\(\int \frac {e^{x/3} (-18 x^3-36 x^5+16 x^6+x^7)+e^{x/3} (54+108 x^2-42 x^3-6 x^4) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx\) [1155]

Optimal result

Integrand size = 88, antiderivative size = 27 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {1}{5} e^{x/3} \left (-\frac {x^3}{3}+\log (2-x)\right )^2 \]

[Out]

1/5*exp(1/3*x)*(ln(2-x)-1/3*x^3)^2

Rubi [F]

\[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx \]

[In]

Int[(E^(x/3)*(-18*x^3 - 36*x^5 + 16*x^6 + x^7) + E^(x/3)*(54 + 108*x^2 - 42*x^3 - 6*x^4)*Log[2 - x] + E^(x/3)*
(-18 + 9*x)*Log[2 - x]^2)/(-270 + 135*x),x]

[Out]

(E^(x/3)*x^6)/45 - (2*E^(x/3)*x^3*Log[2 - x])/15 + (2*E^(2/3)*ExpIntegralEi[(-2 + x)/3]*Log[2 - x])/5 - (2*E^(
2/3)*Defer[Int][ExpIntegralEi[-2/3 + x/3]/(-2 + x), x])/5 + Defer[Int][E^(x/3)*Log[2 - x]^2, x]/15

