+3*x^7-42*x^6+243*x^5-672*x^4+768*x^3),x, algorithm="fricas")

[Out]

(x^2 + (x - 5)*e^(1/3*(3*x^2 + 5)/x) - 6*x)/(x^3 - 7*x^2 + (x^2 - 6*x + 15)*e^(1/3*(3*x^2 + 5)/x) + 16*x)

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giac [B]  time = 0.73, size = 100, normalized size = 2.70 \begin {gather*} \frac {x^{2} + x e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )} - 6 \, x - 5 \, e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )}}{x^{3} + x^{2} e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )} - 7 \, x^{2} - 6 \, x e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )} + 16 \, x + 15 \, e^{\left (\frac {3 \, x^{2} + 5}{3 \, x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^3+30*x^2-45*x)*exp(1/3*(3*x^2+5)/x)^2+(-6*x^4+66*x^3-90*x^2-30*x-50)*exp(1/3*(3*x^2+5)/x)-3*x
^5+36*x^4-78*x^3)/((3*x^5-36*x^4+198*x^3-540*x^2+675*x)*exp(1/3*(3*x^2+5)/x)^2+(6*x^6-78*x^5+438*x^4-1206*x^3+
1440*x^2)*exp(1/3*(3*x^2+5)/x)+3*x^7-42*x^6+243*x^5-672*x^4+768*x^3),x, algorithm="giac")

[Out]

(x^2 + x*e^(1/3*(3*x^2 + 5)/x) - 6*x - 5*e^(1/3*(3*x^2 + 5)/x))/(x^3 + x^2*e^(1/3*(3*x^2 + 5)/x) - 7*x^2 - 6*x
*e^(1/3*(3*x^2 + 5)/x) + 16*x + 15*e^(1/3*(3*x^2 + 5)/x))

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maple [B]  time = 0.10, size = 91, normalized size = 2.46

method	result	size
risch	\(\frac {x -5}{x^{2}-6 x +15}-\frac {10 x}{\left (x^{2}-6 x +15\right ) \left ({\mathrm e}^{\frac {3 x^{2}+5}{3 x}} x^{2}+x^{3}-6 \,{\mathrm e}^{\frac {3 x^{2}+5}{3 x}} x -7 x^{2}+15 \,{\mathrm e}^{\frac {3 x^{2}+5}{3 x}}+16 x \right )}\)	\(91\)
norman	\(\frac {-6 x -5 \,{\mathrm e}^{\frac {3 x^{2}+5}{3 x}}+x^{2}+{\mathrm e}^{\frac {3 x^{2}+5}{3 x}} x}{{\mathrm e}^{\frac {3 x^{2}+5}{3 x}} x^{2}+x^{3}-6 \,{\mathrm e}^{\frac {3 x^{2}+5}{3 x}} x -7 x^{2}+15 \,{\mathrm e}^{\frac {3 x^{2}+5}{3 x}}+16 x}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^3+30*x^2-45*x)*exp(1/3*(3*x^2+5)/x)^2+(-6*x^4+66*x^3-90*x^2-30*x-50)*exp(1/3*(3*x^2+5)/x)-3*x^5+36*
x^4-78*x^3)/((3*x^5-36*x^4+198*x^3-540*x^2+675*x)*exp(1/3*(3*x^2+5)/x)^2+(6*x^6-78*x^5+438*x^4-1206*x^3+1440*x
^2)*exp(1/3*(3*x^2+5)/x)+3*x^7-42*x^6+243*x^5-672*x^4+768*x^3),x,method=_RETURNVERBOSE)

[Out]

(x-5)/(x^2-6*x+15)-10*x/(x^2-6*x+15)/(exp(1/3*(3*x^2+5)/x)*x^2+x^3-6*exp(1/3*(3*x^2+5)/x)*x-7*x^2+15*exp(1/3*(
3*x^2+5)/x)+16*x)