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^{x/3} x^3}{15 (-2+x)}-\frac {4 e^{x/3} x^5}{15 (-2+x)}+\frac {16 e^{x/3} x^6}{135 (-2+x)}+\frac {e^{x/3} x^7}{135 (-2+x)}+\frac {2 e^{x/3} \log (2-x)}{5 (-2+x)}+\frac {4 e^{x/3} x^2 \log (2-x)}{5 (-2+x)}-\frac {14 e^{x/3} x^3 \log (2-x)}{45 (-2+x)}-\frac {2 e^{x/3} x^4 \log (2-x)}{45 (-2+x)}-\frac {2 e^{x/3} \log ^2(2-x)}{15 (-2+x)}+\frac {e^{x/3} x \log ^2(2-x)}{15 (-2+x)}\right ) \, dx \\ & = \frac {1}{135} \int \frac {e^{x/3} x^7}{-2+x} \, dx-\frac {2}{45} \int \frac {e^{x/3} x^4 \log (2-x)}{-2+x} \, dx+\frac {1}{15} \int \frac {e^{x/3} x \log ^2(2-x)}{-2+x} \, dx+\frac {16}{135} \int \frac {e^{x/3} x^6}{-2+x} \, dx-\frac {2}{15} \int \frac {e^{x/3} x^3}{-2+x} \, dx-\frac {2}{15} \int \frac {e^{x/3} \log ^2(2-x)}{-2+x} \, dx-\frac {4}{15} \int \frac {e^{x/3} x^5}{-2+x} \, dx-\frac {14}{45} \int \frac {e^{x/3} x^3 \log (2-x)}{-2+x} \, dx+\frac {2}{5} \int \frac {e^{x/3} \log (2-x)}{-2+x} \, dx+\frac {4}{5} \int \frac {e^{x/3} x^2 \log (2-x)}{-2+x} \, dx \\ & = -\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{135} \int \left (64 e^{x/3}+\frac {128 e^{x/3}}{-2+x}+32 e^{x/3} x+16 e^{x/3} x^2+8 e^{x/3} x^3+4 e^{x/3} x^4+2 e^{x/3} x^5+e^{x/3} x^6\right ) \, dx+\frac {2}{45} \int \frac {-3 e^{x/3} \left (-130+46 x-7 x^2+x^3\right )-16 e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right )}{2-x} \, dx+\frac {1}{15} \int \left (e^{x/3} \log ^2(2-x)+\frac {2 e^{x/3} \log ^2(2-x)}{-2+x}\right ) \, dx+\frac {16}{135} \int \left (32 e^{x/3}+\frac {64 e^{x/3}}{-2+x}+16 e^{x/3} x+8 e^{x/3} x^2+4 e^{x/3} x^3+2 e^{x/3} x^4+e^{x/3} x^5\right ) \, dx-\frac {2}{15} \int \left (4 e^{x/3}+\frac {8 e^{x/3}}{-2+x}+2 e^{x/3} x+e^{x/3} x^2\right ) \, dx-\frac {2}{15} \int \frac {e^{x/3} \log ^2(2-x)}{-2+x} \, dx-\frac {4}{15} \int \left (16 e^{x/3}+\frac {32 e^{x/3}}{-2+x}+8 e^{x/3} x+4 e^{x/3} x^2+2 e^{x/3} x^3+e^{x/3} x^4\right ) \, dx+\frac {14}{45} \int \frac {-3 e^{x/3} \left (16-4 x+x^2\right )-8 e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right )}{2-x} \, dx-\frac {2}{5} \int \frac {e^{2/3} \operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {4}{5} \int \frac {-3 e^{x/3} (-1+x)-4 e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right )}{2-x} \, dx \\ & = -\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{135} \int e^{x/3} x^6 \, dx+\frac {2}{135} \int e^{x/3} x^5 \, dx+\frac {4}{135} \int e^{x/3} x^4 \, dx+\frac {2}{45} \int \left (\frac {3 e^{x/3} \left (-130+46 x-7 x^2+x^3\right )}{-2+x}+\frac {16 e^{2/3} \operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x}\right ) \, dx+\frac {8}{135} \int e^{x/3} x^3 \, dx+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {16}{135} \int e^{x/3} x^2 \, dx+\frac {16}{135} \int e^{x/3} x^5 \, dx-\frac {2}{15} \int e^{x/3} x^2 \, dx+\frac {32}{135} \int e^{x/3} x \, dx+\frac {32}{135} \int e^{x/3} x^4 \, dx-\frac {4}{15} \int e^{x/3} x \, dx-\frac {4}{15} \int e^{x/3} x^4 \, dx+\frac {14}{45} \int \left (\frac {3 e^{x/3} \left (16-4 x+x^2\right )}{-2+x}+\frac {8 e^{2/3} \operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x}\right ) \, dx+\frac {64}{135} \int e^{x/3} \, dx+\frac {64}{135} \int e^{x/3} x^3 \, dx-\frac {8}{15} \int e^{x/3} \, dx-\frac {8}{15} \int e^{x/3} x^3 \, dx-\frac {4}{5} \int \left (\frac {3 e^{x/3} (-1+x)}{-2+x}+\frac {4 e^{2/3} \operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x}\right ) \, dx+\frac {128}{135} \int \frac {e^{x/3}}{-2+x} \, dx+\frac {128}{135} \int e^{x/3} x^2 \, dx-\frac {16}{15} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {16}{15} \int e^{x/3} x^2 \, dx+\frac {256}{135} \int e^{x/3} x \, dx-\frac {32}{15} \int e^{x/3} x \, dx+\frac {512}{135} \int e^{x/3} \, dx-\frac {64}{15} \int e^{x/3} \, dx+\frac {1024}{135} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {128}{15} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = -\frac {8 e^{x/3}}{5}-\frac {4}{5} e^{x/3} x-\frac {2}{5} e^{x/3} x^2+\frac {2}{5} e^{x/3} x^5+\frac {1}{45} e^{x/3} x^6-\frac {16}{15} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right )-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx-\frac {2}{15} \int e^{x/3} x^5 \, dx+\frac {2}{15} \int \frac {e^{x/3} \left (-130+46 x-7 x^2+x^3\right )}{-2+x} \, dx-\frac {2}{9} \int e^{x/3} x^4 \, dx-\frac {16}{45} \int e^{x/3} x^3 \, dx-\frac {8}{15} \int e^{x/3} x^2 \, dx-\frac {32}{45} \int e^{x/3} \, dx-\frac {32}{45} \int e^{x/3} x \, dx+\frac {4}{5} \int e^{x/3} \, dx+\frac {4}{5} \int e^{x/3} x \, dx+\frac {14}{15} \int \frac {e^{x/3} \left (16-4 x+x^2\right )}{-2+x} \, dx-\frac {16}{9} \int e^{x/3} x^4 \, dx-\frac {12}{5} \int \frac {e^{x/3} (-1+x)}{-2+x} \, dx-\frac {128}{45} \int e^{x/3} x^3 \, dx+\frac {16}{5} \int e^{x/3} x^3 \, dx-\frac {64}{15} \int e^{x/3} x^2 \, dx+\frac {24}{5} \int e^{x/3} x^2 \, dx-\frac {256}{45} \int e^{x/3} \, dx-\frac {256}{45} \int e^{x/3} x \, dx+\frac {32}{5} \int e^{x/3} \, dx+\frac {32}{5} \int e^{x/3} x \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = \frac {4 e^{x/3}}{5}+\frac {8}{5} e^{x/3} x-\frac {2}{5} e^{x/3} x^2-6 e^{x/3} x^4+\frac {1}{45} e^{x/3} x^6-\frac {16}{15} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right )-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {2}{15} \int \left (36 e^{x/3}-\frac {58 e^{x/3}}{-2+x}-5 e^{x/3} x+e^{x/3} x^2\right ) \, dx+\frac {14}{15} \int \left (-2 e^{x/3}+\frac {12 e^{x/3}}{-2+x}+e^{x/3} x\right ) \, dx+2 \int e^{x/3} x^4 \, dx+\frac {32}{15} \int e^{x/3} \, dx-\frac {12}{5} \int e^{x/3} \, dx-\frac {12}{5} \int \left (e^{x/3}+\frac {e^{x/3}}{-2+x}\right ) \, dx+\frac {8}{3} \int e^{x/3} x^3 \, dx+\frac {16}{5} \int e^{x/3} x \, dx+\frac {16}{5} \int e^{x/3} x^2 \, dx+\frac {256}{15} \int e^{x/3} \, dx-\frac {96}{5} \int e^{x/3} \, dx+\frac {64}{3} \int e^{x/3} x^3 \, dx+\frac {128}{5} \int e^{x/3} x \, dx+\frac {128}{5} \int e^{x/3} x^2 \, dx-\frac {144}{5} \int e^{x/3} x \, dx-\frac {144}{5} \int e^{x/3} x^2 \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = -\frac {32 e^{x/3}}{5}+\frac {8}{5} e^{x/3} x-\frac {2}{5} e^{x/3} x^2+72 e^{x/3} x^3+\frac {1}{45} e^{x/3} x^6-\frac {16}{15} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right )-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {2}{15} \int e^{x/3} x^2 \, dx-\frac {2}{3} \int e^{x/3} x \, dx+\frac {14}{15} \int e^{x/3} x \, dx-\frac {28}{15} \int e^{x/3} \, dx-\frac {12}{5} \int e^{x/3} \, dx-\frac {12}{5} \int \frac {e^{x/3}}{-2+x} \, dx+\frac {24}{5} \int e^{x/3} \, dx-\frac {116}{15} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {48}{5} \int e^{x/3} \, dx+\frac {56}{5} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {96}{5} \int e^{x/3} x \, dx-24 \int e^{x/3} x^2 \, dx-24 \int e^{x/3} x^3 \, dx-\frac {384}{5} \int e^{x/3} \, dx+\frac {432}{5} \int e^{x/3} \, dx-\frac {768}{5} \int e^{x/3} x \, dx+\frac {864}{5} \int e^{x/3} x \, dx-192 \int e^{x/3} x^2 \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = -\frac {24 e^{x/3}}{5}+\frac {12}{5} e^{x/3} x-648 e^{x/3} x^2+\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx-\frac {4}{5} \int e^{x/3} x \, dx+2 \int e^{x/3} \, dx-\frac {14}{5} \int e^{x/3} \, dx+\frac {288}{5} \int e^{x/3} \, dx+144 \int e^{x/3} x \, dx+216 \int e^{x/3} x^2 \, dx+\frac {2304}{5} \int e^{x/3} \, dx-\frac {2592}{5} \int e^{x/3} \, dx+1152 \int e^{x/3} x \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = -\frac {36 e^{x/3}}{5}+3888 e^{x/3} x+\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {12}{5} \int e^{x/3} \, dx-432 \int e^{x/3} \, dx-1296 \int e^{x/3} x \, dx-3456 \int e^{x/3} \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = -11664 e^{x/3}+\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+3888 \int e^{x/3} \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ & = \frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \operatorname {ExpIntegralEi}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.56 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {1}{45} e^{x/3} \left (x^3-3 \log (2-x)\right )^2 \]