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maxima [A]  time = 0.76, size = 51, normalized size = 1.38 \begin {gather*} \frac {x^{2} + {\left (x - 5\right )} e^{\left (x + \frac {5}{3 \, x}\right )} - 6 \, x}{x^{3} - 7 \, x^{2} + {\left (x^{2} - 6 \, x + 15\right )} e^{\left (x + \frac {5}{3 \, x}\right )} + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^3+30*x^2-45*x)*exp(1/3*(3*x^2+5)/x)^2+(-6*x^4+66*x^3-90*x^2-30*x-50)*exp(1/3*(3*x^2+5)/x)-3*x
^5+36*x^4-78*x^3)/((3*x^5-36*x^4+198*x^3-540*x^2+675*x)*exp(1/3*(3*x^2+5)/x)^2+(6*x^6-78*x^5+438*x^4-1206*x^3+
1440*x^2)*exp(1/3*(3*x^2+5)/x)+3*x^7-42*x^6+243*x^5-672*x^4+768*x^3),x, algorithm="maxima")

[Out]

(x^2 + (x - 5)*e^(x + 5/3/x) - 6*x)/(x^3 - 7*x^2 + (x^2 - 6*x + 15)*e^(x + 5/3/x) + 16*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{\frac {x^2+\frac {5}{3}}{x}}\,\left (6\,x^4-66\,x^3+90\,x^2+30\,x+50\right )+{\mathrm {e}}^{\frac {2\,\left (x^2+\frac {5}{3}\right )}{x}}\,\left (3\,x^3-30\,x^2+45\,x\right )+78\,x^3-36\,x^4+3\,x^5}{{\mathrm {e}}^{\frac {2\,\left (x^2+\frac {5}{3}\right )}{x}}\,\left (3\,x^5-36\,x^4+198\,x^3-540\,x^2+675\,x\right )+{\mathrm {e}}^{\frac {x^2+\frac {5}{3}}{x}}\,\left (6\,x^6-78\,x^5+438\,x^4-1206\,x^3+1440\,x^2\right )+768\,x^3-672\,x^4+243\,x^5-42\,x^6+3\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x^2 + 5/3)/x)*(30*x + 90*x^2 - 66*x^3 + 6*x^4 + 50) + exp((2*(x^2 + 5/3))/x)*(45*x - 30*x^2 + 3*x^3
) + 78*x^3 - 36*x^4 + 3*x^5)/(exp((2*(x^2 + 5/3))/x)*(675*x - 540*x^2 + 198*x^3 - 36*x^4 + 3*x^5) + exp((x^2 +
5/3)/x)*(1440*x^2 - 1206*x^3 + 438*x^4 - 78*x^5 + 6*x^6) + 768*x^3 - 672*x^4 + 243*x^5 - 42*x^6 + 3*x^7),x)

[Out]

int(-(exp((x^2 + 5/3)/x)*(30*x + 90*x^2 - 66*x^3 + 6*x^4 + 50) + exp((2*(x^2 + 5/3))/x)*(45*x - 30*x^2 + 3*x^3
) + 78*x^3 - 36*x^4 + 3*x^5)/(exp((2*(x^2 + 5/3))/x)*(675*x - 540*x^2 + 198*x^3 - 36*x^4 + 3*x^5) + exp((x^2 +
5/3)/x)*(1440*x^2 - 1206*x^3 + 438*x^4 - 78*x^5 + 6*x^6) + 768*x^3 - 672*x^4 + 243*x^5 - 42*x^6 + 3*x^7), x)

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sympy [B]  time = 0.45, size = 66, normalized size = 1.78 \begin {gather*} - \frac {10 x}{x^{5} - 13 x^{4} + 73 x^{3} - 201 x^{2} + 240 x + \left (x^{4} - 12 x^{3} + 66 x^{2} - 180 x + 225\right ) e^{\frac {x^{2} + \frac {5}{3}}{x}}} - \frac {5 - x}{x^{2} - 6 x + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**3+30*x**2-45*x)*exp(1/3*(3*x**2+5)/x)**2+(-6*x**4+66*x**3-90*x**2-30*x-50)*exp(1/3*(3*x**2+5
)/x)-3*x**5+36*x**4-78*x**3)/((3*x**5-36*x**4+198*x**3-540*x**2+675*x)*exp(1/3*(3*x**2+5)/x)**2+(6*x**6-78*x**
5+438*x**4-1206*x**3+1440*x**2)*exp(1/3*(3*x**2+5)/x)+3*x**7-42*x**6+243*x**5-672*x**4+768*x**3),x)

[Out]

-10*x/(x**5 - 13*x**4 + 73*x**3 - 201*x**2 + 240*x + (x**4 - 12*x**3 + 66*x**2 - 180*x + 225)*exp((x**2 + 5/3)
/x)) - (5 - x)/(x**2 - 6*x + 15)

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