[In]

Integrate[(E^(x/3)*(-18*x^3 - 36*x^5 + 16*x^6 + x^7) + E^(x/3)*(54 + 108*x^2 - 42*x^3 - 6*x^4)*Log[2 - x] + E^
(x/3)*(-18 + 9*x)*Log[2 - x]^2)/(-270 + 135*x),x]

[Out]

(E^(x/3)*(x^3 - 3*Log[2 - x])^2)/45

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x}{3}} \ln \left (2-x \right )^{2}}{5}-\frac {2 \,{\mathrm e}^{\frac {x}{3}} \ln \left (2-x \right ) x^{3}}{15}+\frac {{\mathrm e}^{\frac {x}{3}} x^{6}}{45}\)	\(40\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {x}{3}} \ln \left (2-x \right )^{2}}{5}-\frac {2 \,{\mathrm e}^{\frac {x}{3}} \ln \left (2-x \right ) x^{3}}{15}+\frac {{\mathrm e}^{\frac {x}{3}} x^{6}}{45}\)	\(40\)

[In]

int(((9*x-18)*exp(1/3*x)*ln(2-x)^2+(-6*x^4-42*x^3+108*x^2+54)*exp(1/3*x)*ln(2-x)+(x^7+16*x^6-36*x^5-18*x^3)*ex
p(1/3*x))/(135*x-270),x,method=_RETURNVERBOSE)

[Out]

1/5*exp(1/3*x)*ln(2-x)^2-2/15*exp(1/3*x)*ln(2-x)*x^3+1/45*exp(1/3*x)*x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {1}{45} \, x^{6} e^{\left (\frac {1}{3} \, x\right )} - \frac {2}{15} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right ) + \frac {1}{5} \, e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right )^{2} \]

[In]

integrate(((9*x-18)*exp(1/3*x)*log(2-x)^2+(-6*x^4-42*x^3+108*x^2+54)*exp(1/3*x)*log(2-x)+(x^7+16*x^6-36*x^5-18
*x^3)*exp(1/3*x))/(135*x-270),x, algorithm="fricas")

[Out]

1/45*x^6*e^(1/3*x) - 2/15*x^3*e^(1/3*x)*log(-x + 2) + 1/5*e^(1/3*x)*log(-x + 2)^2

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {\left (x^{6} - 6 x^{3} \log {\left (2 - x \right )} + 9 \log {\left (2 - x \right )}^{2}\right ) e^{\frac {x}{3}}}{45} \]

[In]

integrate(((9*x-18)*exp(1/3*x)*ln(2-x)**2+(-6*x**4-42*x**3+108*x**2+54)*exp(1/3*x)*ln(2-x)+(x**7+16*x**6-36*x*
*5-18*x**3)*exp(1/3*x))/(135*x-270),x)

[Out]

(x**6 - 6*x**3*log(2 - x) + 9*log(2 - x)**2)*exp(x/3)/45

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {1}{45} \, {\left (x^{6} - 6 \, x^{3} \log \left (-x + 2\right ) + 9 \, \log \left (-x + 2\right )^{2}\right )} e^{\left (\frac {1}{3} \, x\right )} \]

[In]

integrate(((9*x-18)*exp(1/3*x)*log(2-x)^2+(-6*x^4-42*x^3+108*x^2+54)*exp(1/3*x)*log(2-x)+(x^7+16*x^6-36*x^5-18
*x^3)*exp(1/3*x))/(135*x-270),x, algorithm="maxima")

[Out]

1/45*(x^6 - 6*x^3*log(-x + 2) + 9*log(-x + 2)^2)*e^(1/3*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {1}{45} \, x^{6} e^{\left (\frac {1}{3} \, x\right )} - \frac {2}{15} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right ) + \frac {1}{5} \, e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right )^{2} \]

[In]

integrate(((9*x-18)*exp(1/3*x)*log(2-x)^2+(-6*x^4-42*x^3+108*x^2+54)*exp(1/3*x)*log(2-x)+(x^7+16*x^6-36*x^5-18
*x^3)*exp(1/3*x))/(135*x-270),x, algorithm="giac")

[Out]

1/45*x^6*e^(1/3*x) - 2/15*x^3*e^(1/3*x)*log(-x + 2) + 1/5*e^(1/3*x)*log(-x + 2)^2

Mupad [B] (verification not implemented)

Time = 8.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx=\frac {{\mathrm {e}}^{x/3}\,{\left (3\,\ln \left (2-x\right )-x^3\right )}^2}{45} \]

[In]

int((exp(x/3)*log(2 - x)*(108*x^2 - 42*x^3 - 6*x^4 + 54) - exp(x/3)*(18*x^3 + 36*x^5 - 16*x^6 - x^7) + exp(x/3
)*log(2 - x)^2*(9*x - 18))/(135*x - 270),x)

[Out]

(exp(x/3)*(3*log(2 - x) - x^3)^2)/45